tan-example (used to crash)

Percentage Accurate: 79.1% → 99.7%

Time: 11.3s

Alternatives: 18

Speedup: 1.0×

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

\[\begin{array}{l} \\ x + \left(\tan \left(y + z\right) - \tan a\right) \end{array} \]

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: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(\tan \left(y + z\right) - \tan a\right) \end{array} \]

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

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

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 79.1%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 79.6%

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

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

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 79.1%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 79.6%

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

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

   4. lift-sin.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. lower-*.f64 80.4

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

5. Applied rewrites 80.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

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

Alternative 3: 88.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(y + z\right)\\ t_1 := \mathsf{fma}\left(\cos z, \cos y, \sin z \cdot \sin \left(-y\right)\right)\\ t_2 := \frac{\sin z \cdot \cos y}{t\_1}\\ t_3 := \frac{\cos z \cdot \sin y}{t\_1}\\ \mathbf{if}\;\tan a \leq -0.0005:\\ \;\;\;\;x + \left(\left(t\_2 + \frac{\sin y \cdot \cos z}{t\_0}\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-70}:\\ \;\;\;\;x + \left(\left(t\_2 + t\_3\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(\frac{\cos y \cdot \sin z}{t\_0} + t\_3\right) - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (cos (+ y z)))
        (t_1 (fma (cos z) (cos y) (* (sin z) (sin (- y)))))
        (t_2 (/ (* (sin z) (cos y)) t_1))
        (t_3 (/ (* (cos z) (sin y)) t_1)))
   (if (<= (tan a) -0.0005)
     (+ x (- (+ t_2 (/ (* (sin y) (cos z)) t_0)) (tan a)))
     (if (<= (tan a) 5e-70)
       (+ x (- (+ t_2 t_3) a))
       (+ x (- (+ (/ (* (cos y) (sin z)) t_0) t_3) (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = cos((y + z));
	double t_1 = fma(cos(z), cos(y), (sin(z) * sin(-y)));
	double t_2 = (sin(z) * cos(y)) / t_1;
	double t_3 = (cos(z) * sin(y)) / t_1;
	double tmp;
	if (tan(a) <= -0.0005) {
		tmp = x + ((t_2 + ((sin(y) * cos(z)) / t_0)) - tan(a));
	} else if (tan(a) <= 5e-70) {
		tmp = x + ((t_2 + t_3) - a);
	} else {
		tmp = x + ((((cos(y) * sin(z)) / t_0) + t_3) - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = cos(Float64(y + z))
	t_1 = fma(cos(z), cos(y), Float64(sin(z) * sin(Float64(-y))))
	t_2 = Float64(Float64(sin(z) * cos(y)) / t_1)
	t_3 = Float64(Float64(cos(z) * sin(y)) / t_1)
	tmp = 0.0
	if (tan(a) <= -0.0005)
		tmp = Float64(x + Float64(Float64(t_2 + Float64(Float64(sin(y) * cos(z)) / t_0)) - tan(a)));
	elseif (tan(a) <= 5e-70)
		tmp = Float64(x + Float64(Float64(t_2 + t_3) - a));
	else
		tmp = Float64(x + Float64(Float64(Float64(Float64(cos(y) * sin(z)) / t_0) + t_3) - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[z], $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Sin[z], $MachinePrecision] * N[Sin[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[z], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Cos[z], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.0005], N[(x + N[(N[(t$95$2 + N[(N[(N[Sin[y], $MachinePrecision] * N[Cos[z], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 5e-70], N[(x + N[(N[(t$95$2 + t$95$3), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(N[(N[Cos[y], $MachinePrecision] * N[Sin[z], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + t$95$3), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -5.0000000000000001e-4

   1. Initial program 79.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 79.6%

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

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

      16. lift-+.f64 N/A

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

      17. lift-cos.f64 80.4

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

   9. Applied rewrites 80.4%

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

   if -5.0000000000000001e-4 < (tan.f64 a) < 4.9999999999999998e-70

   1. Initial program 79.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 79.6%

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

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

   10. Applied rewrites 50.6%

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

   if 4.9999999999999998e-70 < (tan.f64 a)

   1. Initial program 79.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 79.6%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.7

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

      4. lift-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. cos-neg-rev N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. cos-diff-rev N/A

