tan-example (used to crash)

Percentage Accurate: 79.3% → 99.7%

Time: 12.9s

Alternatives: 20

Speedup: 0.4×

Specification

\[\left(\left(\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.751224 \cdot 10^{+308}\right)\right) \land \left(-1.776707 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right)\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.751224 \cdot 10^{+308}\right)\]

Math

\[x + \left(\tan \left(y + z\right) - \tan a\right) \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (+ x (- (tan (+ y z)) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (tan((y + z)) - tan(a));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (tan((y + z)) - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + (Math.tan((y + z)) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 79.3% accurate, 1.0× speedup

Math

\[x + \left(\tan \left(y + z\right) - \tan a\right) \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (+ x (- (tan (+ y z)) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (tan((y + z)) - tan(a));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (tan((y + z)) - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + (Math.tan((y + z)) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)\\ x + \left(\left(\frac{\sin z \cdot \cos y}{t\_0} + \frac{\cos z \cdot \sin y}{t\_0}\right) - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (let* ((t_0 (fma (cos z) (cos y) (* (sin z) (sin (- y))))))
  (+
   x
   (-
    (+ (/ (* (sin z) (cos y)) t_0) (/ (* (cos z) (sin y)) t_0))
    (tan a)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = fma(cos(z), cos(y), (sin(z) * sin(-y)));
	return x + ((((sin(z) * cos(y)) / t_0) + ((cos(z) * sin(y)) / t_0)) - tan(a));
}

Julia

function code(x, y, z, a)
	t_0 = fma(cos(z), cos(y), Float64(sin(z) * sin(Float64(-y))))
	return Float64(x + Float64(Float64(Float64(Float64(sin(z) * cos(y)) / t_0) + Float64(Float64(cos(z) * sin(y)) / t_0)) - tan(a)))
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Cos[z], $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Sin[z], $MachinePrecision] * N[Sin[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x + N[(N[(N[(N[(N[Sin[z], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(N[Cos[z], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Step-by-step derivation

   1. lift-tan.f64 N/A

      \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

   2. tan-quot N/A

      \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

   3. lift-+.f64 N/A

      \[\leadsto x + \left(\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

   4. +-commutative N/A

      \[\leadsto x + \left(\frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

   5. sin-sum N/A

      \[\leadsto x + \left(\frac{\color{blue}{\sin z \cdot \cos y + \cos z \cdot \sin y}}{\cos \left(y + z\right)} - \tan a\right) \]

   6. div-add N/A

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\cos z \cdot \sin y}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

   7. *-commutative N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   8. lower-+.f64 N/A

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

   9. lower-/.f64 N/A

      \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   11. lower-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z} \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \color{blue}{\cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   13. lower-cos.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   14. lift-+.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   15. +-commutative N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   16. lower-+.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   17. lower-/.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]

3. Applied rewrites 79.7%

   \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
4. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   2. lift-+.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   3. add-flip N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   4. cos-diff N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   5. cos-neg-rev N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   6. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   7. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   9. lift-sin.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   11. lower-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   12. lower-neg.f64 80.4%

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

5. Applied rewrites 80.4%

   \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]
6. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos \left(z + y\right)}}\right) - \tan a\right) \]

   2. lift-+.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z + y\right)}}\right) - \tan a\right) \]

   3. add-flip N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

   4. cos-diff N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}\right) - \tan a\right) \]

   5. cos-neg-rev N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

   6. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

   7. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

   9. lift-sin.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}\right) - \tan a\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

   11. lower-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

   12. lower-neg.f64 99.7%

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)}\right) - \tan a\right) \]

7. Applied rewrites 99.7%

   \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}}\right) - \tan a\right) \]
8. Add Preprocessing

Alternative 2: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \cos z \cdot \cos y - \sin y \cdot \sin z\\ x + \left(\left(\frac{\sin z \cdot \cos y}{t\_0} + \frac{\cos z \cdot \sin y}{t\_0}\right) - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (let* ((t_0 (- (* (cos z) (cos y)) (* (sin y) (sin z)))))
  (+
   x
   (-
    (+ (/ (* (sin z) (cos y)) t_0) (/ (* (cos z) (sin y)) t_0))
    (tan a)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = (cos(z) * cos(y)) - (sin(y) * sin(z));
	return x + ((((sin(z) * cos(y)) / t_0) + ((cos(z) * sin(y)) / t_0)) - tan(a));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    t_0 = (cos(z) * cos(y)) - (sin(y) * sin(z))
    code = x + ((((sin(z) * cos(y)) / t_0) + ((cos(z) * sin(y)) / t_0)) - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = (Math.cos(z) * Math.cos(y)) - (Math.sin(y) * Math.sin(z));
	return x + ((((Math.sin(z) * Math.cos(y)) / t_0) + ((Math.cos(z) * Math.sin(y)) / t_0)) - Math.tan(a));
}

Python

def code(x, y, z, a):
	t_0 = (math.cos(z) * math.cos(y)) - (math.sin(y) * math.sin(z))
	return x + ((((math.sin(z) * math.cos(y)) / t_0) + ((math.cos(z) * math.sin(y)) / t_0)) - math.tan(a))

Julia

function code(x, y, z, a)
	t_0 = Float64(Float64(cos(z) * cos(y)) - Float64(sin(y) * sin(z)))
	return Float64(x + Float64(Float64(Float64(Float64(sin(z) * cos(y)) / t_0) + Float64(Float64(cos(z) * sin(y)) / t_0)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	t_0 = (cos(z) * cos(y)) - (sin(y) * sin(z));
	tmp = x + ((((sin(z) * cos(y)) / t_0) + ((cos(z) * sin(y)) / t_0)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[(N[Cos[z], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * N[Sin[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x + N[(N[(N[(N[(N[Sin[z], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(N[Cos[z], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Step-by-step derivation

   1. lift-tan.f64 N/A

      \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

   2. tan-quot N/A

      \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

   3. lift-+.f64 N/A

      \[\leadsto x + \left(\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

   4. +-commutative N/A

      \[\leadsto x + \left(\frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

   5. sin-sum N/A

      \[\leadsto x + \left(\frac{\color{blue}{\sin z \cdot \cos y + \cos z \cdot \sin y}}{\cos \left(y + z\right)} - \tan a\right) \]

   6. div-add N/A

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\cos z \cdot \sin y}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

   7. *-commutative N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   8. lower-+.f64 N/A

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

   9. lower-/.f64 N/A

      \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   11. lower-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z} \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \color{blue}{\cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   13. lower-cos.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   14. lift-+.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   15. +-commutative N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   16. lower-+.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   17. lower-/.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]

3. Applied rewrites 79.7%

   \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
4. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   2. lift-+.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   3. add-flip N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   4. cos-diff N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   5. cos-neg-rev N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   6. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   7. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   9. lift-sin.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   11. lower-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   12. lower-neg.f64 80.4%

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

5. Applied rewrites 80.4%

   \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]
6. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos \left(z + y\right)}}\right) - \tan a\right) \]

   2. lift-+.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z + y\right)}}\right) - \tan a\right) \]

   3. add-flip N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

   4. cos-diff N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}\right) - \tan a\right) \]

   5. cos-neg-rev N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

   6. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

   7. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

   9. lift-sin.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}\right) - \tan a\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

   11. lower-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

   12. lower-neg.f64 99.7%

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)}\right) - \tan a\right) \]

7. Applied rewrites 99.7%

   \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}}\right) - \tan a\right) \]
8. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos y + \sin z \cdot \sin \left(-y\right)}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   2. lift-*.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y + \color{blue}{\sin z \cdot \sin \left(-y\right)}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   3. lift-sin.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y + \sin z \cdot \color{blue}{\sin \left(-y\right)}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   4. lift-neg.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y + \sin z \cdot \sin \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   5. sin-neg N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y + \sin z \cdot \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right)}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   6. lift-sin.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y + \sin z \cdot \left(\mathsf{neg}\left(\color{blue}{\sin y}\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   7. distribute-rgt-neg-out N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y + \color{blue}{\left(\mathsf{neg}\left(\sin z \cdot \sin y\right)\right)}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   8. lift-sin.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y + \left(\mathsf{neg}\left(\color{blue}{\sin z} \cdot \sin y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   9. lift-sin.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y + \left(\mathsf{neg}\left(\sin z \cdot \color{blue}{\sin y}\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   10. sub-flip N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos y - \sin z \cdot \sin y}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   11. lower--.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos y - \sin z \cdot \sin y}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   12. lower-*.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos y} - \sin z \cdot \sin y} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   13. lift-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y - \color{blue}{\sin z} \cdot \sin y} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   14. lift-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y - \sin z \cdot \color{blue}{\sin y}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   15. *-commutative N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y - \color{blue}{\sin y \cdot \sin z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   16. lower-*.f64 99.7%

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y - \color{blue}{\sin y \cdot \sin z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

9. Applied rewrites 99.7%

   \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos y - \sin y \cdot \sin z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]
10. Step-by-step derivation

    1. lift-fma.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y - \sin y \cdot \sin z} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos y + \sin z \cdot \sin \left(-y\right)}}\right) - \tan a\right) \]

    2. lift-*.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y - \sin y \cdot \sin z} + \frac{\cos z \cdot \sin y}{\cos z \cdot \cos y + \color{blue}{\sin z \cdot \sin \left(-y\right)}}\right) - \tan a\right) \]

    3. lift-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y - \sin y \cdot \sin z} + \frac{\cos z \cdot \sin y}{\cos z \cdot \cos y + \sin z \cdot \color{blue}{\sin \left(-y\right)}}\right) - \tan a\right) \]

    4. lift-neg.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y - \sin y \cdot \sin z} + \frac{\cos z \cdot \sin y}{\cos z \cdot \cos y + \sin z \cdot \sin \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}}\right) - \tan a\right) \]

    5. sin-neg N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y - \sin y \cdot \sin z} + \frac{\cos z \cdot \sin y}{\cos z \cdot \cos y + \sin z \cdot \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right)}}\right) - \tan a\right) \]

    6. lift-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y - \sin y \cdot \sin z} + \frac{\cos z \cdot \sin y}{\cos z \cdot \cos y + \sin z \cdot \left(\mathsf{neg}\left(\color{blue}{\sin y}\right)\right)}\right) - \tan a\right) \]

    7. distribute-rgt-neg-out N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y - \sin y \cdot \sin z} + \frac{\cos z \cdot \sin y}{\cos z \cdot \cos y + \color{blue}{\left(\mathsf{neg}\left(\sin z \cdot \sin y\right)\right)}}\right) - \tan a\right) \]

    8. lift-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y - \sin y \cdot \sin z} + \frac{\cos z \cdot \sin y}{\cos z \cdot \cos y + \left(\mathsf{neg}\left(\color{blue}{\sin z} \cdot \sin y\right)\right)}\right) - \tan a\right) \]

    9. lift-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y - \sin y \cdot \sin z} + \frac{\cos z \cdot \sin y}{\cos z \cdot \cos y + \left(\mathsf{neg}\left(\sin z \cdot \color{blue}{\sin y}\right)\right)}\right) - \tan a\right) \]

    10. sub-flip N/A

        \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y - \sin y \cdot \sin z} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos y - \sin z \cdot \sin y}}\right) - \tan a\right) \]

    11. lower--.f64 N/A

        \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y - \sin y \cdot \sin z} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos y - \sin z \cdot \sin y}}\right) - \tan a\right) \]

    12. lower-*.f64 N/A

        \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y - \sin y \cdot \sin z} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos y} - \sin z \cdot \sin y}\right) - \tan a\right) \]

    13. lift-sin.f64 N/A

        \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y - \sin y \cdot \sin z} + \frac{\cos z \cdot \sin y}{\cos z \cdot \cos y - \color{blue}{\sin z} \cdot \sin y}\right) - \tan a\right) \]

    14. lift-sin.f64 N/A

        \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y - \sin y \cdot \sin z} + \frac{\cos z \cdot \sin y}{\cos z \cdot \cos y - \sin z \cdot \color{blue}{\sin y}}\right) - \tan a\right) \]

    15. *-commutative N/A

        \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y - \sin y \cdot \sin z} + \frac{\cos z \cdot \sin y}{\cos z \cdot \cos y - \color{blue}{\sin y \cdot \sin z}}\right) - \tan a\right) \]

    16. lower-*.f64 99.7%

        \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y - \sin y \cdot \sin z} + \frac{\cos z \cdot \sin y}{\cos z \cdot \cos y - \color{blue}{\sin y \cdot \sin z}}\right) - \tan a\right) \]

11. Applied rewrites 99.7%

    \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \cos y - \sin y \cdot \sin z} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos y - \sin y \cdot \sin z}}\right) - \tan a\right) \]
12. Add Preprocessing

Alternative 3: 89.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(y, z\right)\right)\\ t_1 := \cos \left(\mathsf{min}\left(y, z\right)\right)\\ t_2 := t\_0 \cdot \sin \left(\mathsf{min}\left(y, z\right)\right)\\ t_3 := \sin \left(\mathsf{max}\left(y, z\right)\right)\\ t_4 := t\_3 \cdot \sin \left(-\mathsf{min}\left(y, z\right)\right)\\ t_5 := \mathsf{fma}\left(t\_0, t\_1, t\_4\right)\\ t_6 := x + \left(\left(\frac{t\_3}{t\_5} + \frac{t\_2}{t\_5}\right) - \tan a\right)\\ t_7 := \mathsf{fma}\left(t\_1, t\_0, t\_4\right)\\ \mathbf{if}\;a \leq -0.86:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;a \leq 0.105:\\ \;\;\;\;x + \mathsf{fma}\left(a, -0.3333333333333333 \cdot {a}^{2} - 1, \frac{t\_1 \cdot t\_3}{t\_7} + \frac{t\_2}{t\_7}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_6\\ \end{array} \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (let* ((t_0 (cos (fmax y z)))
       (t_1 (cos (fmin y z)))
       (t_2 (* t_0 (sin (fmin y z))))
       (t_3 (sin (fmax y z)))
       (t_4 (* t_3 (sin (- (fmin y z)))))
       (t_5 (fma t_0 t_1 t_4))
       (t_6 (+ x (- (+ (/ t_3 t_5) (/ t_2 t_5)) (tan a))))
       (t_7 (fma t_1 t_0 t_4)))
  (if (<= a -0.86)
    t_6
    (if (<= a 0.105)
      (+
       x
       (fma
        a
        (- (* -0.3333333333333333 (pow a 2.0)) 1.0)
        (+ (/ (* t_1 t_3) t_7) (/ t_2 t_7))))
      t_6))))

C

double code(double x, double y, double z, double a) {
	double t_0 = cos(fmax(y, z));
	double t_1 = cos(fmin(y, z));
	double t_2 = t_0 * sin(fmin(y, z));
	double t_3 = sin(fmax(y, z));
	double t_4 = t_3 * sin(-fmin(y, z));
	double t_5 = fma(t_0, t_1, t_4);
	double t_6 = x + (((t_3 / t_5) + (t_2 / t_5)) - tan(a));
	double t_7 = fma(t_1, t_0, t_4);
	double tmp;
	if (a <= -0.86) {
		tmp = t_6;
	} else if (a <= 0.105) {
		tmp = x + fma(a, ((-0.3333333333333333 * pow(a, 2.0)) - 1.0), (((t_1 * t_3) / t_7) + (t_2 / t_7)));
	} else {
		tmp = t_6;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = cos(fmax(y, z))
	t_1 = cos(fmin(y, z))
	t_2 = Float64(t_0 * sin(fmin(y, z)))
	t_3 = sin(fmax(y, z))
	t_4 = Float64(t_3 * sin(Float64(-fmin(y, z))))
	t_5 = fma(t_0, t_1, t_4)
	t_6 = Float64(x + Float64(Float64(Float64(t_3 / t_5) + Float64(t_2 / t_5)) - tan(a)))
	t_7 = fma(t_1, t_0, t_4)
	tmp = 0.0
	if (a <= -0.86)
		tmp = t_6;
	elseif (a <= 0.105)
		tmp = Float64(x + fma(a, Float64(Float64(-0.3333333333333333 * (a ^ 2.0)) - 1.0), Float64(Float64(Float64(t_1 * t_3) / t_7) + Float64(t_2 / t_7))));
	else
		tmp = t_6;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Cos[N[Max[y, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Min[y, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[Sin[N[Min[y, z], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[Max[y, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Sin[(-N[Min[y, z], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$0 * t$95$1 + t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(x + N[(N[(N[(t$95$3 / t$95$5), $MachinePrecision] + N[(t$95$2 / t$95$5), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$1 * t$95$0 + t$95$4), $MachinePrecision]}, If[LessEqual[a, -0.86], t$95$6, If[LessEqual[a, 0.105], N[(x + N[(a * N[(N[(-0.3333333333333333 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] + N[(N[(N[(t$95$1 * t$95$3), $MachinePrecision] / t$95$7), $MachinePrecision] + N[(t$95$2 / t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$6]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.85999999999999999 or 0.105 < a

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. lift-tan.f64 N/A

         \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

      2. tan-quot N/A

         \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

      3. lift-+.f64 N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      4. +-commutative N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      5. sin-sum N/A

         \[\leadsto x + \left(\frac{\color{blue}{\sin z \cdot \cos y + \cos z \cdot \sin y}}{\cos \left(y + z\right)} - \tan a\right) \]

      6. div-add N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\cos z \cdot \sin y}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      7. *-commutative N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      8. lower-+.f64 N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      9. lower-/.f64 N/A

         \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z} \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \color{blue}{\cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      14. lift-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      15. +-commutative N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      16. lower-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      17. lower-/.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]

   3. Applied rewrites 79.7%

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 80.4%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   5. Applied rewrites 80.4%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos \left(z + y\right)}}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z + y\right)}}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 99.7%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)}\right) - \tan a\right) \]

   7. Applied rewrites 99.7%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}}\right) - \tan a\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z}}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]
   9. Step-by-step derivation

      1. lower-sin.f64 80.4%

         \[\leadsto x + \left(\left(\frac{\sin z}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   10. Applied rewrites 80.4%

       \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z}}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   if -0.85999999999999999 < a < 0.105

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. lift-tan.f64 N/A

         \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

      2. tan-quot N/A

         \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

      3. lift-+.f64 N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      4. +-commutative N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      5. sin-sum N/A

         \[\leadsto x + \left(\frac{\color{blue}{\sin z \cdot \cos y + \cos z \cdot \sin y}}{\cos \left(y + z\right)} - \tan a\right) \]