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

      13. lift-neg.f64 N/A

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

      14. add-flip N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-cos.f64 80.3

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

   9. Applied rewrites 80.3%

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

Alternative 4: 88.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin z \cdot \sin \left(-y\right)\\ t_1 := \mathsf{fma}\left(\cos z, \cos y, t\_0\right)\\ t_2 := \cos y \cdot \sin z\\ t_3 := \cos z \cdot \sin y\\ t_4 := \cos \left(y + z\right)\\ t_5 := \mathsf{fma}\left(\cos y, \cos z, t\_0\right)\\ \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;x + \left(\left(\frac{\sin z \cdot \cos y}{t\_1} + \frac{\sin y \cdot \cos z}{t\_4}\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-70}:\\ \;\;\;\;x + \mathsf{fma}\left(a, -0.3333333333333333 \cdot {a}^{2} - 1, \frac{t\_2}{t\_5} + \frac{t\_3}{t\_5}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(\frac{t\_2}{t\_4} + \frac{t\_3}{t\_1}\right) - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (* (sin z) (sin (- y))))
        (t_1 (fma (cos z) (cos y) t_0))
        (t_2 (* (cos y) (sin z)))
        (t_3 (* (cos z) (sin y)))
        (t_4 (cos (+ y z)))
        (t_5 (fma (cos y) (cos z) t_0)))
   (if (<= (tan a) -0.02)
     (+
      x
      (- (+ (/ (* (sin z) (cos y)) t_1) (/ (* (sin y) (cos z)) t_4)) (tan a)))
     (if (<= (tan a) 5e-70)
       (+
        x
        (fma
         a
         (- (* -0.3333333333333333 (pow a 2.0)) 1.0)
         (+ (/ t_2 t_5) (/ t_3 t_5))))
       (+ x (- (+ (/ t_2 t_4) (/ t_3 t_1)) (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = sin(z) * sin(-y);
	double t_1 = fma(cos(z), cos(y), t_0);
	double t_2 = cos(y) * sin(z);
	double t_3 = cos(z) * sin(y);
	double t_4 = cos((y + z));
	double t_5 = fma(cos(y), cos(z), t_0);
	double tmp;
	if (tan(a) <= -0.02) {
		tmp = x + ((((sin(z) * cos(y)) / t_1) + ((sin(y) * cos(z)) / t_4)) - tan(a));
	} else if (tan(a) <= 5e-70) {
		tmp = x + fma(a, ((-0.3333333333333333 * pow(a, 2.0)) - 1.0), ((t_2 / t_5) + (t_3 / t_5)));
	} else {
		tmp = x + (((t_2 / t_4) + (t_3 / t_1)) - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(sin(z) * sin(Float64(-y)))
	t_1 = fma(cos(z), cos(y), t_0)
	t_2 = Float64(cos(y) * sin(z))
	t_3 = Float64(cos(z) * sin(y))
	t_4 = cos(Float64(y + z))
	t_5 = fma(cos(y), cos(z), t_0)
	tmp = 0.0
	if (tan(a) <= -0.02)
		tmp = Float64(x + Float64(Float64(Float64(Float64(sin(z) * cos(y)) / t_1) + Float64(Float64(sin(y) * cos(z)) / t_4)) - tan(a)));
	elseif (tan(a) <= 5e-70)
		tmp = Float64(x + fma(a, Float64(Float64(-0.3333333333333333 * (a ^ 2.0)) - 1.0), Float64(Float64(t_2 / t_5) + Float64(t_3 / t_5))));
	else
		tmp = Float64(x + Float64(Float64(Float64(t_2 / t_4) + Float64(t_3 / t_1)) - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Sin[z], $MachinePrecision] * N[Sin[(-y)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[z], $MachinePrecision] * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[y], $MachinePrecision] * N[Sin[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[z], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[Cos[y], $MachinePrecision] * N[Cos[z], $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.02], N[(x + N[(N[(N[(N[(N[Sin[z], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(N[Sin[y], $MachinePrecision] * N[Cos[z], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 5e-70], N[(x + N[(a * N[(N[(-0.3333333333333333 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] + N[(N[(t$95$2 / t$95$5), $MachinePrecision] + N[(t$95$3 / t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(t$95$2 / t$95$4), $MachinePrecision] + N[(t$95$3 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.0200000000000000004

   1. Initial program 79.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 79.6%

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

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

      16. lift-+.f64 N/A

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

      17. lift-cos.f64 80.4

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

   9. Applied rewrites 80.4%

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

   if -0.0200000000000000004 < (tan.f64 a) < 4.9999999999999998e-70

   1. Initial program 79.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 79.6%

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
   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 50.3%

       \[\leadsto x + \color{blue}{\mathsf{fma}\left(a, -0.3333333333333333 \cdot {a}^{2} - 1, \frac{\cos y \cdot \sin z}{\mathsf{fma}\left(\cos y, \cos z, \sin z \cdot \sin \left(-y\right)\right)} + \frac{\cos z \cdot \sin y}{\mathsf{fma}\left(\cos y, \cos z, \sin z \cdot \sin \left(-y\right)\right)}\right)} \]

   if 4.9999999999999998e-70 < (tan.f64 a)

   1. Initial program 79.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 79.6%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.7

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

      4. lift-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. cos-neg-rev N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. cos-diff-rev N/A

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

      13. lift-neg.f64 N/A

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

      14. add-flip N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-cos.f64 80.3

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

   9. Applied rewrites 80.3%

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

Alternative 5: 88.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin z \cdot \sin \left(-y\right)\\ t_1 := \mathsf{fma}\left(\cos y, \cos z, t\_0\right)\\ t_2 := \cos z \cdot \sin y\\ t_3 := \cos \left(y + z\right)\\ t_4 := \mathsf{fma}\left(\cos z, \cos y, t\_0\right)\\ t_5 := \cos y \cdot \sin z\\ \mathbf{if}\;\tan a \leq -5 \cdot 10^{-8}:\\ \;\;\;\;x + \left(\left(\frac{\sin z \cdot \cos y}{t\_4} + \frac{\sin y \cdot \cos z}{t\_3}\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-70}:\\ \;\;\;\;x + \left(\frac{t\_5}{t\_1} + \frac{t\_2}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(\frac{t\_5}{t\_3} + \frac{t\_2}{t\_4}\right) - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (* (sin z) (sin (- y))))
        (t_1 (fma (cos y) (cos z) t_0))
        (t_2 (* (cos z) (sin y)))
        (t_3 (cos (+ y z)))
        (t_4 (fma (cos z) (cos y) t_0))
        (t_5 (* (cos y) (sin z))))
   (if (<= (tan a) -5e-8)
     (+
      x
      (- (+ (/ (* (sin z) (cos y)) t_4) (/ (* (sin y) (cos z)) t_3)) (tan a)))
     (if (<= (tan a) 5e-70)
       (+ x (+ (/ t_5 t_1) (/ t_2 t_1)))
       (+ x (- (+ (/ t_5 t_3) (/ t_2 t_4)) (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = sin(z) * sin(-y);
	double t_1 = fma(cos(y), cos(z), t_0);
	double t_2 = cos(z) * sin(y);
	double t_3 = cos((y + z));
	double t_4 = fma(cos(z), cos(y), t_0);
	double t_5 = cos(y) * sin(z);
	double tmp;
	if (tan(a) <= -5e-8) {
		tmp = x + ((((sin(z) * cos(y)) / t_4) + ((sin(y) * cos(z)) / t_3)) - tan(a));
	} else if (tan(a) <= 5e-70) {
		tmp = x + ((t_5 / t_1) + (t_2 / t_1));
	} else {
		tmp = x + (((t_5 / t_3) + (t_2 / t_4)) - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(sin(z) * sin(Float64(-y)))
	t_1 = fma(cos(y), cos(z), t_0)
	t_2 = Float64(cos(z) * sin(y))
	t_3 = cos(Float64(y + z))
	t_4 = fma(cos(z), cos(y), t_0)
	t_5 = Float64(cos(y) * sin(z))
	tmp = 0.0
	if (tan(a) <= -5e-8)
		tmp = Float64(x + Float64(Float64(Float64(Float64(sin(z) * cos(y)) / t_4) + Float64(Float64(sin(y) * cos(z)) / t_3)) - tan(a)));
	elseif (tan(a) <= 5e-70)
		tmp = Float64(x + Float64(Float64(t_5 / t_1) + Float64(t_2 / t_1)));
	else
		tmp = Float64(x + Float64(Float64(Float64(t_5 / t_3) + Float64(t_2 / t_4)) - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Sin[z], $MachinePrecision] * N[Sin[(-y)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[y], $MachinePrecision] * N[Cos[z], $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[z], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[z], $MachinePrecision] * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$5 = N[(N[Cos[y], $MachinePrecision] * N[Sin[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -5e-8], N[(x + N[(N[(N[(N[(N[Sin[z], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] + N[(N[(N[Sin[y], $MachinePrecision] * N[Cos[z], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 5e-70], N[(x + N[(N[(t$95$5 / t$95$1), $MachinePrecision] + N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(t$95$5 / t$95$3), $MachinePrecision] + N[(t$95$2 / t$95$4), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 79.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 79.6%

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

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

      16. lift-+.f64 N/A

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

      17. lift-cos.f64 80.4

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

   9. Applied rewrites 80.4%

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

   if -4.9999999999999998e-8 < (tan.f64 a) < 4.9999999999999998e-70

   1. Initial program 79.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 79.6%

      \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
   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 60.5%

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

   if 4.9999999999999998e-70 < (tan.f64 a)

   1. Initial program 79.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 79.6%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.7

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

      4. lift-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. cos-neg-rev N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. cos-diff-rev N/A