      6. div-add N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\cos z \cdot \sin y}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      7. *-commutative N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      8. lower-+.f64 N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      9. lower-/.f64 N/A

         \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z} \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \color{blue}{\cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      14. lift-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      15. +-commutative N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      16. lower-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      17. lower-/.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]

   3. Applied rewrites 79.7%

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 80.4%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   5. Applied rewrites 80.4%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos \left(z + y\right)}}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z + y\right)}}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 99.7%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)}\right) - \tan a\right) \]

   7. Applied rewrites 99.7%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}}\right) - \tan a\right) \]

   8. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{\left(a \cdot \left(\frac{-1}{3} \cdot {a}^{2} - 1\right) + \left(\frac{\cos y \cdot \sin z}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)\right)} \]
   9. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(a, \color{blue}{\frac{-1}{3} \cdot {a}^{2} - 1}, \frac{\cos y \cdot \sin z}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) \]

      2. lower--.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(a, \frac{-1}{3} \cdot {a}^{2} - \color{blue}{1}, \frac{\cos y \cdot \sin z}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(a, \frac{-1}{3} \cdot {a}^{2} - 1, \frac{\cos y \cdot \sin z}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(a, \frac{-1}{3} \cdot {a}^{2} - 1, \frac{\cos y \cdot \sin z}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) \]

   10. Applied rewrites 50.6%

       \[\leadsto x + \color{blue}{\mathsf{fma}\left(a, -0.3333333333333333 \cdot {a}^{2} - 1, \frac{\cos y \cdot \sin z}{\mathsf{fma}\left(\cos y, \cos z, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos y, \cos z, \sin z \cdot \sin \left(-y\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 89.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(y, z\right)\right)\\ t_1 := \cos \left(\mathsf{min}\left(y, z\right)\right)\\ t_2 := t\_0 \cdot \sin \left(\mathsf{min}\left(y, z\right)\right)\\ t_3 := \sin \left(\mathsf{max}\left(y, z\right)\right)\\ t_4 := t\_3 \cdot \sin \left(-\mathsf{min}\left(y, z\right)\right)\\ t_5 := \mathsf{fma}\left(t\_0, t\_1, t\_4\right)\\ t_6 := \mathsf{fma}\left(t\_1, t\_0, t\_4\right)\\ t_7 := \frac{t\_2}{t\_5}\\ \mathbf{if}\;\tan a \leq -5 \cdot 10^{-5}:\\ \;\;\;\;x + \left(\left(\frac{t\_3}{t\_5} + t\_7\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 0.00038:\\ \;\;\;\;x + \mathsf{fma}\left(-1, a, \frac{t\_1 \cdot t\_3}{t\_6} + \frac{t\_2}{t\_6}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(\frac{t\_3}{t\_0} + t\_7\right) - \tan a\right)\\ \end{array} \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (let* ((t_0 (cos (fmax y z)))
       (t_1 (cos (fmin y z)))
       (t_2 (* t_0 (sin (fmin y z))))
       (t_3 (sin (fmax y z)))
       (t_4 (* t_3 (sin (- (fmin y z)))))
       (t_5 (fma t_0 t_1 t_4))
       (t_6 (fma t_1 t_0 t_4))
       (t_7 (/ t_2 t_5)))
  (if (<= (tan a) -5e-5)
    (+ x (- (+ (/ t_3 t_5) t_7) (tan a)))
    (if (<= (tan a) 0.00038)
      (+ x (fma -1.0 a (+ (/ (* t_1 t_3) t_6) (/ t_2 t_6))))
      (+ x (- (+ (/ t_3 t_0) t_7) (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = cos(fmax(y, z));
	double t_1 = cos(fmin(y, z));
	double t_2 = t_0 * sin(fmin(y, z));
	double t_3 = sin(fmax(y, z));
	double t_4 = t_3 * sin(-fmin(y, z));
	double t_5 = fma(t_0, t_1, t_4);
	double t_6 = fma(t_1, t_0, t_4);
	double t_7 = t_2 / t_5;
	double tmp;
	if (tan(a) <= -5e-5) {
		tmp = x + (((t_3 / t_5) + t_7) - tan(a));
	} else if (tan(a) <= 0.00038) {
		tmp = x + fma(-1.0, a, (((t_1 * t_3) / t_6) + (t_2 / t_6)));
	} else {
		tmp = x + (((t_3 / t_0) + t_7) - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = cos(fmax(y, z))
	t_1 = cos(fmin(y, z))
	t_2 = Float64(t_0 * sin(fmin(y, z)))
	t_3 = sin(fmax(y, z))
	t_4 = Float64(t_3 * sin(Float64(-fmin(y, z))))
	t_5 = fma(t_0, t_1, t_4)
	t_6 = fma(t_1, t_0, t_4)
	t_7 = Float64(t_2 / t_5)
	tmp = 0.0
	if (tan(a) <= -5e-5)
		tmp = Float64(x + Float64(Float64(Float64(t_3 / t_5) + t_7) - tan(a)));
	elseif (tan(a) <= 0.00038)
		tmp = Float64(x + fma(-1.0, a, Float64(Float64(Float64(t_1 * t_3) / t_6) + Float64(t_2 / t_6))));
	else
		tmp = Float64(x + Float64(Float64(Float64(t_3 / t_0) + t_7) - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Cos[N[Max[y, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Min[y, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[Sin[N[Min[y, z], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[Max[y, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Sin[(-N[Min[y, z], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$0 * t$95$1 + t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$1 * t$95$0 + t$95$4), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$2 / t$95$5), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -5e-5], N[(x + N[(N[(N[(t$95$3 / t$95$5), $MachinePrecision] + t$95$7), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 0.00038], N[(x + N[(-1.0 * a + N[(N[(N[(t$95$1 * t$95$3), $MachinePrecision] / t$95$6), $MachinePrecision] + N[(t$95$2 / t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(t$95$3 / t$95$0), $MachinePrecision] + t$95$7), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -5.0000000000000002e-5

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. lift-tan.f64 N/A

         \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

      2. tan-quot N/A

         \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

      3. lift-+.f64 N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      4. +-commutative N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      5. sin-sum N/A

         \[\leadsto x + \left(\frac{\color{blue}{\sin z \cdot \cos y + \cos z \cdot \sin y}}{\cos \left(y + z\right)} - \tan a\right) \]

      6. div-add N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\cos z \cdot \sin y}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      7. *-commutative N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      8. lower-+.f64 N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      9. lower-/.f64 N/A

         \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z} \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \color{blue}{\cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      14. lift-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      15. +-commutative N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      16. lower-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      17. lower-/.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]

   3. Applied rewrites 79.7%

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 80.4%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   5. Applied rewrites 80.4%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos \left(z + y\right)}}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z + y\right)}}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 99.7%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)}\right) - \tan a\right) \]

   7. Applied rewrites 99.7%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}}\right) - \tan a\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z}}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]
   9. Step-by-step derivation

      1. lower-sin.f64 80.4%

         \[\leadsto x + \left(\left(\frac{\sin z}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   10. Applied rewrites 80.4%

       \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z}}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   if -5.0000000000000002e-5 < (tan.f64 a) < 3.8000000000000002e-4

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. lift-tan.f64 N/A

         \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

      2. tan-quot N/A

         \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

      3. lift-+.f64 N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      4. +-commutative N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      5. sin-sum N/A

         \[\leadsto x + \left(\frac{\color{blue}{\sin z \cdot \cos y + \cos z \cdot \sin y}}{\cos \left(y + z\right)} - \tan a\right) \]

      6. div-add N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\cos z \cdot \sin y}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      7. *-commutative N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      8. lower-+.f64 N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      9. lower-/.f64 N/A

         \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z} \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \color{blue}{\cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      14. lift-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      15. +-commutative N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      16. lower-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      17. lower-/.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]

   3. Applied rewrites 79.7%

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 80.4%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   5. Applied rewrites 80.4%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos \left(z + y\right)}}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z + y\right)}}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 99.7%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)}\right) - \tan a\right) \]

   7. Applied rewrites 99.7%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}}\right) - \tan a\right) \]

   8. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{\left(-1 \cdot a + \left(\frac{\cos y \cdot \sin z}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)\right)} \]
   9. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(-1, \color{blue}{a}, \frac{\cos y \cdot \sin z}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(-1, a, \frac{\cos y \cdot \sin z}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) \]

   10. Applied rewrites 51.0%

       \[\leadsto x + \color{blue}{\mathsf{fma}\left(-1, a, \frac{\cos y \cdot \sin z}{\mathsf{fma}\left(\cos y, \cos z, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos y, \cos z, \sin z \cdot \sin \left(-y\right)\right)}\right)} \]

   if 3.8000000000000002e-4 < (tan.f64 a)

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. lift-tan.f64 N/A

         \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

      2. tan-quot N/A

         \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

      3. lift-+.f64 N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      4. +-commutative N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      5. sin-sum N/A

         \[\leadsto x + \left(\frac{\color{blue}{\sin z \cdot \cos y + \cos z \cdot \sin y}}{\cos \left(y + z\right)} - \tan a\right) \]

      6. div-add N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\cos z \cdot \sin y}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      7. *-commutative N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      8. lower-+.f64 N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      9. lower-/.f64 N/A

         \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z} \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \color{blue}{\cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      14. lift-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      15. +-commutative N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      16. lower-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      17. lower-/.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]

   3. Applied rewrites 79.7%

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 80.4%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   5. Applied rewrites 80.4%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos \left(z + y\right)}}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z + y\right)}}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 99.7%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)}\right) - \tan a\right) \]

   7. Applied rewrites 99.7%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}}\right) - \tan a\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z}{\cos z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z}{\color{blue}{\cos z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

      2. lower-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z}{\cos \color{blue}{z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

      3. lower-cos.f64 80.4%

         \[\leadsto x + \left(\left(\frac{\sin z}{\cos z} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   10. Applied rewrites 80.4%

       \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z}{\cos z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 89.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(y, z\right)\right)\\ t_1 := \sin \left(\mathsf{min}\left(y, z\right)\right)\\ t_2 := \sin \left(\mathsf{max}\left(y, z\right)\right)\\ t_3 := \cos \left(\mathsf{max}\left(y, z\right)\right)\\ t_4 := \mathsf{fma}\left(t\_3, t\_0, t\_2 \cdot \sin \left(-\mathsf{min}\left(y, z\right)\right)\right)\\ t_5 := t\_3 \cdot t\_1\\ t_6 := \frac{t\_5}{t\_4}\\ t_7 := \mathsf{fma}\left(-1, t\_1 \cdot t\_2, t\_0 \cdot t\_3\right)\\ \mathbf{if}\;\tan a \leq -5 \cdot 10^{-5}:\\ \;\;\;\;x + \left(\left(\frac{t\_2}{t\_4} + t\_6\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x + \left(\frac{t\_0 \cdot t\_2}{t\_7} + \frac{t\_5}{t\_7}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(\frac{t\_2}{t\_3} + t\_6\right) - \tan a\right)\\ \end{array} \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (let* ((t_0 (cos (fmin y z)))
       (t_1 (sin (fmin y z)))
       (t_2 (sin (fmax y z)))
       (t_3 (cos (fmax y z)))
       (t_4 (fma t_3 t_0 (* t_2 (sin (- (fmin y z))))))
       (t_5 (* t_3 t_1))
       (t_6 (/ t_5 t_4))
       (t_7 (fma -1.0 (* t_1 t_2) (* t_0 t_3))))
  (if (<= (tan a) -5e-5)
    (+ x (- (+ (/ t_2 t_4) t_6) (tan a)))
    (if (<= (tan a) 5e-8)
      (+ x (+ (/ (* t_0 t_2) t_7) (/ t_5 t_7)))
      (+ x (- (+ (/ t_2 t_3) t_6) (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = cos(fmin(y, z));
	double t_1 = sin(fmin(y, z));
	double t_2 = sin(fmax(y, z));
	double t_3 = cos(fmax(y, z));
	double t_4 = fma(t_3, t_0, (t_2 * sin(-fmin(y, z))));
	double t_5 = t_3 * t_1;
	double t_6 = t_5 / t_4;
	double t_7 = fma(-1.0, (t_1 * t_2), (t_0 * t_3));
	double tmp;
	if (tan(a) <= -5e-5) {
		tmp = x + (((t_2 / t_4) + t_6) - tan(a));
	} else if (tan(a) <= 5e-8) {
		tmp = x + (((t_0 * t_2) / t_7) + (t_5 / t_7));
	} else {
		tmp = x + (((t_2 / t_3) + t_6) - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = cos(fmin(y, z))
	t_1 = sin(fmin(y, z))
	t_2 = sin(fmax(y, z))
	t_3 = cos(fmax(y, z))
	t_4 = fma(t_3, t_0, Float64(t_2 * sin(Float64(-fmin(y, z)))))
	t_5 = Float64(t_3 * t_1)
	t_6 = Float64(t_5 / t_4)
	t_7 = fma(-1.0, Float64(t_1 * t_2), Float64(t_0 * t_3))
	tmp = 0.0
	if (tan(a) <= -5e-5)
		tmp = Float64(x + Float64(Float64(Float64(t_2 / t_4) + t_6) - tan(a)));
	elseif (tan(a) <= 5e-8)
		tmp = Float64(x + Float64(Float64(Float64(t_0 * t_2) / t_7) + Float64(t_5 / t_7)));
	else
		tmp = Float64(x + Float64(Float64(Float64(t_2 / t_3) + t_6) - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Cos[N[Min[y, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[Min[y, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Max[y, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[Max[y, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * t$95$0 + N[(t$95$2 * N[Sin[(-N[Min[y, z], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 * t$95$1), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 / t$95$4), $MachinePrecision]}, Block[{t$95$7 = N[(-1.0 * N[(t$95$1 * t$95$2), $MachinePrecision] + N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -5e-5], N[(x + N[(N[(N[(t$95$2 / t$95$4), $MachinePrecision] + t$95$6), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 5e-8], N[(x + N[(N[(N[(t$95$0 * t$95$2), $MachinePrecision] / t$95$7), $MachinePrecision] + N[(t$95$5 / t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(t$95$2 / t$95$3), $MachinePrecision] + t$95$6), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -5.0000000000000002e-5

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. lift-tan.f64 N/A

         \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

      2. tan-quot N/A

         \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

      3. lift-+.f64 N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      4. +-commutative N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      5. sin-sum N/A

         \[\leadsto x + \left(\frac{\color{blue}{\sin z \cdot \cos y + \cos z \cdot \sin y}}{\cos \left(y + z\right)} - \tan a\right) \]

      6. div-add N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\cos z \cdot \sin y}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      7. *-commutative N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      8. lower-+.f64 N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      9. lower-/.f64 N/A

         \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z} \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \color{blue}{\cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      14. lift-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      15. +-commutative N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      16. lower-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      17. lower-/.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]

   3. Applied rewrites 79.7%

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 80.4%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   5. Applied rewrites 80.4%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos \left(z + y\right)}}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z + y\right)}}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 99.7%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)}\right) - \tan a\right) \]

   7. Applied rewrites 99.7%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}}\right) - \tan a\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z}}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]
   9. Step-by-step derivation

      1. lower-sin.f64 80.4%

         \[\leadsto x + \left(\left(\frac{\sin z}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   10. Applied rewrites 80.4%

       \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z}}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   if -5.0000000000000002e-5 < (tan.f64 a) < 4.9999999999999998e-8

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. lift-tan.f64 N/A

         \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

      2. tan-quot N/A

         \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

      3. lift-+.f64 N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      4. +-commutative N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      5. sin-sum N/A

         \[\leadsto x + \left(\frac{\color{blue}{\sin z \cdot \cos y + \cos z \cdot \sin y}}{\cos \left(y + z\right)} - \tan a\right) \]

      6. div-add N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\cos z \cdot \sin y}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      7. *-commutative N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      8. lower-+.f64 N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      9. lower-/.f64 N/A

         \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z} \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \color{blue}{\cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      14. lift-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      15. +-commutative N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      16. lower-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      17. lower-/.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]

   3. Applied rewrites 79.7%

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 80.4%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   5. Applied rewrites 80.4%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos \left(z + y\right)}}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z + y\right)}}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 99.7%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)}\right) - \tan a\right) \]

   7. Applied rewrites 99.7%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}}\right) - \tan a\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos y + \sin z \cdot \sin \left(-y\right)}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

      2. +-commutative N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\sin z \cdot \sin \left(-y\right) + \cos z \cdot \cos y}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

      3. lift-*.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\sin z \cdot \sin \left(-y\right)} + \cos z \cdot \cos y} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\sin z, \sin \left(-y\right), \cos z \cdot \cos y\right)}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

      5. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\sin z, \color{blue}{\sin \left(-y\right)}, \cos z \cdot \cos y\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

      6. lift-neg.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\sin z, \sin \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, \cos z \cdot \cos y\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

      7. sin-neg N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\sin z, \color{blue}{\mathsf{neg}\left(\sin y\right)}, \cos z \cdot \cos y\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\sin z, \mathsf{neg}\left(\color{blue}{\sin y}\right), \cos z \cdot \cos y\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\sin z, \color{blue}{-\sin y}, \cos z \cdot \cos y\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

      10. lower-*.f64 99.7%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\sin z, -\sin y, \color{blue}{\cos z \cdot \cos y}\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   9. Applied rewrites 99.7%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\sin z, -\sin y, \cos z \cdot \cos y\right)}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\sin z, -\sin y, \cos z \cdot \cos y\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos y + \sin z \cdot \sin \left(-y\right)}}\right) - \tan a\right) \]

       2. +-commutative N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\sin z, -\sin y, \cos z \cdot \cos y\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\sin z \cdot \sin \left(-y\right) + \cos z \cdot \cos y}}\right) - \tan a\right) \]

       3. lift-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\sin z, -\sin y, \cos z \cdot \cos y\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\sin z \cdot \sin \left(-y\right)} + \cos z \cdot \cos y}\right) - \tan a\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\sin z, -\sin y, \cos z \cdot \cos y\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\sin z, \sin \left(-y\right), \cos z \cdot \cos y\right)}}\right) - \tan a\right) \]

       5. lift-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\sin z, -\sin y, \cos z \cdot \cos y\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\sin z, \color{blue}{\sin \left(-y\right)}, \cos z \cdot \cos y\right)}\right) - \tan a\right) \]