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

      13. lift-neg.f64 N/A

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

      14. add-flip N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-cos.f64 80.3

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

   9. Applied rewrites 80.3%

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

Alternative 6: 80.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 79.6%

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

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

   16. lift-+.f64 N/A

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

   17. lift-cos.f64 80.4

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

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 7: 80.3% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (+
  x
  (-
   (+
    (/ (* (cos y) (sin z)) (cos (+ y 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 + ((((cos(y) * sin(z)) / cos((y + 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(Float64(cos(y) * sin(z)) / cos(Float64(y + 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[(N[Cos[y], $MachinePrecision] * N[Sin[z], $MachinePrecision]), $MachinePrecision] / N[Cos[N[(y + z), $MachinePrecision]], $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 79.1%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 79.6%

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

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.7

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

   4. lift-fma.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. cos-neg-rev N/A

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

   8. lift-neg.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. cos-diff-rev N/A

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

   13. lift-neg.f64 N/A

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

   14. add-flip N/A

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

   15. +-commutative N/A

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

   16. lift-+.f64 N/A

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

   17. lift-cos.f64 80.3

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

9. Applied rewrites 80.3%

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

Alternative 8: 79.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 79.6%

   \[\leadsto x + \left(\color{blue}{\left(\frac{\sin z \cdot \cos y}{\cos \left(z + y\right)} + \frac{\cos z \cdot \sin y}{\cos \left(z + y\right)}\right)} - \tan a\right) \]
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. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. div-add-rev N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 79.6%

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

Alternative 9: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \left(\tan a - \tan \left(z + y\right)\right) \end{array} \]

FPCore

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

C

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

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(a) - tan((z + y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. lift-+.f64 N/A

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

   2. add-flip N/A

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

   3. lower--.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sub-negate-rev N/A

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

   6. lower--.f64 79.1

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 79.1

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

3. Applied rewrites 79.1%

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

Alternative 10: 60.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -0.005:\\ \;\;\;\;x + \mathsf{fma}\left(\sin a, -1, \tan \left(z + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -0.005)
   (+ x (fma (sin a) -1.0 (tan (+ z y))))
   (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -0.005) {
		tmp = x + fma(sin(a), -1.0, tan((z + y)));
	} else {
		tmp = x + (tan(z) - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -0.005)
		tmp = Float64(x + fma(sin(a), -1.0, tan(Float64(z + y))));
	else
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -0.005], N[(x + N[(N[Sin[a], $MachinePrecision] * -1.0 + N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -0.0050000000000000001

   1. Initial program 79.1%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mult-flip N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-cos.f64 79.1

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 79.1

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

   3. Applied rewrites 79.1%

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

   4. Taylor expanded in a around 0

      \[\leadsto x + \mathsf{fma}\left(\sin a, \color{blue}{-1}, \tan \left(z + y\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 51.1%

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

   if -0.0050000000000000001 < (+.f64 y z)

   1. Initial program 79.1%

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

   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 60.3%

      \[\leadsto x + \left(\tan \color{blue}{z} - \tan a\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 60.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -0.005:\\ \;\;\;\;x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -0.005)
   (+ x (/ (sin (+ y z)) (cos (+ y z))))
   (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -0.005) {
		tmp = x + (sin((y + z)) / cos((y + z)));
	} else {
		tmp = x + (tan(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 ((y + z) <= (-0.005d0)) then
        tmp = x + (sin((y + z)) / cos((y + z)))
    else
        tmp = x + (tan(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 ((y + z) <= -0.005) {
		tmp = x + (Math.sin((y + z)) / Math.cos((y + z)));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -0.005:
		tmp = x + (math.sin((y + z)) / math.cos((y + z)))
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -0.005)
		tmp = Float64(x + Float64(sin(Float64(y + z)) / cos(Float64(y + z))));
	else
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -0.005)
		tmp = x + (sin((y + z)) / cos((y + z)));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -0.005], N[(x + N[(N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision] / N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -0.0050000000000000001

   1. Initial program 79.1%

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

   2. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-cos.f64 N/A

         \[\leadsto x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} \]

      6. lower-+.f64 50.1

         \[\leadsto x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} \]

   4. Applied rewrites 50.1%

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

   if -0.0050000000000000001 < (+.f64 y z)