       6. lift-neg.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\sin z, -\sin y, \cos z \cdot \cos y\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\sin z, \sin \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, \cos z \cdot \cos y\right)}\right) - \tan a\right) \]

       7. sin-neg N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\sin z, -\sin y, \cos z \cdot \cos y\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\sin z, \color{blue}{\mathsf{neg}\left(\sin y\right)}, \cos z \cdot \cos y\right)}\right) - \tan a\right) \]

       8. lift-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\sin z, -\sin y, \cos z \cdot \cos y\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\sin z, \mathsf{neg}\left(\color{blue}{\sin y}\right), \cos z \cdot \cos y\right)}\right) - \tan a\right) \]

       9. lower-neg.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\sin z, -\sin y, \cos z \cdot \cos y\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\sin z, \color{blue}{-\sin y}, \cos z \cdot \cos y\right)}\right) - \tan a\right) \]

       10. lower-*.f64 99.7%

           \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\sin z, -\sin y, \cos z \cdot \cos y\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\sin z, -\sin y, \color{blue}{\cos z \cdot \cos y}\right)}\right) - \tan a\right) \]

   11. Applied rewrites 99.7%

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\sin z, -\sin y, \cos z \cdot \cos y\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\sin z, -\sin y, \cos z \cdot \cos y\right)}}\right) - \tan a\right) \]

   12. Taylor expanded in a around 0

       \[\leadsto \color{blue}{x + \left(\frac{\cos y \cdot \sin z}{-1 \cdot \left(\sin y \cdot \sin z\right) + \cos y \cdot \cos z} + \frac{\cos z \cdot \sin y}{-1 \cdot \left(\sin y \cdot \sin z\right) + \cos y \cdot \cos z}\right)} \]
   13. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto x + \color{blue}{\left(\frac{\cos y \cdot \sin z}{-1 \cdot \left(\sin y \cdot \sin z\right) + \cos y \cdot \cos z} + \frac{\cos z \cdot \sin y}{-1 \cdot \left(\sin y \cdot \sin z\right) + \cos y \cdot \cos z}\right)} \]

       2. lower-+.f64 N/A

          \[\leadsto x + \left(\frac{\cos y \cdot \sin z}{-1 \cdot \left(\sin y \cdot \sin z\right) + \cos y \cdot \cos z} + \color{blue}{\frac{\cos z \cdot \sin y}{-1 \cdot \left(\sin y \cdot \sin z\right) + \cos y \cdot \cos z}}\right) \]

   14. Applied rewrites 60.6%

       \[\leadsto \color{blue}{x + \left(\frac{\cos y \cdot \sin z}{\mathsf{fma}\left(-1, \sin y \cdot \sin z, \cos y \cdot \cos z\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(-1, \sin y \cdot \sin z, \cos y \cdot \cos z\right)}\right)} \]

   if 4.9999999999999998e-8 < (tan.f64 a)

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. lift-tan.f64 N/A

         \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

      2. tan-quot N/A

         \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

      3. lift-+.f64 N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      4. +-commutative N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      5. sin-sum N/A

         \[\leadsto x + \left(\frac{\color{blue}{\sin z \cdot \cos y + \cos z \cdot \sin y}}{\cos \left(y + z\right)} - \tan a\right) \]

      6. div-add N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\cos z \cdot \sin y}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      7. *-commutative N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      8. lower-+.f64 N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      9. lower-/.f64 N/A

         \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z} \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \color{blue}{\cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      14. lift-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      15. +-commutative N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      16. lower-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      17. lower-/.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]

   3. Applied rewrites 79.7%

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 80.4%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   5. Applied rewrites 80.4%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos \left(z + y\right)}}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z + y\right)}}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 99.7%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)}\right) - \tan a\right) \]

   7. Applied rewrites 99.7%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}}\right) - \tan a\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z}{\cos z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z}{\color{blue}{\cos z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

      2. lower-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z}{\cos \color{blue}{z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

      3. lower-cos.f64 80.4%

         \[\leadsto x + \left(\left(\frac{\sin z}{\cos z} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   10. Applied rewrites 80.4%

       \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z}{\cos z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 89.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(y, z\right)\right)\\ t_1 := \sin \left(\mathsf{max}\left(y, z\right)\right)\\ t_2 := t\_1 \cdot \sin \left(-\mathsf{min}\left(y, z\right)\right)\\ t_3 := \cos \left(\mathsf{max}\left(y, z\right)\right)\\ t_4 := \mathsf{fma}\left(t\_3, t\_0, t\_2\right)\\ t_5 := t\_3 \cdot \sin \left(\mathsf{min}\left(y, z\right)\right)\\ t_6 := \frac{t\_5}{t\_4}\\ t_7 := \mathsf{fma}\left(t\_0, t\_3, t\_2\right)\\ \mathbf{if}\;\tan a \leq -5 \cdot 10^{-5}:\\ \;\;\;\;x + \left(\left(\frac{t\_1}{t\_4} + t\_6\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x + \left(\frac{t\_0 \cdot t\_1}{t\_7} + \frac{t\_5}{t\_7}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(\frac{t\_1}{t\_3} + t\_6\right) - \tan a\right)\\ \end{array} \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (let* ((t_0 (cos (fmin y z)))
       (t_1 (sin (fmax y z)))
       (t_2 (* t_1 (sin (- (fmin y z)))))
       (t_3 (cos (fmax y z)))
       (t_4 (fma t_3 t_0 t_2))
       (t_5 (* t_3 (sin (fmin y z))))
       (t_6 (/ t_5 t_4))
       (t_7 (fma t_0 t_3 t_2)))
  (if (<= (tan a) -5e-5)
    (+ x (- (+ (/ t_1 t_4) t_6) (tan a)))
    (if (<= (tan a) 5e-8)
      (+ x (+ (/ (* t_0 t_1) t_7) (/ t_5 t_7)))
      (+ x (- (+ (/ t_1 t_3) t_6) (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = cos(fmin(y, z));
	double t_1 = sin(fmax(y, z));
	double t_2 = t_1 * sin(-fmin(y, z));
	double t_3 = cos(fmax(y, z));
	double t_4 = fma(t_3, t_0, t_2);
	double t_5 = t_3 * sin(fmin(y, z));
	double t_6 = t_5 / t_4;
	double t_7 = fma(t_0, t_3, t_2);
	double tmp;
	if (tan(a) <= -5e-5) {
		tmp = x + (((t_1 / t_4) + t_6) - tan(a));
	} else if (tan(a) <= 5e-8) {
		tmp = x + (((t_0 * t_1) / t_7) + (t_5 / t_7));
	} else {
		tmp = x + (((t_1 / t_3) + t_6) - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = cos(fmin(y, z))
	t_1 = sin(fmax(y, z))
	t_2 = Float64(t_1 * sin(Float64(-fmin(y, z))))
	t_3 = cos(fmax(y, z))
	t_4 = fma(t_3, t_0, t_2)
	t_5 = Float64(t_3 * sin(fmin(y, z)))
	t_6 = Float64(t_5 / t_4)
	t_7 = fma(t_0, t_3, t_2)
	tmp = 0.0
	if (tan(a) <= -5e-5)
		tmp = Float64(x + Float64(Float64(Float64(t_1 / t_4) + t_6) - tan(a)));
	elseif (tan(a) <= 5e-8)
		tmp = Float64(x + Float64(Float64(Float64(t_0 * t_1) / t_7) + Float64(t_5 / t_7)));
	else
		tmp = Float64(x + Float64(Float64(Float64(t_1 / t_3) + t_6) - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Cos[N[Min[y, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[Max[y, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sin[(-N[Min[y, z], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[Max[y, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * t$95$0 + t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 * N[Sin[N[Min[y, z], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 / t$95$4), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$0 * t$95$3 + t$95$2), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -5e-5], N[(x + N[(N[(N[(t$95$1 / t$95$4), $MachinePrecision] + t$95$6), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 5e-8], N[(x + N[(N[(N[(t$95$0 * t$95$1), $MachinePrecision] / t$95$7), $MachinePrecision] + N[(t$95$5 / t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(t$95$1 / t$95$3), $MachinePrecision] + t$95$6), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -5.0000000000000002e-5

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. lift-tan.f64 N/A

         \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

      2. tan-quot N/A

         \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

      3. lift-+.f64 N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      4. +-commutative N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      5. sin-sum N/A

         \[\leadsto x + \left(\frac{\color{blue}{\sin z \cdot \cos y + \cos z \cdot \sin y}}{\cos \left(y + z\right)} - \tan a\right) \]

      6. div-add N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\cos z \cdot \sin y}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      7. *-commutative N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      8. lower-+.f64 N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      9. lower-/.f64 N/A

         \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z} \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \color{blue}{\cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      14. lift-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      15. +-commutative N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      16. lower-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      17. lower-/.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]

   3. Applied rewrites 79.7%

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 80.4%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   5. Applied rewrites 80.4%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos \left(z + y\right)}}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z + y\right)}}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 99.7%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)}\right) - \tan a\right) \]

   7. Applied rewrites 99.7%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}}\right) - \tan a\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z}}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]
   9. Step-by-step derivation

      1. lower-sin.f64 80.4%

         \[\leadsto x + \left(\left(\frac{\sin z}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   10. Applied rewrites 80.4%

       \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z}}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   if -5.0000000000000002e-5 < (tan.f64 a) < 4.9999999999999998e-8

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. lift-tan.f64 N/A

         \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

      2. tan-quot N/A

         \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

      3. lift-+.f64 N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      4. +-commutative N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      5. sin-sum N/A

         \[\leadsto x + \left(\frac{\color{blue}{\sin z \cdot \cos y + \cos z \cdot \sin y}}{\cos \left(y + z\right)} - \tan a\right) \]

      6. div-add N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\cos z \cdot \sin y}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      7. *-commutative N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      8. lower-+.f64 N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      9. lower-/.f64 N/A

         \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z} \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \color{blue}{\cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      14. lift-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      15. +-commutative N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      16. lower-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      17. lower-/.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]

   3. Applied rewrites 79.7%

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 80.4%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   5. Applied rewrites 80.4%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos \left(z + y\right)}}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z + y\right)}}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 99.7%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)}\right) - \tan a\right) \]

   7. Applied rewrites 99.7%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}}\right) - \tan a\right) \]

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + \left(\frac{\cos y \cdot \sin z}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + \color{blue}{\left(\frac{\cos y \cdot \sin z}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto x + \left(\frac{\cos y \cdot \sin z}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \color{blue}{\frac{\cos z \cdot \sin y}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}\right) \]

   10. Applied rewrites 60.6%

       \[\leadsto \color{blue}{x + \left(\frac{\cos y \cdot \sin z}{\mathsf{fma}\left(\cos y, \cos z, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos y, \cos z, \sin z \cdot \sin \left(-y\right)\right)}\right)} \]

   if 4.9999999999999998e-8 < (tan.f64 a)

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. lift-tan.f64 N/A

         \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

      2. tan-quot N/A

         \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

      3. lift-+.f64 N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      4. +-commutative N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      5. sin-sum N/A

         \[\leadsto x + \left(\frac{\color{blue}{\sin z \cdot \cos y + \cos z \cdot \sin y}}{\cos \left(y + z\right)} - \tan a\right) \]

      6. div-add N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\cos z \cdot \sin y}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      7. *-commutative N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      8. lower-+.f64 N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      9. lower-/.f64 N/A

         \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z} \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \color{blue}{\cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      14. lift-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      15. +-commutative N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      16. lower-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      17. lower-/.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]

   3. Applied rewrites 79.7%

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 80.4%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   5. Applied rewrites 80.4%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos \left(z + y\right)}}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z + y\right)}}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 99.7%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)}\right) - \tan a\right) \]

   7. Applied rewrites 99.7%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}}\right) - \tan a\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z}{\cos z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z}{\color{blue}{\cos z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

      2. lower-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z}{\cos \color{blue}{z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

      3. lower-cos.f64 80.4%

         \[\leadsto x + \left(\left(\frac{\sin z}{\cos z} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   10. Applied rewrites 80.4%

       \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z}{\cos z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 89.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(y, z\right)\right)\\ t_1 := \sin \left(\mathsf{min}\left(y, z\right)\right)\\ t_2 := \sin \left(\mathsf{max}\left(y, z\right)\right)\\ t_3 := t\_2 \cdot \sin \left(-\mathsf{min}\left(y, z\right)\right)\\ t_4 := \cos \left(\mathsf{min}\left(y, z\right)\right)\\ t_5 := \mathsf{fma}\left(t\_4, t\_0, t\_3\right)\\ t_6 := \mathsf{fma}\left(t\_0, t\_4, t\_3\right)\\ t_7 := t\_0 \cdot t\_1\\ \mathbf{if}\;\tan a \leq -5 \cdot 10^{-5}:\\ \;\;\;\;x + \left(\left(\frac{t\_2 \cdot t\_4}{t\_6} + \frac{t\_1}{t\_4}\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x + \left(\frac{t\_4 \cdot t\_2}{t\_5} + \frac{t\_7}{t\_5}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(\frac{t\_2}{t\_0} + \frac{t\_7}{t\_6}\right) - \tan a\right)\\ \end{array} \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (let* ((t_0 (cos (fmax y z)))
       (t_1 (sin (fmin y z)))
       (t_2 (sin (fmax y z)))
       (t_3 (* t_2 (sin (- (fmin y z)))))
       (t_4 (cos (fmin y z)))
       (t_5 (fma t_4 t_0 t_3))
       (t_6 (fma t_0 t_4 t_3))
       (t_7 (* t_0 t_1)))
  (if (<= (tan a) -5e-5)
    (+ x (- (+ (/ (* t_2 t_4) t_6) (/ t_1 t_4)) (tan a)))
    (if (<= (tan a) 5e-8)
      (+ x (+ (/ (* t_4 t_2) t_5) (/ t_7 t_5)))
      (+ x (- (+ (/ t_2 t_0) (/ t_7 t_6)) (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = cos(fmax(y, z));
	double t_1 = sin(fmin(y, z));
	double t_2 = sin(fmax(y, z));
	double t_3 = t_2 * sin(-fmin(y, z));
	double t_4 = cos(fmin(y, z));
	double t_5 = fma(t_4, t_0, t_3);
	double t_6 = fma(t_0, t_4, t_3);
	double t_7 = t_0 * t_1;
	double tmp;
	if (tan(a) <= -5e-5) {
		tmp = x + ((((t_2 * t_4) / t_6) + (t_1 / t_4)) - tan(a));
	} else if (tan(a) <= 5e-8) {
		tmp = x + (((t_4 * t_2) / t_5) + (t_7 / t_5));
	} else {
		tmp = x + (((t_2 / t_0) + (t_7 / t_6)) - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = cos(fmax(y, z))
	t_1 = sin(fmin(y, z))
	t_2 = sin(fmax(y, z))
	t_3 = Float64(t_2 * sin(Float64(-fmin(y, z))))
	t_4 = cos(fmin(y, z))
	t_5 = fma(t_4, t_0, t_3)
	t_6 = fma(t_0, t_4, t_3)
	t_7 = Float64(t_0 * t_1)
	tmp = 0.0
	if (tan(a) <= -5e-5)
		tmp = Float64(x + Float64(Float64(Float64(Float64(t_2 * t_4) / t_6) + Float64(t_1 / t_4)) - tan(a)));
	elseif (tan(a) <= 5e-8)
		tmp = Float64(x + Float64(Float64(Float64(t_4 * t_2) / t_5) + Float64(t_7 / t_5)));
	else
		tmp = Float64(x + Float64(Float64(Float64(t_2 / t_0) + Float64(t_7 / t_6)) - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Cos[N[Max[y, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[Min[y, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Max[y, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Sin[(-N[Min[y, z], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[Min[y, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * t$95$0 + t$95$3), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$0 * t$95$4 + t$95$3), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -5e-5], N[(x + N[(N[(N[(N[(t$95$2 * t$95$4), $MachinePrecision] / t$95$6), $MachinePrecision] + N[(t$95$1 / t$95$4), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 5e-8], N[(x + N[(N[(N[(t$95$4 * t$95$2), $MachinePrecision] / t$95$5), $MachinePrecision] + N[(t$95$7 / t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(t$95$2 / t$95$0), $MachinePrecision] + N[(t$95$7 / t$95$6), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -5.0000000000000002e-5

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. lift-tan.f64 N/A

         \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

      2. tan-quot N/A

         \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

      3. lift-+.f64 N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      4. +-commutative N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      5. sin-sum N/A

         \[\leadsto x + \left(\frac{\color{blue}{\sin z \cdot \cos y + \cos z \cdot \sin y}}{\cos \left(y + z\right)} - \tan a\right) \]

      6. div-add N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\cos z \cdot \sin y}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      7. *-commutative N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      8. lower-+.f64 N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      9. lower-/.f64 N/A

         \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z} \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \color{blue}{\cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      14. lift-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      15. +-commutative N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      16. lower-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      17. lower-/.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]

   3. Applied rewrites 79.7%

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 80.4%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   5. Applied rewrites 80.4%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos \left(z + y\right)}}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z + y\right)}}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 99.7%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)}\right) - \tan a\right) \]

   7. Applied rewrites 99.7%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}}\right) - \tan a\right) \]

   8. Taylor expanded in z around 0

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \color{blue}{\frac{\sin y}{\cos y}}\right) - \tan a\right) \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\sin y}{\color{blue}{\cos y}}\right) - \tan a\right) \]

      2. lower-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\sin y}{\cos \color{blue}{y}}\right) - \tan a\right) \]

      3. lower-cos.f64 80.3%

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\sin y}{\cos y}\right) - \tan a\right) \]

   10. Applied rewrites 80.3%

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \color{blue}{\frac{\sin y}{\cos y}}\right) - \tan a\right) \]

   if -5.0000000000000002e-5 < (tan.f64 a) < 4.9999999999999998e-8

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. lift-tan.f64 N/A

         \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

      2. tan-quot N/A

         \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

      3. lift-+.f64 N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      4. +-commutative N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      5. sin-sum N/A

         \[\leadsto x + \left(\frac{\color{blue}{\sin z \cdot \cos y + \cos z \cdot \sin y}}{\cos \left(y + z\right)} - \tan a\right) \]

      6. div-add N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\cos z \cdot \sin y}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      7. *-commutative N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      8. lower-+.f64 N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      9. lower-/.f64 N/A

         \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z} \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \color{blue}{\cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      14. lift-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      15. +-commutative N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      16. lower-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      17. lower-/.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]