   1. Initial program 79.1%

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

   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 60.3%

      \[\leadsto x + \left(\tan \color{blue}{z} - \tan a\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 59.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ t_1 := x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\\ \mathbf{if}\;t\_0 \leq -0.005:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.002:\\ \;\;\;\;x + \left(\left(y + z \cdot \left(1 + z \cdot \left(y + 0.3333333333333333 \cdot z\right)\right)\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))) (t_1 (+ x (/ (sin (+ y z)) (cos (+ y z))))))
   (if (<= t_0 -0.005)
     t_1
     (if (<= t_0 0.002)
       (+
        x
        (- (+ y (* z (+ 1.0 (* z (+ y (* 0.3333333333333333 z)))))) (tan a)))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double t_1 = x + (sin((y + z)) / cos((y + z)));
	double tmp;
	if (t_0 <= -0.005) {
		tmp = t_1;
	} else if (t_0 <= 0.002) {
		tmp = x + ((y + (z * (1.0 + (z * (y + (0.3333333333333333 * z)))))) - tan(a));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = tan((y + z))
    t_1 = x + (sin((y + z)) / cos((y + z)))
    if (t_0 <= (-0.005d0)) then
        tmp = t_1
    else if (t_0 <= 0.002d0) then
        tmp = x + ((y + (z * (1.0d0 + (z * (y + (0.3333333333333333d0 * z)))))) - tan(a))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan((y + z));
	double t_1 = x + (Math.sin((y + z)) / Math.cos((y + z)));
	double tmp;
	if (t_0 <= -0.005) {
		tmp = t_1;
	} else if (t_0 <= 0.002) {
		tmp = x + ((y + (z * (1.0 + (z * (y + (0.3333333333333333 * z)))))) - Math.tan(a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan((y + z))
	t_1 = x + (math.sin((y + z)) / math.cos((y + z)))
	tmp = 0
	if t_0 <= -0.005:
		tmp = t_1
	elif t_0 <= 0.002:
		tmp = x + ((y + (z * (1.0 + (z * (y + (0.3333333333333333 * z)))))) - math.tan(a))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	t_1 = Float64(x + Float64(sin(Float64(y + z)) / cos(Float64(y + z))))
	tmp = 0.0
	if (t_0 <= -0.005)
		tmp = t_1;
	elseif (t_0 <= 0.002)
		tmp = Float64(x + Float64(Float64(y + Float64(z * Float64(1.0 + Float64(z * Float64(y + Float64(0.3333333333333333 * z)))))) - tan(a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan((y + z));
	t_1 = x + (sin((y + z)) / cos((y + z)));
	tmp = 0.0;
	if (t_0 <= -0.005)
		tmp = t_1;
	elseif (t_0 <= 0.002)
		tmp = x + ((y + (z * (1.0 + (z * (y + (0.3333333333333333 * z)))))) - tan(a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision] / N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.005], t$95$1, If[LessEqual[t$95$0, 0.002], N[(x + N[(N[(y + N[(z * N[(1.0 + N[(z * N[(y + N[(0.3333333333333333 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 (+.f64 y z)) < -0.0050000000000000001 or 2e-3 < (tan.f64 (+.f64 y z))

   1. Initial program 79.1%

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

   2. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-cos.f64 N/A

         \[\leadsto x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} \]

      6. lower-+.f64 50.1

         \[\leadsto x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} \]

   4. Applied rewrites 50.1%

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

   if -0.0050000000000000001 < (tan.f64 (+.f64 y z)) < 2e-3

   1. Initial program 79.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 79.6%

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

      \[\leadsto x + \left(\color{blue}{\left(y \cdot \left(1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}\right) + \frac{\sin z}{\cos z}\right)} - \tan a\right) \]
   9. 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 51.1

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

   10. Applied rewrites 51.1%

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

   11. Taylor expanded in z around 0

       \[\leadsto x + \left(\left(y + \color{blue}{z \cdot \left(1 + z \cdot \left(y + \frac{1}{3} \cdot z\right)\right)}\right) - \tan a\right) \]
   12. 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 27.2

          \[\leadsto x + \left(\left(y + z \cdot \left(1 + z \cdot \left(y + 0.3333333333333333 \cdot z\right)\right)\right) - \tan a\right) \]

   13. Applied rewrites 27.2%

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

Alternative 13: 51.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -72000:\\ \;\;\;\;x + \left(\left(y + z \cdot \left(1 + z \cdot \left(y + 0.3333333333333333 \cdot z\right)\right)\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 1.6:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a \cdot \left(1 + 0.3333333333333333 \cdot {a}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -72000.0)
   (+ x (- (+ y (* z (+ 1.0 (* z (+ y (* 0.3333333333333333 z)))))) (tan a)))
   (if (<= a 1.6)
     (+ x (- (tan (+ y z)) (* a (+ 1.0 (* 0.3333333333333333 (pow a 2.0))))))
     (* x 1.0))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -72000.0) {
		tmp = x + ((y + (z * (1.0 + (z * (y + (0.3333333333333333 * z)))))) - tan(a));
	} else if (a <= 1.6) {
		tmp = x + (tan((y + z)) - (a * (1.0 + (0.3333333333333333 * pow(a, 2.0)))));
	} else {
		tmp = x * 1.0;
	}
	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 (a <= (-72000.0d0)) then
        tmp = x + ((y + (z * (1.0d0 + (z * (y + (0.3333333333333333d0 * z)))))) - tan(a))
    else if (a <= 1.6d0) then
        tmp = x + (tan((y + z)) - (a * (1.0d0 + (0.3333333333333333d0 * (a ** 2.0d0)))))
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -72000.0) {
		tmp = x + ((y + (z * (1.0 + (z * (y + (0.3333333333333333 * z)))))) - Math.tan(a));
	} else if (a <= 1.6) {
		tmp = x + (Math.tan((y + z)) - (a * (1.0 + (0.3333333333333333 * Math.pow(a, 2.0)))));
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -72000.0:
		tmp = x + ((y + (z * (1.0 + (z * (y + (0.3333333333333333 * z)))))) - math.tan(a))
	elif a <= 1.6:
		tmp = x + (math.tan((y + z)) - (a * (1.0 + (0.3333333333333333 * math.pow(a, 2.0)))))
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -72000.0)
		tmp = Float64(x + Float64(Float64(y + Float64(z * Float64(1.0 + Float64(z * Float64(y + Float64(0.3333333333333333 * z)))))) - tan(a)));
	elseif (a <= 1.6)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - Float64(a * Float64(1.0 + Float64(0.3333333333333333 * (a ^ 2.0))))));
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (a <= -72000.0)
		tmp = x + ((y + (z * (1.0 + (z * (y + (0.3333333333333333 * z)))))) - tan(a));
	elseif (a <= 1.6)
		tmp = x + (tan((y + z)) - (a * (1.0 + (0.3333333333333333 * (a ^ 2.0)))));
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -72000.0], N[(x + N[(N[(y + N[(z * N[(1.0 + N[(z * N[(y + N[(0.3333333333333333 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.6], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(a * N[(1.0 + N[(0.3333333333333333 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -72000