   3. Applied rewrites 79.7%

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 80.4%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   5. Applied rewrites 80.4%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos \left(z + y\right)}}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z + y\right)}}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 99.7%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)}\right) - \tan a\right) \]

   7. Applied rewrites 99.7%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}}\right) - \tan a\right) \]

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + \left(\frac{\cos y \cdot \sin z}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + \color{blue}{\left(\frac{\cos y \cdot \sin z}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto x + \left(\frac{\cos y \cdot \sin z}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \color{blue}{\frac{\cos z \cdot \sin y}{\cos y \cdot \cos z + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}\right) \]

   10. Applied rewrites 60.6%

       \[\leadsto \color{blue}{x + \left(\frac{\cos y \cdot \sin z}{\mathsf{fma}\left(\cos y, \cos z, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos y, \cos z, \sin z \cdot \sin \left(-y\right)\right)}\right)} \]

   if 4.9999999999999998e-8 < (tan.f64 a)

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. lift-tan.f64 N/A

         \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

      2. tan-quot N/A

         \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

      3. lift-+.f64 N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      4. +-commutative N/A

         \[\leadsto x + \left(\frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

      5. sin-sum N/A

         \[\leadsto x + \left(\frac{\color{blue}{\sin z \cdot \cos y + \cos z \cdot \sin y}}{\cos \left(y + z\right)} - \tan a\right) \]

      6. div-add N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\cos z \cdot \sin y}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      7. *-commutative N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      8. lower-+.f64 N/A

         \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

      9. lower-/.f64 N/A

         \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z} \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \color{blue}{\cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      14. lift-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      15. +-commutative N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      16. lower-+.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

      17. lower-/.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]

   3. Applied rewrites 79.7%

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 80.4%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   5. Applied rewrites 80.4%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos \left(z + y\right)}}\right) - \tan a\right) \]

      2. lift-+.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z + y\right)}}\right) - \tan a\right) \]

      3. add-flip N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      4. cos-diff N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}\right) - \tan a\right) \]

      5. cos-neg-rev N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}\right) - \tan a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

      12. lower-neg.f64 99.7%

          \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)}\right) - \tan a\right) \]

   7. Applied rewrites 99.7%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}}\right) - \tan a\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z}{\cos z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z}{\color{blue}{\cos z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

      2. lower-sin.f64 N/A

         \[\leadsto x + \left(\left(\frac{\sin z}{\cos \color{blue}{z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

      3. lower-cos.f64 80.4%

         \[\leadsto x + \left(\left(\frac{\sin z}{\cos z} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   10. Applied rewrites 80.4%

       \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z}{\cos z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 80.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(y, z\right)\right)\\ t_1 := \sin \left(\mathsf{max}\left(y, z\right)\right)\\ t_2 := \cos \left(\mathsf{min}\left(y, z\right)\right)\\ x + \left(\left(\frac{t\_1 \cdot t\_2}{\mathsf{fma}\left(t\_0, t\_2, t\_1 \cdot \sin \left(-\mathsf{min}\left(y, z\right)\right)\right)} + \frac{\sin \left(\mathsf{min}\left(y, z\right)\right) \cdot t\_0}{\cos \left(\mathsf{min}\left(y, z\right) + \mathsf{max}\left(y, z\right)\right)}\right) - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (let* ((t_0 (cos (fmax y z)))
       (t_1 (sin (fmax y z)))
       (t_2 (cos (fmin y z))))
  (+
   x
   (-
    (+
     (/ (* t_1 t_2) (fma t_0 t_2 (* t_1 (sin (- (fmin y z))))))
     (/ (* (sin (fmin y z)) t_0) (cos (+ (fmin y z) (fmax y z)))))
    (tan a)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = cos(fmax(y, z));
	double t_1 = sin(fmax(y, z));
	double t_2 = cos(fmin(y, z));
	return x + ((((t_1 * t_2) / fma(t_0, t_2, (t_1 * sin(-fmin(y, z))))) + ((sin(fmin(y, z)) * t_0) / cos((fmin(y, z) + fmax(y, z))))) - tan(a));
}

Julia

function code(x, y, z, a)
	t_0 = cos(fmax(y, z))
	t_1 = sin(fmax(y, z))
	t_2 = cos(fmin(y, z))
	return Float64(x + Float64(Float64(Float64(Float64(t_1 * t_2) / fma(t_0, t_2, Float64(t_1 * sin(Float64(-fmin(y, z)))))) + Float64(Float64(sin(fmin(y, z)) * t_0) / cos(Float64(fmin(y, z) + fmax(y, z))))) - tan(a)))
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Cos[N[Max[y, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[Max[y, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[Min[y, z], $MachinePrecision]], $MachinePrecision]}, N[(x + N[(N[(N[(N[(t$95$1 * t$95$2), $MachinePrecision] / N[(t$95$0 * t$95$2 + N[(t$95$1 * N[Sin[(-N[Min[y, z], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[N[Min[y, z], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] / N[Cos[N[(N[Min[y, z], $MachinePrecision] + N[Max[y, z], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 79.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Step-by-step derivation

   1. lift-tan.f64 N/A

      \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

   2. tan-quot N/A

      \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

   3. lift-+.f64 N/A

      \[\leadsto x + \left(\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

   4. +-commutative N/A

      \[\leadsto x + \left(\frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

   5. sin-sum N/A

      \[\leadsto x + \left(\frac{\color{blue}{\sin z \cdot \cos y + \cos z \cdot \sin y}}{\cos \left(y + z\right)} - \tan a\right) \]

   6. div-add N/A

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\cos z \cdot \sin y}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

   7. *-commutative N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   8. lower-+.f64 N/A

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

   9. lower-/.f64 N/A

      \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   11. lower-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z} \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \color{blue}{\cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   13. lower-cos.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   14. lift-+.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   15. +-commutative N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   16. lower-+.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   17. lower-/.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]

3. Applied rewrites 79.7%

   \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
4. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   2. lift-+.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   3. add-flip N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   4. cos-diff N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   5. cos-neg-rev N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   6. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   7. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   9. lift-sin.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   11. lower-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   12. lower-neg.f64 80.4%

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

5. Applied rewrites 80.4%

   \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]
6. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos \left(z + y\right)}}\right) - \tan a\right) \]

   2. lift-+.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z + y\right)}}\right) - \tan a\right) \]

   3. add-flip N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

   4. cos-diff N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}\right) - \tan a\right) \]

   5. cos-neg-rev N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

   6. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

   7. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

   9. lift-sin.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}\right) - \tan a\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

   11. lower-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

   12. lower-neg.f64 99.7%

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)}\right) - \tan a\right) \]

7. Applied rewrites 99.7%

   \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}}\right) - \tan a\right) \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\color{blue}{\cos z \cdot \sin y}}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   2. *-commutative N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   3. lower-*.f64 99.7%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   4. lift-fma.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\sin y \cdot \cos z}{\color{blue}{\cos z \cdot \cos y + \sin z \cdot \sin \left(-y\right)}}\right) - \tan a\right) \]

   5. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\sin y \cdot \cos z}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(-y\right)}\right) - \tan a\right) \]

   6. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\sin y \cdot \cos z}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(-y\right)}\right) - \tan a\right) \]

   7. cos-neg-rev N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\sin y \cdot \cos z}{\cos z \cdot \color{blue}{\cos \left(\mathsf{neg}\left(y\right)\right)} + \sin z \cdot \sin \left(-y\right)}\right) - \tan a\right) \]

   8. lift-neg.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\sin y \cdot \cos z}{\cos z \cdot \cos \color{blue}{\left(-y\right)} + \sin z \cdot \sin \left(-y\right)}\right) - \tan a\right) \]

   9. lift-*.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\sin y \cdot \cos z}{\cos z \cdot \cos \left(-y\right) + \color{blue}{\sin z \cdot \sin \left(-y\right)}}\right) - \tan a\right) \]

   10. lift-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\sin y \cdot \cos z}{\cos z \cdot \cos \left(-y\right) + \color{blue}{\sin z} \cdot \sin \left(-y\right)}\right) - \tan a\right) \]

   11. lift-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\sin y \cdot \cos z}{\cos z \cdot \cos \left(-y\right) + \sin z \cdot \color{blue}{\sin \left(-y\right)}}\right) - \tan a\right) \]

   12. cos-diff-rev N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\sin y \cdot \cos z}{\color{blue}{\cos \left(z - \left(-y\right)\right)}}\right) - \tan a\right) \]

   13. lift-neg.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\sin y \cdot \cos z}{\cos \left(z - \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

   14. add-flip N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\sin y \cdot \cos z}{\cos \color{blue}{\left(z + y\right)}}\right) - \tan a\right) \]

   15. lift-+.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\sin y \cdot \cos z}{\cos \color{blue}{\left(z + y\right)}}\right) - \tan a\right) \]

   16. lift-cos.f64 80.4%

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\sin y \cdot \cos z}{\color{blue}{\cos \left(z + y\right)}}\right) - \tan a\right) \]

   17. lift-+.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\sin y \cdot \cos z}{\cos \color{blue}{\left(z + y\right)}}\right) - \tan a\right) \]

   18. +-commutative N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\sin y \cdot \cos z}{\cos \color{blue}{\left(y + z\right)}}\right) - \tan a\right) \]

   19. lower-+.f64 80.4%

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\sin y \cdot \cos z}{\cos \color{blue}{\left(y + z\right)}}\right) - \tan a\right) \]

9. Applied rewrites 80.4%

   \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]
10. Add Preprocessing

Alternative 9: 80.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\mathsf{max}\left(y, z\right)\right)\\ t_1 := \cos \left(\mathsf{min}\left(y, z\right)\right)\\ x + \left(\left(\frac{t\_0 \cdot t\_1}{\mathsf{fma}\left(\cos \left(\mathsf{max}\left(y, z\right)\right), t\_1, t\_0 \cdot \sin \left(-\mathsf{min}\left(y, z\right)\right)\right)} + \frac{\sin \left(\mathsf{min}\left(y, z\right)\right)}{t\_1}\right) - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (let* ((t_0 (sin (fmax y z))) (t_1 (cos (fmin y z))))
  (+
   x
   (-
    (+
     (/
      (* t_0 t_1)
      (fma (cos (fmax y z)) t_1 (* t_0 (sin (- (fmin y z))))))
     (/ (sin (fmin y z)) t_1))
    (tan a)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = sin(fmax(y, z));
	double t_1 = cos(fmin(y, z));
	return x + ((((t_0 * t_1) / fma(cos(fmax(y, z)), t_1, (t_0 * sin(-fmin(y, z))))) + (sin(fmin(y, z)) / t_1)) - tan(a));
}

Julia

function code(x, y, z, a)
	t_0 = sin(fmax(y, z))
	t_1 = cos(fmin(y, z))
	return Float64(x + Float64(Float64(Float64(Float64(t_0 * t_1) / fma(cos(fmax(y, z)), t_1, Float64(t_0 * sin(Float64(-fmin(y, z)))))) + Float64(sin(fmin(y, z)) / t_1)) - tan(a)))
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Sin[N[Max[y, z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Min[y, z], $MachinePrecision]], $MachinePrecision]}, N[(x + N[(N[(N[(N[(t$95$0 * t$95$1), $MachinePrecision] / N[(N[Cos[N[Max[y, z], $MachinePrecision]], $MachinePrecision] * t$95$1 + N[(t$95$0 * N[Sin[(-N[Min[y, z], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[N[Min[y, z], $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 79.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Step-by-step derivation

   1. lift-tan.f64 N/A

      \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

   2. tan-quot N/A

      \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

   3. lift-+.f64 N/A

      \[\leadsto x + \left(\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

   4. +-commutative N/A

      \[\leadsto x + \left(\frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

   5. sin-sum N/A

      \[\leadsto x + \left(\frac{\color{blue}{\sin z \cdot \cos y + \cos z \cdot \sin y}}{\cos \left(y + z\right)} - \tan a\right) \]

   6. div-add N/A

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\cos z \cdot \sin y}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

   7. *-commutative N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   8. lower-+.f64 N/A

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

   9. lower-/.f64 N/A

      \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   11. lower-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z} \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \color{blue}{\cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   13. lower-cos.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   14. lift-+.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   15. +-commutative N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   16. lower-+.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   17. lower-/.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]

3. Applied rewrites 79.7%

   \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
4. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   2. lift-+.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   3. add-flip N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   4. cos-diff N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   5. cos-neg-rev N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   6. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   7. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   9. lift-sin.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   11. lower-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   12. lower-neg.f64 80.4%

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

5. Applied rewrites 80.4%

   \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]
6. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos \left(z + y\right)}}\right) - \tan a\right) \]

   2. lift-+.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z + y\right)}}\right) - \tan a\right) \]

   3. add-flip N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

   4. cos-diff N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}\right) - \tan a\right) \]

   5. cos-neg-rev N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

   6. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

   7. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

   9. lift-sin.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}\right) - \tan a\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

   11. lower-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

   12. lower-neg.f64 99.7%

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)}\right) - \tan a\right) \]

7. Applied rewrites 99.7%

   \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}}\right) - \tan a\right) \]

8. Taylor expanded in z around 0

   \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \color{blue}{\frac{\sin y}{\cos y}}\right) - \tan a\right) \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\sin y}{\color{blue}{\cos y}}\right) - \tan a\right) \]

   2. lower-sin.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\sin y}{\cos \color{blue}{y}}\right) - \tan a\right) \]

   3. lower-cos.f64 80.3%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\sin y}{\cos y}\right) - \tan a\right) \]

10. Applied rewrites 80.3%

    \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \color{blue}{\frac{\sin y}{\cos y}}\right) - \tan a\right) \]
11. Add Preprocessing

Alternative 10: 80.3% accurate, 0.2× speedup

Math

\[x + \left(\left(\frac{\sin z}{\cos z} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (+
 x
 (-
  (+
   (/ (sin z) (cos z))
   (/
    (* (cos z) (sin y))
    (fma (cos z) (cos y) (* (sin z) (sin (- y))))))
  (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (((sin(z) / cos(z)) + ((cos(z) * sin(y)) / fma(cos(z), cos(y), (sin(z) * sin(-y))))) - tan(a));
}

Julia

function code(x, y, z, a)
	return Float64(x + Float64(Float64(Float64(sin(z) / cos(z)) + Float64(Float64(cos(z) * sin(y)) / fma(cos(z), cos(y), Float64(sin(z) * sin(Float64(-y)))))) - tan(a)))
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(N[(N[Sin[z], $MachinePrecision] / N[Cos[z], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[z], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[z], $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Sin[z], $MachinePrecision] * N[Sin[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Step-by-step derivation

   1. lift-tan.f64 N/A

      \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

   2. tan-quot N/A

      \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

   3. lift-+.f64 N/A

      \[\leadsto x + \left(\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

   4. +-commutative N/A

      \[\leadsto x + \left(\frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

   5. sin-sum N/A

      \[\leadsto x + \left(\frac{\color{blue}{\sin z \cdot \cos y + \cos z \cdot \sin y}}{\cos \left(y + z\right)} - \tan a\right) \]

   6. div-add N/A

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\cos z \cdot \sin y}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

   7. *-commutative N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   8. lower-+.f64 N/A

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

   9. lower-/.f64 N/A

      \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   11. lower-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z} \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \color{blue}{\cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   13. lower-cos.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   14. lift-+.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   15. +-commutative N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   16. lower-+.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   17. lower-/.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]

3. Applied rewrites 79.7%

   \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
4. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   2. lift-+.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   3. add-flip N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   4. cos-diff N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   5. cos-neg-rev N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   6. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   7. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   9. lift-sin.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   11. lower-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   12. lower-neg.f64 80.4%

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

5. Applied rewrites 80.4%

   \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) \]
6. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos \left(z + y\right)}}\right) - \tan a\right) \]

   2. lift-+.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z + y\right)}}\right) - \tan a\right) \]

   3. add-flip N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos \color{blue}{\left(z - \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

   4. cos-diff N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z \cdot \cos \left(\mathsf{neg}\left(y\right)\right) + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}}\right) - \tan a\right) \]

   5. cos-neg-rev N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

   6. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\cos z} \cdot \cos y + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

   7. lift-cos.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\cos z \cdot \color{blue}{\cos y} + \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right) - \tan a\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) - \tan a\right) \]

   9. lift-sin.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z} \cdot \sin \left(\mathsf{neg}\left(y\right)\right)\right)}\right) - \tan a\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \color{blue}{\sin z \cdot \sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

   11. lower-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \color{blue}{\sin \left(\mathsf{neg}\left(y\right)\right)}\right)}\right) - \tan a\right) \]

   12. lower-neg.f64 99.7%

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \color{blue}{\left(-y\right)}\right)}\right) - \tan a\right) \]

7. Applied rewrites 99.7%

   \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\color{blue}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}}\right) - \tan a\right) \]

8. Taylor expanded in y around 0

   \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z}{\cos z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z}{\color{blue}{\cos z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   2. lower-sin.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z}{\cos \color{blue}{z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

   3. lower-cos.f64 80.4%

      \[\leadsto x + \left(\left(\frac{\sin z}{\cos z} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]

10. Applied rewrites 80.4%

    \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z}{\cos z}} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)}\right) - \tan a\right) \]
11. Add Preprocessing

Alternative 11: 79.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(y + z\right)\\ \mathsf{fma}\left(\frac{\cos y}{t\_0}, \sin z, \left(\frac{\sin y \cdot \cos z}{t\_0} - \tan a\right) + x\right) \end{array} \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (let* ((t_0 (cos (+ y z))))
  (fma
   (/ (cos y) t_0)
   (sin z)
   (+ (- (/ (* (sin y) (cos z)) t_0) (tan a)) x))))

C

double code(double x, double y, double z, double a) {
	double t_0 = cos((y + z));
	return fma((cos(y) / t_0), sin(z), ((((sin(y) * cos(z)) / t_0) - tan(a)) + x));
}

Julia

function code(x, y, z, a)
	t_0 = cos(Float64(y + z))
	return fma(Float64(cos(y) / t_0), sin(z), Float64(Float64(Float64(Float64(sin(y) * cos(z)) / t_0) - tan(a)) + x))
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Cos[y], $MachinePrecision] / t$95$0), $MachinePrecision] * N[Sin[z], $MachinePrecision] + N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * N[Cos[z], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Step-by-step derivation

   1. lift-tan.f64 N/A

      \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

   2. tan-quot N/A

      \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

   3. lift-+.f64 N/A

      \[\leadsto x + \left(\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

   4. +-commutative N/A

      \[\leadsto x + \left(\frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

   5. sin-sum N/A

      \[\leadsto x + \left(\frac{\color{blue}{\sin z \cdot \cos y + \cos z \cdot \sin y}}{\cos \left(y + z\right)} - \tan a\right) \]

   6. div-add N/A

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\cos z \cdot \sin y}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

   7. *-commutative N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   8. lower-+.f64 N/A

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

   9. lower-/.f64 N/A

      \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   11. lower-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z} \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \color{blue}{\cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   13. lower-cos.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   14. lift-+.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   15. +-commutative N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   16. lower-+.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   17. lower-/.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]

3. Applied rewrites 79.7%

   \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
4. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right) + x} \]

   3. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right) - \tan a\right)} + x \]

   4. lift-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) + x \]

   5. associate--l+ N/A

      \[\leadsto \color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \left(\frac{\cos z \cdot \sin y}{\cos \left(z + y\right)} - \tan a\right)\right)} + x \]

   6. associate-+l+ N/A

      \[\leadsto \color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \left(\left(\frac{\cos z \cdot \sin y}{\cos \left(z + y\right)} - \tan a\right) + x\right)} \]

   7. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)}} + \left(\left(\frac{\cos z \cdot \sin y}{\cos \left(z + y\right)} - \tan a\right) + x\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(z + y\right)} + \left(\left(\frac{\cos z \cdot \sin y}{\cos \left(z + y\right)} - \tan a\right) + x\right) \]

   9. associate-/l* N/A

      \[\leadsto \color{blue}{\sin z \cdot \frac{\cos y}{\cos \left(z + y\right)}} + \left(\left(\frac{\cos z \cdot \sin y}{\cos \left(z + y\right)} - \tan a\right) + x\right) \]

5. Applied rewrites 79.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\cos y}{\cos \left(y + z\right)}, \sin z, \left(\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)} - \tan a\right) + x\right)} \]
6. Add Preprocessing

Alternative 12: 79.7% accurate, 0.4× speedup

Math

\[x + \left(\frac{\mathsf{fma}\left(-\sin y, \cos z, \left(-\cos y\right) \cdot \sin z\right)}{-\cos \left(y + z\right)} - \tan a\right) \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (+
 x
 (-
  (/
   (fma (- (sin y)) (cos z) (* (- (cos y)) (sin z)))
   (- (cos (+ y z))))
  (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + ((fma(-sin(y), cos(z), (-cos(y) * sin(z))) / -cos((y + z))) - tan(a));
}

Julia

function code(x, y, z, a)
	return Float64(x + Float64(Float64(fma(Float64(-sin(y)), cos(z), Float64(Float64(-cos(y)) * sin(z))) / Float64(-cos(Float64(y + z)))) - tan(a)))
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(N[((-N[Sin[y], $MachinePrecision]) * N[Cos[z], $MachinePrecision] + N[((-N[Cos[y], $MachinePrecision]) * N[Sin[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Step-by-step derivation

   1. lift-tan.f64 N/A

      \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

   2. tan-quot N/A

      \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

   3. lift-+.f64 N/A

      \[\leadsto x + \left(\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

   4. +-commutative N/A

      \[\leadsto x + \left(\frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

   5. sin-sum N/A

      \[\leadsto x + \left(\frac{\color{blue}{\sin z \cdot \cos y + \cos z \cdot \sin y}}{\cos \left(y + z\right)} - \tan a\right) \]

   6. div-add N/A

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\cos z \cdot \sin y}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

   7. *-commutative N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   8. lower-+.f64 N/A

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

   9. lower-/.f64 N/A

      \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   11. lower-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z} \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \color{blue}{\cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   13. lower-cos.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   14. lift-+.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   15. +-commutative N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   16. lower-+.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   17. lower-/.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]

3. Applied rewrites 79.7%

   \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
4. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]

   2. +-commutative N/A

      \[\leadsto x + \left(\color{blue}{\left(\frac{\cos z \cdot \sin y}{\cos \left(z + y\right)} + \frac{\sin z \cdot \cos y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]

   3. lift-/.f64 N/A

      \[\leadsto x + \left(\left(\color{blue}{\frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}} + \frac{\sin z \cdot \cos y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   4. frac-2neg N/A

      \[\leadsto x + \left(\left(\color{blue}{\frac{\mathsf{neg}\left(\cos z \cdot \sin y\right)}{\mathsf{neg}\left(\cos \left(z + y\right)\right)}} + \frac{\sin z \cdot \cos y}{\cos \left(z + y\right)}\right) - \tan a\right) \]

   5. lift-/.f64 N/A

      \[\leadsto x + \left(\left(\frac{\mathsf{neg}\left(\cos z \cdot \sin y\right)}{\mathsf{neg}\left(\cos \left(z + y\right)\right)} + \color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)}}\right) - \tan a\right) \]

   6. frac-2neg N/A

      \[\leadsto x + \left(\left(\frac{\mathsf{neg}\left(\cos z \cdot \sin y\right)}{\mathsf{neg}\left(\cos \left(z + y\right)\right)} + \color{blue}{\frac{\mathsf{neg}\left(\sin z \cdot \cos y\right)}{\mathsf{neg}\left(\cos \left(z + y\right)\right)}}\right) - \tan a\right) \]

   7. div-add-rev N/A

      \[\leadsto x + \left(\color{blue}{\frac{\left(\mathsf{neg}\left(\cos z \cdot \sin y\right)\right) + \left(\mathsf{neg}\left(\sin z \cdot \cos y\right)\right)}{\mathsf{neg}\left(\cos \left(z + y\right)\right)}} - \tan a\right) \]

   8. lower-/.f64 N/A

      \[\leadsto x + \left(\color{blue}{\frac{\left(\mathsf{neg}\left(\cos z \cdot \sin y\right)\right) + \left(\mathsf{neg}\left(\sin z \cdot \cos y\right)\right)}{\mathsf{neg}\left(\cos \left(z + y\right)\right)}} - \tan a\right) \]

5. Applied rewrites 79.7%

   \[\leadsto x + \left(\color{blue}{\frac{\mathsf{fma}\left(-\sin y, \cos z, \left(-\cos y\right) \cdot \sin z\right)}{-\cos \left(y + z\right)}} - \tan a\right) \]
6. Add Preprocessing

Alternative 13: 79.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{min}\left(y, z\right)\right)\\ x + \left(\left(\frac{\sin \left(\mathsf{max}\left(y, z\right)\right) \cdot t\_0}{\cos \left(\mathsf{max}\left(y, z\right) + \mathsf{min}\left(y, z\right)\right)} + \frac{\sin \left(\mathsf{min}\left(y, z\right)\right)}{t\_0}\right) - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (let* ((t_0 (cos (fmin y z))))
  (+
   x
   (-
    (+
     (/ (* (sin (fmax y z)) t_0) (cos (+ (fmax y z) (fmin y z))))
     (/ (sin (fmin y z)) t_0))
    (tan a)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = cos(fmin(y, z));
	return x + ((((sin(fmax(y, z)) * t_0) / cos((fmax(y, z) + fmin(y, z)))) + (sin(fmin(y, z)) / t_0)) - tan(a));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    t_0 = cos(fmin(y, z))
    code = x + ((((sin(fmax(y, z)) * t_0) / cos((fmax(y, z) + fmin(y, z)))) + (sin(fmin(y, z)) / t_0)) - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.cos(fmin(y, z));
	return x + ((((Math.sin(fmax(y, z)) * t_0) / Math.cos((fmax(y, z) + fmin(y, z)))) + (Math.sin(fmin(y, z)) / t_0)) - Math.tan(a));
}

Python

def code(x, y, z, a):
	t_0 = math.cos(fmin(y, z))
	return x + ((((math.sin(fmax(y, z)) * t_0) / math.cos((fmax(y, z) + fmin(y, z)))) + (math.sin(fmin(y, z)) / t_0)) - math.tan(a))

Julia

function code(x, y, z, a)
	t_0 = cos(fmin(y, z))
	return Float64(x + Float64(Float64(Float64(Float64(sin(fmax(y, z)) * t_0) / cos(Float64(fmax(y, z) + fmin(y, z)))) + Float64(sin(fmin(y, z)) / t_0)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	t_0 = cos(min(y, z));
	tmp = x + ((((sin(max(y, z)) * t_0) / cos((max(y, z) + min(y, z)))) + (sin(min(y, z)) / t_0)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Cos[N[Min[y, z], $MachinePrecision]], $MachinePrecision]}, N[(x + N[(N[(N[(N[(N[Sin[N[Max[y, z], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] / N[Cos[N[(N[Max[y, z], $MachinePrecision] + N[Min[y, z], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[N[Min[y, z], $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Step-by-step derivation

   1. lift-tan.f64 N/A

      \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

   2. tan-quot N/A

      \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

   3. lift-+.f64 N/A

      \[\leadsto x + \left(\frac{\sin \color{blue}{\left(y + z\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

   4. +-commutative N/A

      \[\leadsto x + \left(\frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} - \tan a\right) \]

   5. sin-sum N/A

      \[\leadsto x + \left(\frac{\color{blue}{\sin z \cdot \cos y + \cos z \cdot \sin y}}{\cos \left(y + z\right)} - \tan a\right) \]

   6. div-add N/A

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\cos z \cdot \sin y}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

   7. *-commutative N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\color{blue}{\sin y \cdot \cos z}}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   8. lower-+.f64 N/A

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right)} - \tan a\right) \]

   9. lower-/.f64 N/A

      \[\leadsto x + \left(\left(\color{blue}{\frac{\sin z \cdot \cos y}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z \cdot \cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   11. lower-sin.f64 N/A

       \[\leadsto x + \left(\left(\frac{\color{blue}{\sin z} \cdot \cos y}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \color{blue}{\cos y}}{\cos \left(y + z\right)} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   13. lower-cos.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\color{blue}{\cos \left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   14. lift-+.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(y + z\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   15. +-commutative N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   16. lower-+.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \color{blue}{\left(z + y\right)}} + \frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}\right) - \tan a\right) \]

   17. lower-/.f64 N/A

       \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y \cdot \cos z}{\cos \left(y + z\right)}}\right) - \tan a\right) \]

3. Applied rewrites 79.7%

   \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]

4. Taylor expanded in z around 0

   \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y}{\cos y}}\right) - \tan a\right) \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\sin y}{\color{blue}{\cos y}}\right) - \tan a\right) \]

   2. lower-sin.f64 N/A

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\sin y}{\cos \color{blue}{y}}\right) - \tan a\right) \]

   3. lower-cos.f64 79.3%

      \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\sin y}{\cos y}\right) - \tan a\right) \]

6. Applied rewrites 79.3%

   \[\leadsto x + \left(\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \color{blue}{\frac{\sin y}{\cos y}}\right) - \tan a\right) \]
7. Add Preprocessing

Alternative 14: 68.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(y, z\right) + \mathsf{max}\left(y, z\right) \leq -0.0001:\\ \;\;\;\;1 \cdot \left(x + \frac{\sin \left(\mathsf{min}\left(y, z\right)\right)}{\cos \left(\mathsf{min}\left(y, z\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(\mathsf{max}\left(y, z\right)\right) - \tan a\right)\\ \end{array} \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (if (<= (+ (fmin y z) (fmax y z)) -0.0001)
  (* 1.0 (+ x (/ (sin (fmin y z)) (cos (fmin y z)))))
  (+ x (- (tan (fmax y z)) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((fmin(y, z) + fmax(y, z)) <= -0.0001) {
		tmp = 1.0 * (x + (sin(fmin(y, z)) / cos(fmin(y, z))));
	} else {
		tmp = x + (tan(fmax(y, z)) - tan(a));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((fmin(y, z) + fmax(y, z)) <= (-0.0001d0)) then
        tmp = 1.0d0 * (x + (sin(fmin(y, z)) / cos(fmin(y, z))))
    else
        tmp = x + (tan(fmax(y, z)) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((fmin(y, z) + fmax(y, z)) <= -0.0001) {
		tmp = 1.0 * (x + (Math.sin(fmin(y, z)) / Math.cos(fmin(y, z))));
	} else {
		tmp = x + (Math.tan(fmax(y, z)) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (fmin(y, z) + fmax(y, z)) <= -0.0001:
		tmp = 1.0 * (x + (math.sin(fmin(y, z)) / math.cos(fmin(y, z))))
	else:
		tmp = x + (math.tan(fmax(y, z)) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(fmin(y, z) + fmax(y, z)) <= -0.0001)
		tmp = Float64(1.0 * Float64(x + Float64(sin(fmin(y, z)) / cos(fmin(y, z)))));
	else
		tmp = Float64(x + Float64(tan(fmax(y, z)) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((min(y, z) + max(y, z)) <= -0.0001)
		tmp = 1.0 * (x + (sin(min(y, z)) / cos(min(y, z))));
	else
		tmp = x + (tan(max(y, z)) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(N[Min[y, z], $MachinePrecision] + N[Max[y, z], $MachinePrecision]), $MachinePrecision], -0.0001], N[(1.0 * N[(x + N[(N[Sin[N[Min[y, z], $MachinePrecision]], $MachinePrecision] / N[Cos[N[Min[y, z], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[Max[y, z], $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -1e-4

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\tan \left(y + z\right) - \tan a\right)} \]

      2. lift--.f64 N/A

         \[\leadsto x + \color{blue}{\left(\tan \left(y + z\right) - \tan a\right)} \]

      3. associate-+r- N/A

         \[\leadsto \color{blue}{\left(x + \tan \left(y + z\right)\right) - \tan a} \]

      4. sub-to-mult N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right) \cdot \left(x + \tan \left(y + z\right)\right)} \]

      5. lower-unsound-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right) \cdot \left(x + \tan \left(y + z\right)\right)} \]

      6. lower-unsound--.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right)} \cdot \left(x + \tan \left(y + z\right)\right) \]

      7. lower-unsound-/.f64 N/A

         \[\leadsto \left(1 - \color{blue}{\frac{\tan a}{x + \tan \left(y + z\right)}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \left(1 - \frac{\tan a}{\color{blue}{\tan \left(y + z\right) + x}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(1 - \frac{\tan a}{\color{blue}{\tan \left(y + z\right) + x}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(y + z\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(z + y\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(z + y\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \color{blue}{\left(\tan \left(y + z\right) + x\right)} \]

      14. lower-+.f64 79.2%

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \color{blue}{\left(\tan \left(y + z\right) + x\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(y + z\right)} + x\right) \]

      16. +-commutative N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

      17. lower-+.f64 79.2%

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

   3. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \left(z + y\right) + x\right)} \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \cdot \left(\tan \left(z + y\right) + x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 50.3%

      \[\leadsto \color{blue}{1} \cdot \left(\tan \left(z + y\right) + x\right) \]

   7. Taylor expanded in z around 0

      \[\leadsto 1 \cdot \color{blue}{\left(x + \frac{\sin y}{\cos y}\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 \cdot \left(x + \color{blue}{\frac{\sin y}{\cos y}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto 1 \cdot \left(x + \frac{\sin y}{\color{blue}{\cos y}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto 1 \cdot \left(x + \frac{\sin y}{\cos \color{blue}{y}}\right) \]

      4. lower-cos.f64 40.2%

         \[\leadsto 1 \cdot \left(x + \frac{\sin y}{\cos y}\right) \]

   9. Applied rewrites 40.2%

      \[\leadsto 1 \cdot \color{blue}{\left(x + \frac{\sin y}{\cos y}\right)} \]

   if -1e-4 < (+.f64 y z)

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

   2. Taylor expanded in y around 0

      \[\leadsto x + \left(\tan \color{blue}{z} - \tan a\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 60.2%