   1. Initial program 79.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 79.6%

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

      \[\leadsto x + \left(\color{blue}{\left(y \cdot \left(1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}\right) + \frac{\sin z}{\cos z}\right)} - \tan a\right) \]
   9. 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 51.1

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

   10. Applied rewrites 51.1%

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

   11. Taylor expanded in z around 0

       \[\leadsto x + \left(\left(y + \color{blue}{z \cdot \left(1 + z \cdot \left(y + \frac{1}{3} \cdot z\right)\right)}\right) - \tan a\right) \]
   12. 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 27.2

          \[\leadsto x + \left(\left(y + z \cdot \left(1 + z \cdot \left(y + 0.3333333333333333 \cdot z\right)\right)\right) - \tan a\right) \]

   13. Applied rewrites 27.2%

       \[\leadsto x + \left(\left(y + \color{blue}{z \cdot \left(1 + z \cdot \left(y + 0.3333333333333333 \cdot z\right)\right)}\right) - \tan a\right) \]

   if -72000 < a < 1.6000000000000001

   1. Initial program 79.1%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \left(1 + \color{blue}{\frac{1}{3} \cdot {a}^{2}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \left(1 + \frac{1}{3} \cdot \color{blue}{{a}^{2}}\right)\right) \]

      4. lower-pow.f64 40.1

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \left(1 + 0.3333333333333333 \cdot {a}^{\color{blue}{2}}\right)\right) \]

   4. Applied rewrites 40.1%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + 0.3333333333333333 \cdot {a}^{2}\right)}\right) \]

   if 1.6000000000000001 < a

   1. Initial program 79.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 78.8

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

   4. Applied rewrites 78.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto x \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 31.8%

      \[\leadsto x \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 51.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -72000:\\ \;\;\;\;x + \left(\left(y + z \cdot \left(1 + z \cdot \left(y + 0.3333333333333333 \cdot z\right)\right)\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 1.6:\\ \;\;\;\;x - \left(a - \tan \left(y + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -72000.0)
   (+ x (- (+ y (* z (+ 1.0 (* z (+ y (* 0.3333333333333333 z)))))) (tan a)))
   (if (<= a 1.6) (- x (- a (tan (+ y z)))) (* x 1.0))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -72000.0) {
		tmp = x + ((y + (z * (1.0 + (z * (y + (0.3333333333333333 * z)))))) - tan(a));
	} else if (a <= 1.6) {
		tmp = x - (a - tan((y + z)));
	} else {
		tmp = x * 1.0;
	}
	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 (a <= (-72000.0d0)) then
        tmp = x + ((y + (z * (1.0d0 + (z * (y + (0.3333333333333333d0 * z)))))) - tan(a))
    else if (a <= 1.6d0) then
        tmp = x - (a - tan((y + z)))
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -72000.0) {
		tmp = x + ((y + (z * (1.0 + (z * (y + (0.3333333333333333 * z)))))) - Math.tan(a));
	} else if (a <= 1.6) {
		tmp = x - (a - Math.tan((y + z)));
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -72000.0:
		tmp = x + ((y + (z * (1.0 + (z * (y + (0.3333333333333333 * z)))))) - math.tan(a))
	elif a <= 1.6:
		tmp = x - (a - math.tan((y + z)))
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -72000.0)
		tmp = Float64(x + Float64(Float64(y + Float64(z * Float64(1.0 + Float64(z * Float64(y + Float64(0.3333333333333333 * z)))))) - tan(a)));
	elseif (a <= 1.6)
		tmp = Float64(x - Float64(a - tan(Float64(y + z))));
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (a <= -72000.0)
		tmp = x + ((y + (z * (1.0 + (z * (y + (0.3333333333333333 * z)))))) - tan(a));
	elseif (a <= 1.6)
		tmp = x - (a - tan((y + z)));
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -72000.0], N[(x + N[(N[(y + N[(z * N[(1.0 + N[(z * N[(y + N[(0.3333333333333333 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.6], N[(x - N[(a - N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -72000