      \[\leadsto x + \left(\tan \color{blue}{z} - \tan a\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 59.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(y, z\right) + \mathsf{max}\left(y, z\right)\\ \mathbf{if}\;t\_0 \leq -0.0001:\\ \;\;\;\;1 \cdot \left(x + \frac{\sin \left(\mathsf{min}\left(y, z\right)\right)}{\cos \left(\mathsf{min}\left(y, z\right)\right)}\right)\\ \mathbf{elif}\;t\_0 \leq 0.002:\\ \;\;\;\;x + \left(\left(\mathsf{min}\left(y, z\right) + \mathsf{max}\left(y, z\right) \cdot \left(1 + \mathsf{max}\left(y, z\right) \cdot \left(\mathsf{min}\left(y, z\right) + 0.3333333333333333 \cdot \mathsf{max}\left(y, z\right)\right)\right)\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\tan \left(\frac{t\_0}{\mathsf{max}\left(y, z\right)} \cdot \mathsf{max}\left(y, z\right)\right) + x\right)\\ \end{array} \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (let* ((t_0 (+ (fmin y z) (fmax y z))))
  (if (<= t_0 -0.0001)
    (* 1.0 (+ x (/ (sin (fmin y z)) (cos (fmin y z)))))
    (if (<= t_0 0.002)
      (+
       x
       (-
        (+
         (fmin y z)
         (*
          (fmax y z)
          (+
           1.0
           (*
            (fmax y z)
            (+ (fmin y z) (* 0.3333333333333333 (fmax y z)))))))
        (tan a)))
      (* 1.0 (+ (tan (* (/ t_0 (fmax y z)) (fmax y z))) x))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = fmin(y, z) + fmax(y, z);
	double tmp;
	if (t_0 <= -0.0001) {
		tmp = 1.0 * (x + (sin(fmin(y, z)) / cos(fmin(y, z))));
	} else if (t_0 <= 0.002) {
		tmp = x + ((fmin(y, z) + (fmax(y, z) * (1.0 + (fmax(y, z) * (fmin(y, z) + (0.3333333333333333 * fmax(y, z))))))) - tan(a));
	} else {
		tmp = 1.0 * (tan(((t_0 / fmax(y, z)) * fmax(y, z))) + x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = fmin(y, z) + fmax(y, z)
    if (t_0 <= (-0.0001d0)) then
        tmp = 1.0d0 * (x + (sin(fmin(y, z)) / cos(fmin(y, z))))
    else if (t_0 <= 0.002d0) then
        tmp = x + ((fmin(y, z) + (fmax(y, z) * (1.0d0 + (fmax(y, z) * (fmin(y, z) + (0.3333333333333333d0 * fmax(y, z))))))) - tan(a))
    else
        tmp = 1.0d0 * (tan(((t_0 / fmax(y, z)) * fmax(y, z))) + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = fmin(y, z) + fmax(y, z);
	double tmp;
	if (t_0 <= -0.0001) {
		tmp = 1.0 * (x + (Math.sin(fmin(y, z)) / Math.cos(fmin(y, z))));
	} else if (t_0 <= 0.002) {
		tmp = x + ((fmin(y, z) + (fmax(y, z) * (1.0 + (fmax(y, z) * (fmin(y, z) + (0.3333333333333333 * fmax(y, z))))))) - Math.tan(a));
	} else {
		tmp = 1.0 * (Math.tan(((t_0 / fmax(y, z)) * fmax(y, z))) + x);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = fmin(y, z) + fmax(y, z)
	tmp = 0
	if t_0 <= -0.0001:
		tmp = 1.0 * (x + (math.sin(fmin(y, z)) / math.cos(fmin(y, z))))
	elif t_0 <= 0.002:
		tmp = x + ((fmin(y, z) + (fmax(y, z) * (1.0 + (fmax(y, z) * (fmin(y, z) + (0.3333333333333333 * fmax(y, z))))))) - math.tan(a))
	else:
		tmp = 1.0 * (math.tan(((t_0 / fmax(y, z)) * fmax(y, z))) + x)
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(fmin(y, z) + fmax(y, z))
	tmp = 0.0
	if (t_0 <= -0.0001)
		tmp = Float64(1.0 * Float64(x + Float64(sin(fmin(y, z)) / cos(fmin(y, z)))));
	elseif (t_0 <= 0.002)
		tmp = Float64(x + Float64(Float64(fmin(y, z) + Float64(fmax(y, z) * Float64(1.0 + Float64(fmax(y, z) * Float64(fmin(y, z) + Float64(0.3333333333333333 * fmax(y, z))))))) - tan(a)));
	else
		tmp = Float64(1.0 * Float64(tan(Float64(Float64(t_0 / fmax(y, z)) * fmax(y, z))) + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = min(y, z) + max(y, z);
	tmp = 0.0;
	if (t_0 <= -0.0001)
		tmp = 1.0 * (x + (sin(min(y, z)) / cos(min(y, z))));
	elseif (t_0 <= 0.002)
		tmp = x + ((min(y, z) + (max(y, z) * (1.0 + (max(y, z) * (min(y, z) + (0.3333333333333333 * max(y, z))))))) - tan(a));
	else
		tmp = 1.0 * (tan(((t_0 / max(y, z)) * max(y, z))) + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Min[y, z], $MachinePrecision] + N[Max[y, z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.0001], N[(1.0 * N[(x + N[(N[Sin[N[Min[y, z], $MachinePrecision]], $MachinePrecision] / N[Cos[N[Min[y, z], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.002], N[(x + N[(N[(N[Min[y, z], $MachinePrecision] + N[(N[Max[y, z], $MachinePrecision] * N[(1.0 + N[(N[Max[y, z], $MachinePrecision] * N[(N[Min[y, z], $MachinePrecision] + N[(0.3333333333333333 * N[Max[y, z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[Tan[N[(N[(t$95$0 / N[Max[y, z], $MachinePrecision]), $MachinePrecision] * N[Max[y, z], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -1e-4

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\tan \left(y + z\right) - \tan a\right)} \]

      2. lift--.f64 N/A

         \[\leadsto x + \color{blue}{\left(\tan \left(y + z\right) - \tan a\right)} \]

      3. associate-+r- N/A

         \[\leadsto \color{blue}{\left(x + \tan \left(y + z\right)\right) - \tan a} \]

      4. sub-to-mult N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right) \cdot \left(x + \tan \left(y + z\right)\right)} \]

      5. lower-unsound-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right) \cdot \left(x + \tan \left(y + z\right)\right)} \]

      6. lower-unsound--.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right)} \cdot \left(x + \tan \left(y + z\right)\right) \]

      7. lower-unsound-/.f64 N/A

         \[\leadsto \left(1 - \color{blue}{\frac{\tan a}{x + \tan \left(y + z\right)}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \left(1 - \frac{\tan a}{\color{blue}{\tan \left(y + z\right) + x}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(1 - \frac{\tan a}{\color{blue}{\tan \left(y + z\right) + x}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(y + z\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(z + y\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(z + y\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \color{blue}{\left(\tan \left(y + z\right) + x\right)} \]

      14. lower-+.f64 79.2%

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \color{blue}{\left(\tan \left(y + z\right) + x\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(y + z\right)} + x\right) \]

      16. +-commutative N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

      17. lower-+.f64 79.2%

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

   3. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \left(z + y\right) + x\right)} \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \cdot \left(\tan \left(z + y\right) + x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 50.3%

      \[\leadsto \color{blue}{1} \cdot \left(\tan \left(z + y\right) + x\right) \]

   7. Taylor expanded in z around 0

      \[\leadsto 1 \cdot \color{blue}{\left(x + \frac{\sin y}{\cos y}\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 \cdot \left(x + \color{blue}{\frac{\sin y}{\cos y}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto 1 \cdot \left(x + \frac{\sin y}{\color{blue}{\cos y}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto 1 \cdot \left(x + \frac{\sin y}{\cos \color{blue}{y}}\right) \]

      4. lower-cos.f64 40.2%

         \[\leadsto 1 \cdot \left(x + \frac{\sin y}{\cos y}\right) \]

   9. Applied rewrites 40.2%

      \[\leadsto 1 \cdot \color{blue}{\left(x + \frac{\sin y}{\cos y}\right)} \]

   if -1e-4 < (+.f64 y z) < 2e-3

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

   2. Taylor expanded in y around 0

      \[\leadsto x + \left(\color{blue}{\left(y \cdot \left(1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}\right) + \frac{\sin z}{\cos z}\right)} - \tan a\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, \color{blue}{1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      2. lower--.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - \color{blue}{-1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \color{blue}{\frac{{\sin z}^{2}}{{\cos z}^{2}}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      4. lower-/.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{\color{blue}{{\cos z}^{2}}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\color{blue}{\cos z}}^{2}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos \color{blue}{z}}^{2}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{\color{blue}{2}}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      8. lower-cos.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      9. lower-/.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      11. lower-cos.f64 50.8%

          \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

   4. Applied rewrites 50.8%

      \[\leadsto x + \left(\color{blue}{\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}, \frac{\sin z}{\cos z}\right)} - \tan a\right) \]

   5. Taylor expanded in z around 0

      \[\leadsto x + \left(\left(y + \color{blue}{z \cdot \left(1 + z \cdot \left(y + \frac{1}{3} \cdot z\right)\right)}\right) - \tan a\right) \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + \left(\left(y + z \cdot \color{blue}{\left(1 + z \cdot \left(y + \frac{1}{3} \cdot z\right)\right)}\right) - \tan a\right) \]

      2. lower-*.f64 N/A

         \[\leadsto x + \left(\left(y + z \cdot \left(1 + \color{blue}{z \cdot \left(y + \frac{1}{3} \cdot z\right)}\right)\right) - \tan a\right) \]

      3. lower-+.f64 N/A

         \[\leadsto x + \left(\left(y + z \cdot \left(1 + z \cdot \color{blue}{\left(y + \frac{1}{3} \cdot z\right)}\right)\right) - \tan a\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x + \left(\left(y + z \cdot \left(1 + z \cdot \left(y + \color{blue}{\frac{1}{3} \cdot z}\right)\right)\right) - \tan a\right) \]

      5. lower-+.f64 N/A

         \[\leadsto x + \left(\left(y + z \cdot \left(1 + z \cdot \left(y + \frac{1}{3} \cdot \color{blue}{z}\right)\right)\right) - \tan a\right) \]

      6. lower-*.f64 26.0%

         \[\leadsto x + \left(\left(y + z \cdot \left(1 + z \cdot \left(y + 0.3333333333333333 \cdot z\right)\right)\right) - \tan a\right) \]

   7. Applied rewrites 26.0%

      \[\leadsto x + \left(\left(y + \color{blue}{z \cdot \left(1 + z \cdot \left(y + 0.3333333333333333 \cdot z\right)\right)}\right) - \tan a\right) \]

   if 2e-3 < (+.f64 y z)

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\tan \left(y + z\right) - \tan a\right)} \]

      2. lift--.f64 N/A

         \[\leadsto x + \color{blue}{\left(\tan \left(y + z\right) - \tan a\right)} \]

      3. associate-+r- N/A

         \[\leadsto \color{blue}{\left(x + \tan \left(y + z\right)\right) - \tan a} \]

      4. sub-to-mult N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right) \cdot \left(x + \tan \left(y + z\right)\right)} \]

      5. lower-unsound-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right) \cdot \left(x + \tan \left(y + z\right)\right)} \]

      6. lower-unsound--.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right)} \cdot \left(x + \tan \left(y + z\right)\right) \]

      7. lower-unsound-/.f64 N/A

         \[\leadsto \left(1 - \color{blue}{\frac{\tan a}{x + \tan \left(y + z\right)}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \left(1 - \frac{\tan a}{\color{blue}{\tan \left(y + z\right) + x}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(1 - \frac{\tan a}{\color{blue}{\tan \left(y + z\right) + x}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(y + z\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(z + y\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(z + y\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \color{blue}{\left(\tan \left(y + z\right) + x\right)} \]

      14. lower-+.f64 79.2%

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \color{blue}{\left(\tan \left(y + z\right) + x\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(y + z\right)} + x\right) \]

      16. +-commutative N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

      17. lower-+.f64 79.2%

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

   3. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \left(z + y\right) + x\right)} \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \cdot \left(\tan \left(z + y\right) + x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 50.3%

      \[\leadsto \color{blue}{1} \cdot \left(\tan \left(z + y\right) + x\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

      2. sum-to-mult N/A

         \[\leadsto 1 \cdot \left(\tan \color{blue}{\left(\left(1 + \frac{y}{z}\right) \cdot z\right)} + x\right) \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \color{blue}{\left(\left(1 + \frac{y}{z}\right) \cdot z\right)} + x\right) \]

      4. lower-unsound-+.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\left(1 + \frac{y}{z}\right)} \cdot z\right) + x\right) \]

      5. lower-unsound-/.f64 42.1%

         \[\leadsto 1 \cdot \left(\tan \left(\left(1 + \color{blue}{\frac{y}{z}}\right) \cdot z\right) + x\right) \]

      6. lower-+.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\left(1 + \frac{y}{z}\right)} \cdot z\right) + x\right) \]

      7. +-commutative N/A

         \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\left(\frac{y}{z} + 1\right)} \cdot z\right) + x\right) \]

      8. sum-to-mult-rev N/A

         \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\left(\left(1 + \frac{1}{\frac{y}{z}}\right) \cdot \frac{y}{z}\right)} \cdot z\right) + x\right) \]

      9. lift-/.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \left(\left(\left(1 + \color{blue}{\frac{1}{\frac{y}{z}}}\right) \cdot \frac{y}{z}\right) \cdot z\right) + x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\left(\color{blue}{\left(1 + \frac{1}{\frac{y}{z}}\right)} \cdot \frac{y}{z}\right) \cdot z\right) + x\right) \]

      11. lift-/.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\left(\left(1 + \frac{1}{\frac{y}{z}}\right) \cdot \color{blue}{\frac{y}{z}}\right) \cdot z\right) + x\right) \]

      12. associate-*r/ N/A

          \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\frac{\left(1 + \frac{1}{\frac{y}{z}}\right) \cdot y}{z}} \cdot z\right) + x\right) \]

      13. lift-+.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\color{blue}{\left(1 + \frac{1}{\frac{y}{z}}\right)} \cdot y}{z} \cdot z\right) + x\right) \]

      14. lift-/.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\left(1 + \color{blue}{\frac{1}{\frac{y}{z}}}\right) \cdot y}{z} \cdot z\right) + x\right) \]

      15. lift-/.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\left(1 + \frac{1}{\color{blue}{\frac{y}{z}}}\right) \cdot y}{z} \cdot z\right) + x\right) \]

      16. div-flip-rev N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\left(1 + \color{blue}{\frac{z}{y}}\right) \cdot y}{z} \cdot z\right) + x\right) \]

      17. sum-to-mult-rev N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\color{blue}{y + z}}{z} \cdot z\right) + x\right) \]

      18. +-commutative N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\color{blue}{z + y}}{z} \cdot z\right) + x\right) \]

      19. lift-+.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\color{blue}{z + y}}{z} \cdot z\right) + x\right) \]

      20. lower-/.f64 42.2%

          \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\frac{z + y}{z}} \cdot z\right) + x\right) \]

      21. lift-+.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\color{blue}{z + y}}{z} \cdot z\right) + x\right) \]

      22. +-commutative N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\color{blue}{y + z}}{z} \cdot z\right) + x\right) \]

      23. lower-+.f64 42.2%

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\color{blue}{y + z}}{z} \cdot z\right) + x\right) \]

   8. Applied rewrites 42.2%

      \[\leadsto 1 \cdot \left(\tan \color{blue}{\left(\frac{y + z}{z} \cdot z\right)} + x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 59.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(y, z\right) + \mathsf{max}\left(y, z\right)\\ \mathbf{if}\;t\_0 \leq -0.0001:\\ \;\;\;\;1 \cdot \left(\tan \left(\left(\frac{\mathsf{max}\left(y, z\right)}{\mathsf{min}\left(y, z\right)} - -1\right) \cdot \mathsf{min}\left(y, z\right)\right) + x\right)\\ \mathbf{elif}\;t\_0 \leq 0.002:\\ \;\;\;\;x + \left(\left(\mathsf{min}\left(y, z\right) + \mathsf{max}\left(y, z\right) \cdot \left(1 + \mathsf{max}\left(y, z\right) \cdot \left(\mathsf{min}\left(y, z\right) + 0.3333333333333333 \cdot \mathsf{max}\left(y, z\right)\right)\right)\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\tan \left(\frac{t\_0}{\mathsf{max}\left(y, z\right)} \cdot \mathsf{max}\left(y, z\right)\right) + x\right)\\ \end{array} \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (let* ((t_0 (+ (fmin y z) (fmax y z))))
  (if (<= t_0 -0.0001)
    (*
     1.0
     (+ (tan (* (- (/ (fmax y z) (fmin y z)) -1.0) (fmin y z))) x))
    (if (<= t_0 0.002)
      (+
       x
       (-
        (+
         (fmin y z)
         (*
          (fmax y z)
          (+
           1.0
           (*
            (fmax y z)
            (+ (fmin y z) (* 0.3333333333333333 (fmax y z)))))))
        (tan a)))
      (* 1.0 (+ (tan (* (/ t_0 (fmax y z)) (fmax y z))) x))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = fmin(y, z) + fmax(y, z);
	double tmp;
	if (t_0 <= -0.0001) {
		tmp = 1.0 * (tan((((fmax(y, z) / fmin(y, z)) - -1.0) * fmin(y, z))) + x);
	} else if (t_0 <= 0.002) {
		tmp = x + ((fmin(y, z) + (fmax(y, z) * (1.0 + (fmax(y, z) * (fmin(y, z) + (0.3333333333333333 * fmax(y, z))))))) - tan(a));
	} else {
		tmp = 1.0 * (tan(((t_0 / fmax(y, z)) * fmax(y, z))) + x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = fmin(y, z) + fmax(y, z)
    if (t_0 <= (-0.0001d0)) then
        tmp = 1.0d0 * (tan((((fmax(y, z) / fmin(y, z)) - (-1.0d0)) * fmin(y, z))) + x)
    else if (t_0 <= 0.002d0) then
        tmp = x + ((fmin(y, z) + (fmax(y, z) * (1.0d0 + (fmax(y, z) * (fmin(y, z) + (0.3333333333333333d0 * fmax(y, z))))))) - tan(a))
    else
        tmp = 1.0d0 * (tan(((t_0 / fmax(y, z)) * fmax(y, z))) + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = fmin(y, z) + fmax(y, z);
	double tmp;
	if (t_0 <= -0.0001) {
		tmp = 1.0 * (Math.tan((((fmax(y, z) / fmin(y, z)) - -1.0) * fmin(y, z))) + x);
	} else if (t_0 <= 0.002) {
		tmp = x + ((fmin(y, z) + (fmax(y, z) * (1.0 + (fmax(y, z) * (fmin(y, z) + (0.3333333333333333 * fmax(y, z))))))) - Math.tan(a));
	} else {
		tmp = 1.0 * (Math.tan(((t_0 / fmax(y, z)) * fmax(y, z))) + x);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = fmin(y, z) + fmax(y, z)
	tmp = 0
	if t_0 <= -0.0001:
		tmp = 1.0 * (math.tan((((fmax(y, z) / fmin(y, z)) - -1.0) * fmin(y, z))) + x)
	elif t_0 <= 0.002:
		tmp = x + ((fmin(y, z) + (fmax(y, z) * (1.0 + (fmax(y, z) * (fmin(y, z) + (0.3333333333333333 * fmax(y, z))))))) - math.tan(a))
	else:
		tmp = 1.0 * (math.tan(((t_0 / fmax(y, z)) * fmax(y, z))) + x)
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(fmin(y, z) + fmax(y, z))
	tmp = 0.0
	if (t_0 <= -0.0001)
		tmp = Float64(1.0 * Float64(tan(Float64(Float64(Float64(fmax(y, z) / fmin(y, z)) - -1.0) * fmin(y, z))) + x));
	elseif (t_0 <= 0.002)
		tmp = Float64(x + Float64(Float64(fmin(y, z) + Float64(fmax(y, z) * Float64(1.0 + Float64(fmax(y, z) * Float64(fmin(y, z) + Float64(0.3333333333333333 * fmax(y, z))))))) - tan(a)));
	else
		tmp = Float64(1.0 * Float64(tan(Float64(Float64(t_0 / fmax(y, z)) * fmax(y, z))) + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = min(y, z) + max(y, z);
	tmp = 0.0;
	if (t_0 <= -0.0001)
		tmp = 1.0 * (tan((((max(y, z) / min(y, z)) - -1.0) * min(y, z))) + x);
	elseif (t_0 <= 0.002)
		tmp = x + ((min(y, z) + (max(y, z) * (1.0 + (max(y, z) * (min(y, z) + (0.3333333333333333 * max(y, z))))))) - tan(a));
	else
		tmp = 1.0 * (tan(((t_0 / max(y, z)) * max(y, z))) + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Min[y, z], $MachinePrecision] + N[Max[y, z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.0001], N[(1.0 * N[(N[Tan[N[(N[(N[(N[Max[y, z], $MachinePrecision] / N[Min[y, z], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] * N[Min[y, z], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.002], N[(x + N[(N[(N[Min[y, z], $MachinePrecision] + N[(N[Max[y, z], $MachinePrecision] * N[(1.0 + N[(N[Max[y, z], $MachinePrecision] * N[(N[Min[y, z], $MachinePrecision] + N[(0.3333333333333333 * N[Max[y, z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[Tan[N[(N[(t$95$0 / N[Max[y, z], $MachinePrecision]), $MachinePrecision] * N[Max[y, z], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -1e-4