   1. Initial program 79.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 79.6%

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

      \[\leadsto x + \left(\color{blue}{\left(y \cdot \left(1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}\right) + \frac{\sin z}{\cos z}\right)} - \tan a\right) \]
   9. 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 51.1

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

   10. Applied rewrites 51.1%

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

   11. Taylor expanded in z around 0

       \[\leadsto x + \left(\left(y + \color{blue}{z \cdot \left(1 + z \cdot \left(y + \frac{1}{3} \cdot z\right)\right)}\right) - \tan a\right) \]
   12. 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 27.2

          \[\leadsto x + \left(\left(y + z \cdot \left(1 + z \cdot \left(y + 0.3333333333333333 \cdot z\right)\right)\right) - \tan a\right) \]

   13. Applied rewrites 27.2%

       \[\leadsto x + \left(\left(y + \color{blue}{z \cdot \left(1 + z \cdot \left(y + 0.3333333333333333 \cdot z\right)\right)}\right) - \tan a\right) \]

   if -72000 < a < 1.6000000000000001

   1. Initial program 79.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 40.5%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f64 40.5

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

   6. Applied rewrites 40.5%

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

   if 1.6000000000000001 < a

   1. Initial program 79.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 78.8

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

   4. Applied rewrites 78.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto x \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 31.8%

      \[\leadsto x \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 51.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -72000:\\ \;\;\;\;x + \left(\left(y + z \cdot \left(1 + y \cdot z\right)\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 1.6:\\ \;\;\;\;x - \left(a - \tan \left(y + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -72000.0)
   (+ x (- (+ y (* z (+ 1.0 (* y z)))) (tan a)))
   (if (<= a 1.6) (- x (- a (tan (+ y z)))) (* x 1.0))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -72000.0) {
		tmp = x + ((y + (z * (1.0 + (y * z)))) - tan(a));
	} else if (a <= 1.6) {
		tmp = x - (a - tan((y + z)));
	} else {
		tmp = x * 1.0;
	}
	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 (a <= (-72000.0d0)) then
        tmp = x + ((y + (z * (1.0d0 + (y * z)))) - tan(a))
    else if (a <= 1.6d0) then
        tmp = x - (a - tan((y + z)))
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -72000.0) {
		tmp = x + ((y + (z * (1.0 + (y * z)))) - Math.tan(a));
	} else if (a <= 1.6) {
		tmp = x - (a - Math.tan((y + z)));
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -72000.0:
		tmp = x + ((y + (z * (1.0 + (y * z)))) - math.tan(a))
	elif a <= 1.6:
		tmp = x - (a - math.tan((y + z)))
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -72000.0)
		tmp = Float64(x + Float64(Float64(y + Float64(z * Float64(1.0 + Float64(y * z)))) - tan(a)));
	elseif (a <= 1.6)
		tmp = Float64(x - Float64(a - tan(Float64(y + z))));
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (a <= -72000.0)
		tmp = x + ((y + (z * (1.0 + (y * z)))) - tan(a));
	elseif (a <= 1.6)
		tmp = x - (a - tan((y + z)));
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -72000.0], N[(x + N[(N[(y + N[(z * N[(1.0 + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.6], N[(x - N[(a - N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -72000

   1. Initial program 79.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 79.6%

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

      \[\leadsto x + \left(\color{blue}{\left(y \cdot \left(1 - -1 \cdot \frac{{\sin z}^{2}}{{\cos z}^{2}}\right) + \frac{\sin z}{\cos z}\right)} - \tan a\right) \]
   9. 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 51.1

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

   10. Applied rewrites 51.1%

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

   11. Taylor expanded in z around 0

       \[\leadsto x + \left(\left(y + \color{blue}{z \cdot \left(1 + y \cdot z\right)}\right) - \tan a\right) \]
   12. 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 27.3

          \[\leadsto x + \left(\left(y + z \cdot \left(1 + y \cdot z\right)\right) - \tan a\right) \]

   13. Applied rewrites 27.3%

       \[\leadsto x + \left(\left(y + \color{blue}{z \cdot \left(1 + y \cdot z\right)}\right) - \tan a\right) \]

   if -72000 < a < 1.6000000000000001

   1. Initial program 79.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 40.5%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f64 40.5