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\tan \left(y + z\right) - \tan a\right)} \]

      2. lift--.f64 N/A

         \[\leadsto x + \color{blue}{\left(\tan \left(y + z\right) - \tan a\right)} \]

      3. associate-+r- N/A

         \[\leadsto \color{blue}{\left(x + \tan \left(y + z\right)\right) - \tan a} \]

      4. sub-to-mult N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right) \cdot \left(x + \tan \left(y + z\right)\right)} \]

      5. lower-unsound-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right) \cdot \left(x + \tan \left(y + z\right)\right)} \]

      6. lower-unsound--.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right)} \cdot \left(x + \tan \left(y + z\right)\right) \]

      7. lower-unsound-/.f64 N/A

         \[\leadsto \left(1 - \color{blue}{\frac{\tan a}{x + \tan \left(y + z\right)}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \left(1 - \frac{\tan a}{\color{blue}{\tan \left(y + z\right) + x}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(1 - \frac{\tan a}{\color{blue}{\tan \left(y + z\right) + x}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(y + z\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(z + y\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(z + y\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \color{blue}{\left(\tan \left(y + z\right) + x\right)} \]

      14. lower-+.f64 79.2%

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \color{blue}{\left(\tan \left(y + z\right) + x\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(y + z\right)} + x\right) \]

      16. +-commutative N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

      17. lower-+.f64 79.2%

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

   3. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \left(z + y\right) + x\right)} \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \cdot \left(\tan \left(z + y\right) + x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 50.3%

      \[\leadsto \color{blue}{1} \cdot \left(\tan \left(z + y\right) + x\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

      2. +-commutative N/A

         \[\leadsto 1 \cdot \left(\tan \color{blue}{\left(y + z\right)} + x\right) \]

      3. sum-to-mult-rev N/A

         \[\leadsto 1 \cdot \left(\tan \color{blue}{\left(\left(1 + \frac{z}{y}\right) \cdot y\right)} + x\right) \]

      4. div-flip-rev N/A

         \[\leadsto 1 \cdot \left(\tan \left(\left(1 + \color{blue}{\frac{1}{\frac{y}{z}}}\right) \cdot y\right) + x\right) \]

      5. lift-/.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \left(\left(1 + \frac{1}{\color{blue}{\frac{y}{z}}}\right) \cdot y\right) + x\right) \]

      6. lift-/.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \left(\left(1 + \color{blue}{\frac{1}{\frac{y}{z}}}\right) \cdot y\right) + x\right) \]

      7. lift-+.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\left(1 + \frac{1}{\frac{y}{z}}\right)} \cdot y\right) + x\right) \]

      8. lower-*.f64 41.3%

         \[\leadsto 1 \cdot \left(\tan \color{blue}{\left(\left(1 + \frac{1}{\frac{y}{z}}\right) \cdot y\right)} + x\right) \]

      9. lift-+.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\left(1 + \frac{1}{\frac{y}{z}}\right)} \cdot y\right) + x\right) \]

      10. +-commutative N/A

          \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\left(\frac{1}{\frac{y}{z}} + 1\right)} \cdot y\right) + x\right) \]

      11. add-flip N/A

          \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\left(\frac{1}{\frac{y}{z}} - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot y\right) + x\right) \]

      12. metadata-eval N/A

          \[\leadsto 1 \cdot \left(\tan \left(\left(\frac{1}{\frac{y}{z}} - \color{blue}{-1}\right) \cdot y\right) + x\right) \]

      13. lower--.f64 41.3%

          \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\left(\frac{1}{\frac{y}{z}} - -1\right)} \cdot y\right) + x\right) \]

      14. lift-/.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\left(\color{blue}{\frac{1}{\frac{y}{z}}} - -1\right) \cdot y\right) + x\right) \]

      15. lift-/.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\left(\frac{1}{\color{blue}{\frac{y}{z}}} - -1\right) \cdot y\right) + x\right) \]

      16. div-flip-rev N/A

          \[\leadsto 1 \cdot \left(\tan \left(\left(\color{blue}{\frac{z}{y}} - -1\right) \cdot y\right) + x\right) \]

      17. lower-/.f64 42.3%

          \[\leadsto 1 \cdot \left(\tan \left(\left(\color{blue}{\frac{z}{y}} - -1\right) \cdot y\right) + x\right) \]

   8. Applied rewrites 42.3%

      \[\leadsto 1 \cdot \left(\tan \color{blue}{\left(\left(\frac{z}{y} - -1\right) \cdot y\right)} + x\right) \]

   if -1e-4 < (+.f64 y z) < 2e-3

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

   2. Taylor expanded in y around 0

      \[\leadsto x + \left(\color{blue}{\left(y \cdot \left(1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}\right) + \frac{\sin z}{\cos z}\right)} - \tan a\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, \color{blue}{1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      2. lower--.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - \color{blue}{-1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \color{blue}{\frac{{\sin z}^{2}}{{\cos z}^{2}}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      4. lower-/.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{\color{blue}{{\cos z}^{2}}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\color{blue}{\cos z}}^{2}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos \color{blue}{z}}^{2}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{\color{blue}{2}}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      8. lower-cos.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      9. lower-/.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      11. lower-cos.f64 50.8%

          \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

   4. Applied rewrites 50.8%

      \[\leadsto x + \left(\color{blue}{\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}, \frac{\sin z}{\cos z}\right)} - \tan a\right) \]

   5. Taylor expanded in z around 0

      \[\leadsto x + \left(\left(y + \color{blue}{z \cdot \left(1 + z \cdot \left(y + \frac{1}{3} \cdot z\right)\right)}\right) - \tan a\right) \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + \left(\left(y + z \cdot \color{blue}{\left(1 + z \cdot \left(y + \frac{1}{3} \cdot z\right)\right)}\right) - \tan a\right) \]

      2. lower-*.f64 N/A

         \[\leadsto x + \left(\left(y + z \cdot \left(1 + \color{blue}{z \cdot \left(y + \frac{1}{3} \cdot z\right)}\right)\right) - \tan a\right) \]

      3. lower-+.f64 N/A

         \[\leadsto x + \left(\left(y + z \cdot \left(1 + z \cdot \color{blue}{\left(y + \frac{1}{3} \cdot z\right)}\right)\right) - \tan a\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x + \left(\left(y + z \cdot \left(1 + z \cdot \left(y + \color{blue}{\frac{1}{3} \cdot z}\right)\right)\right) - \tan a\right) \]

      5. lower-+.f64 N/A

         \[\leadsto x + \left(\left(y + z \cdot \left(1 + z \cdot \left(y + \frac{1}{3} \cdot \color{blue}{z}\right)\right)\right) - \tan a\right) \]

      6. lower-*.f64 26.0%

         \[\leadsto x + \left(\left(y + z \cdot \left(1 + z \cdot \left(y + 0.3333333333333333 \cdot z\right)\right)\right) - \tan a\right) \]

   7. Applied rewrites 26.0%

      \[\leadsto x + \left(\left(y + \color{blue}{z \cdot \left(1 + z \cdot \left(y + 0.3333333333333333 \cdot z\right)\right)}\right) - \tan a\right) \]

   if 2e-3 < (+.f64 y z)

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\tan \left(y + z\right) - \tan a\right)} \]

      2. lift--.f64 N/A

         \[\leadsto x + \color{blue}{\left(\tan \left(y + z\right) - \tan a\right)} \]

      3. associate-+r- N/A

         \[\leadsto \color{blue}{\left(x + \tan \left(y + z\right)\right) - \tan a} \]

      4. sub-to-mult N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right) \cdot \left(x + \tan \left(y + z\right)\right)} \]

      5. lower-unsound-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right) \cdot \left(x + \tan \left(y + z\right)\right)} \]

      6. lower-unsound--.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right)} \cdot \left(x + \tan \left(y + z\right)\right) \]

      7. lower-unsound-/.f64 N/A

         \[\leadsto \left(1 - \color{blue}{\frac{\tan a}{x + \tan \left(y + z\right)}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \left(1 - \frac{\tan a}{\color{blue}{\tan \left(y + z\right) + x}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(1 - \frac{\tan a}{\color{blue}{\tan \left(y + z\right) + x}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(y + z\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(z + y\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(z + y\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \color{blue}{\left(\tan \left(y + z\right) + x\right)} \]

      14. lower-+.f64 79.2%

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \color{blue}{\left(\tan \left(y + z\right) + x\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(y + z\right)} + x\right) \]

      16. +-commutative N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

      17. lower-+.f64 79.2%

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

   3. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \left(z + y\right) + x\right)} \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \cdot \left(\tan \left(z + y\right) + x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 50.3%

      \[\leadsto \color{blue}{1} \cdot \left(\tan \left(z + y\right) + x\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

      2. sum-to-mult N/A

         \[\leadsto 1 \cdot \left(\tan \color{blue}{\left(\left(1 + \frac{y}{z}\right) \cdot z\right)} + x\right) \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \color{blue}{\left(\left(1 + \frac{y}{z}\right) \cdot z\right)} + x\right) \]

      4. lower-unsound-+.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\left(1 + \frac{y}{z}\right)} \cdot z\right) + x\right) \]

      5. lower-unsound-/.f64 42.1%

         \[\leadsto 1 \cdot \left(\tan \left(\left(1 + \color{blue}{\frac{y}{z}}\right) \cdot z\right) + x\right) \]

      6. lower-+.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\left(1 + \frac{y}{z}\right)} \cdot z\right) + x\right) \]

      7. +-commutative N/A

         \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\left(\frac{y}{z} + 1\right)} \cdot z\right) + x\right) \]

      8. sum-to-mult-rev N/A

         \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\left(\left(1 + \frac{1}{\frac{y}{z}}\right) \cdot \frac{y}{z}\right)} \cdot z\right) + x\right) \]

      9. lift-/.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \left(\left(\left(1 + \color{blue}{\frac{1}{\frac{y}{z}}}\right) \cdot \frac{y}{z}\right) \cdot z\right) + x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\left(\color{blue}{\left(1 + \frac{1}{\frac{y}{z}}\right)} \cdot \frac{y}{z}\right) \cdot z\right) + x\right) \]

      11. lift-/.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\left(\left(1 + \frac{1}{\frac{y}{z}}\right) \cdot \color{blue}{\frac{y}{z}}\right) \cdot z\right) + x\right) \]

      12. associate-*r/ N/A

          \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\frac{\left(1 + \frac{1}{\frac{y}{z}}\right) \cdot y}{z}} \cdot z\right) + x\right) \]

      13. lift-+.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\color{blue}{\left(1 + \frac{1}{\frac{y}{z}}\right)} \cdot y}{z} \cdot z\right) + x\right) \]

      14. lift-/.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\left(1 + \color{blue}{\frac{1}{\frac{y}{z}}}\right) \cdot y}{z} \cdot z\right) + x\right) \]

      15. lift-/.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\left(1 + \frac{1}{\color{blue}{\frac{y}{z}}}\right) \cdot y}{z} \cdot z\right) + x\right) \]

      16. div-flip-rev N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\left(1 + \color{blue}{\frac{z}{y}}\right) \cdot y}{z} \cdot z\right) + x\right) \]

      17. sum-to-mult-rev N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\color{blue}{y + z}}{z} \cdot z\right) + x\right) \]

      18. +-commutative N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\color{blue}{z + y}}{z} \cdot z\right) + x\right) \]

      19. lift-+.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\color{blue}{z + y}}{z} \cdot z\right) + x\right) \]

      20. lower-/.f64 42.2%

          \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\frac{z + y}{z}} \cdot z\right) + x\right) \]

      21. lift-+.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\color{blue}{z + y}}{z} \cdot z\right) + x\right) \]

      22. +-commutative N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\color{blue}{y + z}}{z} \cdot z\right) + x\right) \]

      23. lower-+.f64 42.2%

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\color{blue}{y + z}}{z} \cdot z\right) + x\right) \]

   8. Applied rewrites 42.2%

      \[\leadsto 1 \cdot \left(\tan \color{blue}{\left(\frac{y + z}{z} \cdot z\right)} + x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 59.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(y, z\right) + \mathsf{max}\left(y, z\right)\\ \mathbf{if}\;t\_0 \leq -0.0001:\\ \;\;\;\;1 \cdot \left(\tan \left(\left(\frac{\mathsf{max}\left(y, z\right)}{\mathsf{min}\left(y, z\right)} - -1\right) \cdot \mathsf{min}\left(y, z\right)\right) + x\right)\\ \mathbf{elif}\;t\_0 \leq 0.002:\\ \;\;\;\;x + \left(\left(\mathsf{min}\left(y, z\right) + \mathsf{max}\left(y, z\right) \cdot \left(1 + \mathsf{min}\left(y, z\right) \cdot \mathsf{max}\left(y, z\right)\right)\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\tan \left(\frac{t\_0}{\mathsf{max}\left(y, z\right)} \cdot \mathsf{max}\left(y, z\right)\right) + x\right)\\ \end{array} \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (let* ((t_0 (+ (fmin y z) (fmax y z))))
  (if (<= t_0 -0.0001)
    (*
     1.0
     (+ (tan (* (- (/ (fmax y z) (fmin y z)) -1.0) (fmin y z))) x))
    (if (<= t_0 0.002)
      (+
       x
       (-
        (+
         (fmin y z)
         (* (fmax y z) (+ 1.0 (* (fmin y z) (fmax y z)))))
        (tan a)))
      (* 1.0 (+ (tan (* (/ t_0 (fmax y z)) (fmax y z))) x))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = fmin(y, z) + fmax(y, z);
	double tmp;
	if (t_0 <= -0.0001) {
		tmp = 1.0 * (tan((((fmax(y, z) / fmin(y, z)) - -1.0) * fmin(y, z))) + x);
	} else if (t_0 <= 0.002) {
		tmp = x + ((fmin(y, z) + (fmax(y, z) * (1.0 + (fmin(y, z) * fmax(y, z))))) - tan(a));
	} else {
		tmp = 1.0 * (tan(((t_0 / fmax(y, z)) * fmax(y, z))) + x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = fmin(y, z) + fmax(y, z)
    if (t_0 <= (-0.0001d0)) then
        tmp = 1.0d0 * (tan((((fmax(y, z) / fmin(y, z)) - (-1.0d0)) * fmin(y, z))) + x)
    else if (t_0 <= 0.002d0) then
        tmp = x + ((fmin(y, z) + (fmax(y, z) * (1.0d0 + (fmin(y, z) * fmax(y, z))))) - tan(a))
    else
        tmp = 1.0d0 * (tan(((t_0 / fmax(y, z)) * fmax(y, z))) + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = fmin(y, z) + fmax(y, z);
	double tmp;
	if (t_0 <= -0.0001) {
		tmp = 1.0 * (Math.tan((((fmax(y, z) / fmin(y, z)) - -1.0) * fmin(y, z))) + x);
	} else if (t_0 <= 0.002) {
		tmp = x + ((fmin(y, z) + (fmax(y, z) * (1.0 + (fmin(y, z) * fmax(y, z))))) - Math.tan(a));
	} else {
		tmp = 1.0 * (Math.tan(((t_0 / fmax(y, z)) * fmax(y, z))) + x);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = fmin(y, z) + fmax(y, z)
	tmp = 0
	if t_0 <= -0.0001:
		tmp = 1.0 * (math.tan((((fmax(y, z) / fmin(y, z)) - -1.0) * fmin(y, z))) + x)
	elif t_0 <= 0.002:
		tmp = x + ((fmin(y, z) + (fmax(y, z) * (1.0 + (fmin(y, z) * fmax(y, z))))) - math.tan(a))
	else:
		tmp = 1.0 * (math.tan(((t_0 / fmax(y, z)) * fmax(y, z))) + x)
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(fmin(y, z) + fmax(y, z))
	tmp = 0.0
	if (t_0 <= -0.0001)
		tmp = Float64(1.0 * Float64(tan(Float64(Float64(Float64(fmax(y, z) / fmin(y, z)) - -1.0) * fmin(y, z))) + x));
	elseif (t_0 <= 0.002)
		tmp = Float64(x + Float64(Float64(fmin(y, z) + Float64(fmax(y, z) * Float64(1.0 + Float64(fmin(y, z) * fmax(y, z))))) - tan(a)));
	else
		tmp = Float64(1.0 * Float64(tan(Float64(Float64(t_0 / fmax(y, z)) * fmax(y, z))) + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = min(y, z) + max(y, z);
	tmp = 0.0;
	if (t_0 <= -0.0001)
		tmp = 1.0 * (tan((((max(y, z) / min(y, z)) - -1.0) * min(y, z))) + x);
	elseif (t_0 <= 0.002)
		tmp = x + ((min(y, z) + (max(y, z) * (1.0 + (min(y, z) * max(y, z))))) - tan(a));
	else
		tmp = 1.0 * (tan(((t_0 / max(y, z)) * max(y, z))) + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Min[y, z], $MachinePrecision] + N[Max[y, z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.0001], N[(1.0 * N[(N[Tan[N[(N[(N[(N[Max[y, z], $MachinePrecision] / N[Min[y, z], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] * N[Min[y, z], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.002], N[(x + N[(N[(N[Min[y, z], $MachinePrecision] + N[(N[Max[y, z], $MachinePrecision] * N[(1.0 + N[(N[Min[y, z], $MachinePrecision] * N[Max[y, z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[Tan[N[(N[(t$95$0 / N[Max[y, z], $MachinePrecision]), $MachinePrecision] * N[Max[y, z], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -1e-4

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\tan \left(y + z\right) - \tan a\right)} \]

      2. lift--.f64 N/A

         \[\leadsto x + \color{blue}{\left(\tan \left(y + z\right) - \tan a\right)} \]

      3. associate-+r- N/A

         \[\leadsto \color{blue}{\left(x + \tan \left(y + z\right)\right) - \tan a} \]

      4. sub-to-mult N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right) \cdot \left(x + \tan \left(y + z\right)\right)} \]

      5. lower-unsound-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right) \cdot \left(x + \tan \left(y + z\right)\right)} \]

      6. lower-unsound--.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right)} \cdot \left(x + \tan \left(y + z\right)\right) \]

      7. lower-unsound-/.f64 N/A

         \[\leadsto \left(1 - \color{blue}{\frac{\tan a}{x + \tan \left(y + z\right)}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \left(1 - \frac{\tan a}{\color{blue}{\tan \left(y + z\right) + x}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(1 - \frac{\tan a}{\color{blue}{\tan \left(y + z\right) + x}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(y + z\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(z + y\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(z + y\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \color{blue}{\left(\tan \left(y + z\right) + x\right)} \]

      14. lower-+.f64 79.2%

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \color{blue}{\left(\tan \left(y + z\right) + x\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(y + z\right)} + x\right) \]

      16. +-commutative N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

      17. lower-+.f64 79.2%

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

   3. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \left(z + y\right) + x\right)} \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \cdot \left(\tan \left(z + y\right) + x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 50.3%

      \[\leadsto \color{blue}{1} \cdot \left(\tan \left(z + y\right) + x\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

      2. +-commutative N/A

         \[\leadsto 1 \cdot \left(\tan \color{blue}{\left(y + z\right)} + x\right) \]

      3. sum-to-mult-rev N/A

         \[\leadsto 1 \cdot \left(\tan \color{blue}{\left(\left(1 + \frac{z}{y}\right) \cdot y\right)} + x\right) \]

      4. div-flip-rev N/A

         \[\leadsto 1 \cdot \left(\tan \left(\left(1 + \color{blue}{\frac{1}{\frac{y}{z}}}\right) \cdot y\right) + x\right) \]

      5. lift-/.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \left(\left(1 + \frac{1}{\color{blue}{\frac{y}{z}}}\right) \cdot y\right) + x\right) \]

      6. lift-/.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \left(\left(1 + \color{blue}{\frac{1}{\frac{y}{z}}}\right) \cdot y\right) + x\right) \]

      7. lift-+.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\left(1 + \frac{1}{\frac{y}{z}}\right)} \cdot y\right) + x\right) \]

      8. lower-*.f64 41.3%

         \[\leadsto 1 \cdot \left(\tan \color{blue}{\left(\left(1 + \frac{1}{\frac{y}{z}}\right) \cdot y\right)} + x\right) \]

      9. lift-+.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\left(1 + \frac{1}{\frac{y}{z}}\right)} \cdot y\right) + x\right) \]

      10. +-commutative N/A

          \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\left(\frac{1}{\frac{y}{z}} + 1\right)} \cdot y\right) + x\right) \]

      11. add-flip N/A

          \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\left(\frac{1}{\frac{y}{z}} - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot y\right) + x\right) \]

      12. metadata-eval N/A

          \[\leadsto 1 \cdot \left(\tan \left(\left(\frac{1}{\frac{y}{z}} - \color{blue}{-1}\right) \cdot y\right) + x\right) \]

      13. lower--.f64 41.3%

          \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\left(\frac{1}{\frac{y}{z}} - -1\right)} \cdot y\right) + x\right) \]

      14. lift-/.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\left(\color{blue}{\frac{1}{\frac{y}{z}}} - -1\right) \cdot y\right) + x\right) \]

      15. lift-/.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\left(\frac{1}{\color{blue}{\frac{y}{z}}} - -1\right) \cdot y\right) + x\right) \]

      16. div-flip-rev N/A

          \[\leadsto 1 \cdot \left(\tan \left(\left(\color{blue}{\frac{z}{y}} - -1\right) \cdot y\right) + x\right) \]

      17. lower-/.f64 42.3%

          \[\leadsto 1 \cdot \left(\tan \left(\left(\color{blue}{\frac{z}{y}} - -1\right) \cdot y\right) + x\right) \]

   8. Applied rewrites 42.3%

      \[\leadsto 1 \cdot \left(\tan \color{blue}{\left(\left(\frac{z}{y} - -1\right) \cdot y\right)} + x\right) \]

   if -1e-4 < (+.f64 y z) < 2e-3

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

   2. Taylor expanded in y around 0

      \[\leadsto x + \left(\color{blue}{\left(y \cdot \left(1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}\right) + \frac{\sin z}{\cos z}\right)} - \tan a\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, \color{blue}{1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      2. lower--.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - \color{blue}{-1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \color{blue}{\frac{{\sin z}^{2}}{{\cos z}^{2}}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      4. lower-/.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{\color{blue}{{\cos z}^{2}}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\color{blue}{\cos z}}^{2}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos \color{blue}{z}}^{2}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{\color{blue}{2}}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      8. lower-cos.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      9. lower-/.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

      11. lower-cos.f64 50.8%

          \[\leadsto x + \left(\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}, \frac{\sin z}{\cos z}\right) - \tan a\right) \]

   4. Applied rewrites 50.8%

      \[\leadsto x + \left(\color{blue}{\mathsf{fma}\left(y, 1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}, \frac{\sin z}{\cos z}\right)} - \tan a\right) \]

   5. Taylor expanded in z around 0

      \[\leadsto x + \left(\left(y + \color{blue}{z \cdot \left(1 + y \cdot z\right)}\right) - \tan a\right) \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + \left(\left(y + z \cdot \color{blue}{\left(1 + y \cdot z\right)}\right) - \tan a\right) \]

      2. lower-*.f64 N/A

         \[\leadsto x + \left(\left(y + z \cdot \left(1 + \color{blue}{y \cdot z}\right)\right) - \tan a\right) \]

      3. lower-+.f64 N/A

         \[\leadsto x + \left(\left(y + z \cdot \left(1 + y \cdot \color{blue}{z}\right)\right) - \tan a\right) \]

      4. lower-*.f64 26.1%

         \[\leadsto x + \left(\left(y + z \cdot \left(1 + y \cdot z\right)\right) - \tan a\right) \]

   7. Applied rewrites 26.1%

      \[\leadsto x + \left(\left(y + \color{blue}{z \cdot \left(1 + y \cdot z\right)}\right) - \tan a\right) \]

   if 2e-3 < (+.f64 y z)

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\tan \left(y + z\right) - \tan a\right)} \]

      2. lift--.f64 N/A

         \[\leadsto x + \color{blue}{\left(\tan \left(y + z\right) - \tan a\right)} \]

      3. associate-+r- N/A

         \[\leadsto \color{blue}{\left(x + \tan \left(y + z\right)\right) - \tan a} \]

      4. sub-to-mult N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right) \cdot \left(x + \tan \left(y + z\right)\right)} \]

      5. lower-unsound-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right) \cdot \left(x + \tan \left(y + z\right)\right)} \]

      6. lower-unsound--.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right)} \cdot \left(x + \tan \left(y + z\right)\right) \]

      7. lower-unsound-/.f64 N/A

         \[\leadsto \left(1 - \color{blue}{\frac{\tan a}{x + \tan \left(y + z\right)}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \left(1 - \frac{\tan a}{\color{blue}{\tan \left(y + z\right) + x}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(1 - \frac{\tan a}{\color{blue}{\tan \left(y + z\right) + x}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(y + z\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(z + y\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(z + y\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \color{blue}{\left(\tan \left(y + z\right) + x\right)} \]

      14. lower-+.f64 79.2%

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \color{blue}{\left(\tan \left(y + z\right) + x\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(y + z\right)} + x\right) \]

      16. +-commutative N/A

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

      17. lower-+.f64 79.2%

          \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

   3. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \left(z + y\right) + x\right)} \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \cdot \left(\tan \left(z + y\right) + x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 50.3%

      \[\leadsto \color{blue}{1} \cdot \left(\tan \left(z + y\right) + x\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

      2. sum-to-mult N/A

         \[\leadsto 1 \cdot \left(\tan \color{blue}{\left(\left(1 + \frac{y}{z}\right) \cdot z\right)} + x\right) \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \color{blue}{\left(\left(1 + \frac{y}{z}\right) \cdot z\right)} + x\right) \]

      4. lower-unsound-+.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\left(1 + \frac{y}{z}\right)} \cdot z\right) + x\right) \]

      5. lower-unsound-/.f64 42.1%

         \[\leadsto 1 \cdot \left(\tan \left(\left(1 + \color{blue}{\frac{y}{z}}\right) \cdot z\right) + x\right) \]

      6. lower-+.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\left(1 + \frac{y}{z}\right)} \cdot z\right) + x\right) \]

      7. +-commutative N/A

         \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\left(\frac{y}{z} + 1\right)} \cdot z\right) + x\right) \]

      8. sum-to-mult-rev N/A

         \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\left(\left(1 + \frac{1}{\frac{y}{z}}\right) \cdot \frac{y}{z}\right)} \cdot z\right) + x\right) \]

      9. lift-/.f64 N/A

         \[\leadsto 1 \cdot \left(\tan \left(\left(\left(1 + \color{blue}{\frac{1}{\frac{y}{z}}}\right) \cdot \frac{y}{z}\right) \cdot z\right) + x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\left(\color{blue}{\left(1 + \frac{1}{\frac{y}{z}}\right)} \cdot \frac{y}{z}\right) \cdot z\right) + x\right) \]

      11. lift-/.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\left(\left(1 + \frac{1}{\frac{y}{z}}\right) \cdot \color{blue}{\frac{y}{z}}\right) \cdot z\right) + x\right) \]

      12. associate-*r/ N/A

          \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\frac{\left(1 + \frac{1}{\frac{y}{z}}\right) \cdot y}{z}} \cdot z\right) + x\right) \]

      13. lift-+.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\color{blue}{\left(1 + \frac{1}{\frac{y}{z}}\right)} \cdot y}{z} \cdot z\right) + x\right) \]

      14. lift-/.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\left(1 + \color{blue}{\frac{1}{\frac{y}{z}}}\right) \cdot y}{z} \cdot z\right) + x\right) \]

      15. lift-/.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\left(1 + \frac{1}{\color{blue}{\frac{y}{z}}}\right) \cdot y}{z} \cdot z\right) + x\right) \]

      16. div-flip-rev N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\left(1 + \color{blue}{\frac{z}{y}}\right) \cdot y}{z} \cdot z\right) + x\right) \]

      17. sum-to-mult-rev N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\color{blue}{y + z}}{z} \cdot z\right) + x\right) \]

      18. +-commutative N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\color{blue}{z + y}}{z} \cdot z\right) + x\right) \]

      19. lift-+.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\color{blue}{z + y}}{z} \cdot z\right) + x\right) \]

      20. lower-/.f64 42.2%

          \[\leadsto 1 \cdot \left(\tan \left(\color{blue}{\frac{z + y}{z}} \cdot z\right) + x\right) \]

      21. lift-+.f64 N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\color{blue}{z + y}}{z} \cdot z\right) + x\right) \]

      22. +-commutative N/A

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\color{blue}{y + z}}{z} \cdot z\right) + x\right) \]

      23. lower-+.f64 42.2%

          \[\leadsto 1 \cdot \left(\tan \left(\frac{\color{blue}{y + z}}{z} \cdot z\right) + x\right) \]

   8. Applied rewrites 42.2%

      \[\leadsto 1 \cdot \left(\tan \color{blue}{\left(\frac{y + z}{z} \cdot z\right)} + x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 50.3% accurate, 1.8× speedup

Math

\[1 \cdot \left(\tan \left(z + y\right) + x\right) \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (* 1.0 (+ (tan (+ z y)) x)))

C

double code(double x, double y, double z, double a) {
	return 1.0 * (tan((z + y)) + x);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = 1.0d0 * (tan((z + y)) + x)
end function

Java

public static double code(double x, double y, double z, double a) {
	return 1.0 * (Math.tan((z + y)) + x);
}

Python

def code(x, y, z, a):
	return 1.0 * (math.tan((z + y)) + x)

Julia

function code(x, y, z, a)
	return Float64(1.0 * Float64(tan(Float64(z + y)) + x))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = 1.0 * (tan((z + y)) + x);
end

Wolfram

code[x_, y_, z_, a_] := N[(1.0 * N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \left(\tan \left(y + z\right) - \tan a\right)} \]

   2. lift--.f64 N/A

      \[\leadsto x + \color{blue}{\left(\tan \left(y + z\right) - \tan a\right)} \]

   3. associate-+r- N/A

      \[\leadsto \color{blue}{\left(x + \tan \left(y + z\right)\right) - \tan a} \]

   4. sub-to-mult N/A

      \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right) \cdot \left(x + \tan \left(y + z\right)\right)} \]

   5. lower-unsound-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right) \cdot \left(x + \tan \left(y + z\right)\right)} \]

   6. lower-unsound--.f64 N/A

      \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right)} \cdot \left(x + \tan \left(y + z\right)\right) \]

   7. lower-unsound-/.f64 N/A

      \[\leadsto \left(1 - \color{blue}{\frac{\tan a}{x + \tan \left(y + z\right)}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \left(1 - \frac{\tan a}{\color{blue}{\tan \left(y + z\right) + x}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

   9. lower-+.f64 N/A

      \[\leadsto \left(1 - \frac{\tan a}{\color{blue}{\tan \left(y + z\right) + x}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

   10. lift-+.f64 N/A

       \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(y + z\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(z + y\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

   12. lower-+.f64 N/A

       \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(z + y\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \color{blue}{\left(\tan \left(y + z\right) + x\right)} \]

   14. lower-+.f64 79.2%

       \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \color{blue}{\left(\tan \left(y + z\right) + x\right)} \]

   15. lift-+.f64 N/A

       \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(y + z\right)} + x\right) \]

   16. +-commutative N/A

       \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

   17. lower-+.f64 79.2%

       \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

3. Applied rewrites 79.2%

   \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \left(z + y\right) + x\right)} \]

4. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1} \cdot \left(\tan \left(z + y\right) + x\right) \]
5. Step-by-step derivation

6. Applied rewrites 50.3%

   \[\leadsto \color{blue}{1} \cdot \left(\tan \left(z + y\right) + x\right) \]
7. Add Preprocessing

Alternative 19: 40.8% accurate, 1.8× speedup

Math

\[1 \cdot \left(\tan \left(\mathsf{max}\left(y, z\right)\right) + x\right) \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (* 1.0 (+ (tan (fmax y z)) x)))

C

double code(double x, double y, double z, double a) {
	return 1.0 * (tan(fmax(y, z)) + x);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = 1.0d0 * (tan(fmax(y, z)) + x)
end function

Java

public static double code(double x, double y, double z, double a) {
	return 1.0 * (Math.tan(fmax(y, z)) + x);
}

Python

def code(x, y, z, a):
	return 1.0 * (math.tan(fmax(y, z)) + x)

Julia

function code(x, y, z, a)
	return Float64(1.0 * Float64(tan(fmax(y, z)) + x))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = 1.0 * (tan(max(y, z)) + x);
end

Wolfram

code[x_, y_, z_, a_] := N[(1.0 * N[(N[Tan[N[Max[y, z], $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \left(\tan \left(y + z\right) - \tan a\right)} \]

   2. lift--.f64 N/A

      \[\leadsto x + \color{blue}{\left(\tan \left(y + z\right) - \tan a\right)} \]

   3. associate-+r- N/A

      \[\leadsto \color{blue}{\left(x + \tan \left(y + z\right)\right) - \tan a} \]

   4. sub-to-mult N/A

      \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right) \cdot \left(x + \tan \left(y + z\right)\right)} \]

   5. lower-unsound-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right) \cdot \left(x + \tan \left(y + z\right)\right)} \]

   6. lower-unsound--.f64 N/A

      \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{x + \tan \left(y + z\right)}\right)} \cdot \left(x + \tan \left(y + z\right)\right) \]

   7. lower-unsound-/.f64 N/A

      \[\leadsto \left(1 - \color{blue}{\frac{\tan a}{x + \tan \left(y + z\right)}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \left(1 - \frac{\tan a}{\color{blue}{\tan \left(y + z\right) + x}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

   9. lower-+.f64 N/A

      \[\leadsto \left(1 - \frac{\tan a}{\color{blue}{\tan \left(y + z\right) + x}}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

   10. lift-+.f64 N/A

       \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(y + z\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(z + y\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

   12. lower-+.f64 N/A

       \[\leadsto \left(1 - \frac{\tan a}{\tan \color{blue}{\left(z + y\right)} + x}\right) \cdot \left(x + \tan \left(y + z\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \color{blue}{\left(\tan \left(y + z\right) + x\right)} \]

   14. lower-+.f64 79.2%

       \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \color{blue}{\left(\tan \left(y + z\right) + x\right)} \]

   15. lift-+.f64 N/A

       \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(y + z\right)} + x\right) \]

   16. +-commutative N/A

       \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

   17. lower-+.f64 79.2%

       \[\leadsto \left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \color{blue}{\left(z + y\right)} + x\right) \]

3. Applied rewrites 79.2%

   \[\leadsto \color{blue}{\left(1 - \frac{\tan a}{\tan \left(z + y\right) + x}\right) \cdot \left(\tan \left(z + y\right) + x\right)} \]

4. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1} \cdot \left(\tan \left(z + y\right) + x\right) \]
5. Step-by-step derivation

6. Applied rewrites 50.3%

   \[\leadsto \color{blue}{1} \cdot \left(\tan \left(z + y\right) + x\right) \]

7. Taylor expanded in y around 0

   \[\leadsto 1 \cdot \left(\tan \color{blue}{z} + x\right) \]
8. Step-by-step derivation

9. Applied rewrites 40.9%

   \[\leadsto 1 \cdot \left(\tan \color{blue}{z} + x\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (x y z a)
  :name "tan-example (used to crash)"
  :precision binary64
  :pre (and (and (and (or (== x 0.0) (and (<= 0.5884142 x) (<= x 505.5909))) (or (and (<= -1.796658e+308 y) (<= y -9.425585e-310)) (and (<= 1.284938e-309 y) (<= y 1.751224e+308)))) (or (and (<= -1.776707e+308 z) (<= z -8.599796e-310)) (and (<= 3.293145e-311 z) (<= z 1.725154e+308)))) (or (and (<= -1.796658e+308 a) (<= a -9.425585e-310)) (and (<= 1.284938e-309 a) (<= a 1.751224e+308))))
  (+ x (- (tan (+ y z)) (tan a))))