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

   6. Applied rewrites 40.5%

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

   if 1.6000000000000001 < a

   1. Initial program 79.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 78.8

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

   4. Applied rewrites 78.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto x \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 31.8%

      \[\leadsto x \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 50.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.09:\\ \;\;\;\;x \cdot 1\\ \mathbf{elif}\;a \leq 1.6:\\ \;\;\;\;x - \left(a - \tan \left(y + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -0.09)
   (* x 1.0)
   (if (<= a 1.6) (- x (- a (tan (+ y z)))) (* x 1.0))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -0.09) {
		tmp = x * 1.0;
	} else if (a <= 1.6) {
		tmp = x - (a - tan((y + z)));
	} else {
		tmp = x * 1.0;
	}
	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 (a <= (-0.09d0)) then
        tmp = x * 1.0d0
    else if (a <= 1.6d0) then
        tmp = x - (a - tan((y + z)))
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -0.09) {
		tmp = x * 1.0;
	} else if (a <= 1.6) {
		tmp = x - (a - Math.tan((y + z)));
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -0.09:
		tmp = x * 1.0
	elif a <= 1.6:
		tmp = x - (a - math.tan((y + z)))
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -0.09)
		tmp = Float64(x * 1.0);
	elseif (a <= 1.6)
		tmp = Float64(x - Float64(a - tan(Float64(y + z))));
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (a <= -0.09)
		tmp = x * 1.0;
	elseif (a <= 1.6)
		tmp = x - (a - tan((y + z)));
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -0.09], N[(x * 1.0), $MachinePrecision], If[LessEqual[a, 1.6], N[(x - N[(a - N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.089999999999999997 or 1.6000000000000001 < a

   1. Initial program 79.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 78.8

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

   4. Applied rewrites 78.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto x \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 31.8%

      \[\leadsto x \cdot 1 \]

   if -0.089999999999999997 < a < 1.6000000000000001

   1. Initial program 79.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 40.5%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f64 40.5

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

   6. Applied rewrites 40.5%

      \[\leadsto \color{blue}{x - \left(a - \tan \left(y + z\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 40.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.65:\\ \;\;\;\;x \cdot 1\\ \mathbf{elif}\;a \leq 1.55:\\ \;\;\;\;x - \left(a - \tan z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.65) (* x 1.0) (if (<= a 1.55) (- x (- a (tan z))) (* x 1.0))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.65) {
		tmp = x * 1.0;
	} else if (a <= 1.55) {
		tmp = x - (a - tan(z));
	} else {
		tmp = x * 1.0;
	}
	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 (a <= (-1.65d0)) then
        tmp = x * 1.0d0
    else if (a <= 1.55d0) then
        tmp = x - (a - tan(z))
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.65) {
		tmp = x * 1.0;
	} else if (a <= 1.55) {
		tmp = x - (a - Math.tan(z));
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -1.65:
		tmp = x * 1.0
	elif a <= 1.55:
		tmp = x - (a - math.tan(z))
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.65)
		tmp = Float64(x * 1.0);
	elseif (a <= 1.55)
		tmp = Float64(x - Float64(a - tan(z)));
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (a <= -1.65)
		tmp = x * 1.0;
	elseif (a <= 1.55)
		tmp = x - (a - tan(z));
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.65], N[(x * 1.0), $MachinePrecision], If[LessEqual[a, 1.55], N[(x - N[(a - N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.6499999999999999 or 1.55000000000000004 < a

   1. Initial program 79.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 78.8

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

   4. Applied rewrites 78.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto x \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 31.8%

      \[\leadsto x \cdot 1 \]

   if -1.6499999999999999 < a < 1.55000000000000004

   1. Initial program 79.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 40.5%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f64 40.5

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

   6. Applied rewrites 40.5%

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

   7. Taylor expanded in y around 0

      \[\leadsto x - \left(a - \tan \color{blue}{z}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 31.1%

      \[\leadsto x - \left(a - \tan \color{blue}{z}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 31.8% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 1 \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (* x 1.0))

C

double code(double x, double y, double z, double a) {
	return x * 1.0;
}

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 * 1.0d0
end function

Java

public static double code(double x, double y, double z, double a) {
	return x * 1.0;
}

Python

def code(x, y, z, a):
	return x * 1.0

Julia

function code(x, y, z, a)
	return Float64(x * 1.0)
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x * 1.0;
end

Wolfram

code[x_, y_, z_, a_] := N[(x * 1.0), $MachinePrecision]

Derivation

1. Initial program 79.1%

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

2. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 78.8

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

4. Applied rewrites 78.8%

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

5. Taylor expanded in x around inf

   \[\leadsto x \cdot 1 \]
6. Step-by-step derivation

7. Applied rewrites 31.8%

   \[\leadsto x \cdot 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025147 
(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))))