tan-example (used to crash)

Percentage Accurate: 79.8% → 99.5%

Time: 6.2s

Alternatives: 9

Speedup: 1.0×

Specification

\[\left(\left(\left(x = 0 \lor \frac{2942071}{5000000} \leq x \land x \leq \frac{5055909}{10000}\right) \land \left(-179665800000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 \leq y \land y \leq \frac{-1885117}{2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000} \lor \frac{642469}{500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000} \leq y \land y \leq 175122400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000\right)\right) \land \left(-177670700000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 \leq z \land z \leq \frac{-2149949}{2500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000} \lor \frac{658629}{20000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000} \leq z \land z \leq 172515400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000\right)\right) \land \left(-179665800000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 \leq a \land a \leq \frac{-1885117}{2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000} \lor \frac{642469}{500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000} \leq a \land a \leq 175122400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000\right)\]

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\mathsf{min}\left(y, z\right)\right)\\ t_1 := \sin \left(\mathsf{max}\left(y, z\right)\right)\\ t_2 := \cos \left(\mathsf{max}\left(y, z\right)\right)\\ t_3 := \cos \left(\mathsf{min}\left(y, z\right)\right)\\ t_4 := t\_3 \cdot t\_2\\ t_5 := \left(1 - \frac{t\_0 \cdot t\_1}{t\_4}\right) \cdot t\_4\\ \left(\frac{t\_3}{t\_5} \cdot t\_1 + \left(x + \frac{t\_0}{t\_5} \cdot t\_2\right)\right) - \tan a \end{array} \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (let* ((t_0 (sin (fmin y z)))
       (t_1 (sin (fmax y z)))
       (t_2 (cos (fmax y z)))
       (t_3 (cos (fmin y z)))
       (t_4 (* t_3 t_2))
       (t_5 (* (- 1 (/ (* t_0 t_1) t_4)) t_4)))
  (- (+ (* (/ t_3 t_5) t_1) (+ x (* (/ t_0 t_5) t_2))) (tan a))))

C

double code(double x, double y, double z, double a) {
	double t_0 = sin(fmin(y, z));
	double t_1 = sin(fmax(y, z));
	double t_2 = cos(fmax(y, z));
	double t_3 = cos(fmin(y, z));
	double t_4 = t_3 * t_2;
	double t_5 = (1.0 - ((t_0 * t_1) / t_4)) * t_4;
	return (((t_3 / t_5) * t_1) + (x + ((t_0 / t_5) * t_2))) - 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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    t_0 = sin(fmin(y, z))
    t_1 = sin(fmax(y, z))
    t_2 = cos(fmax(y, z))
    t_3 = cos(fmin(y, z))
    t_4 = t_3 * t_2
    t_5 = (1.0d0 - ((t_0 * t_1) / t_4)) * t_4
    code = (((t_3 / t_5) * t_1) + (x + ((t_0 / t_5) * t_2))) - tan(a)
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.sin(fmin(y, z));
	double t_1 = Math.sin(fmax(y, z));
	double t_2 = Math.cos(fmax(y, z));
	double t_3 = Math.cos(fmin(y, z));
	double t_4 = t_3 * t_2;
	double t_5 = (1.0 - ((t_0 * t_1) / t_4)) * t_4;
	return (((t_3 / t_5) * t_1) + (x + ((t_0 / t_5) * t_2))) - Math.tan(a);
}

Python

def code(x, y, z, a):
	t_0 = math.sin(fmin(y, z))
	t_1 = math.sin(fmax(y, z))
	t_2 = math.cos(fmax(y, z))
	t_3 = math.cos(fmin(y, z))
	t_4 = t_3 * t_2
	t_5 = (1.0 - ((t_0 * t_1) / t_4)) * t_4
	return (((t_3 / t_5) * t_1) + (x + ((t_0 / t_5) * t_2))) - math.tan(a)

Julia

function code(x, y, z, a)
	t_0 = sin(fmin(y, z))
	t_1 = sin(fmax(y, z))
	t_2 = cos(fmax(y, z))
	t_3 = cos(fmin(y, z))
	t_4 = Float64(t_3 * t_2)
	t_5 = Float64(Float64(1.0 - Float64(Float64(t_0 * t_1) / t_4)) * t_4)
	return Float64(Float64(Float64(Float64(t_3 / t_5) * t_1) + Float64(x + Float64(Float64(t_0 / t_5) * t_2))) - tan(a))
end

MATLAB

function tmp = code(x, y, z, a)
	t_0 = sin(min(y, z));
	t_1 = sin(max(y, z));
	t_2 = cos(max(y, z));
	t_3 = cos(min(y, z));
	t_4 = t_3 * t_2;
	t_5 = (1.0 - ((t_0 * t_1) / t_4)) * t_4;
	tmp = (((t_3 / t_5) * t_1) + (x + ((t_0 / t_5) * t_2))) - tan(a);
end

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. sub-to-fraction N/A

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

   7. mult-flip N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. lower-sin.f64 N/A

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

   18. lower-/.f64 N/A

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

   19. lower-cos.f64 79.6%

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

3. Applied rewrites 79.6%

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

5. Applied rewrites 80.3%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. lower--.f64 N/A

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

7. Applied rewrites 80.3%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-to-mult N/A

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

   5. lower-unsound-*.f64 N/A

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

   6. lower-unsound--.f64 N/A

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

   7. lower-unsound-/.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. lift-cos.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-cos.f64 N/A

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

   17. lift-cos.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 81.0%

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

9. Applied rewrites 81.0%

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

    1. lift-cos.f64 N/A

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

    2. lift-+.f64 N/A

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

    3. cos-sum N/A

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

    4. sub-to-mult N/A

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

    5. lower-unsound-*.f64 N/A

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

    6. lower-unsound--.f64 N/A

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

    7. lower-unsound-/.f64 N/A

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

    8. lift-sin.f64 N/A

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

    9. lift-sin.f64 N/A

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

    10. *-commutative N/A

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

    11. lower-*.f64 N/A

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

    12. lift-cos.f64 N/A

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

    13. lift-cos.f64 N/A

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

    14. *-commutative N/A

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

    15. lower-*.f64 N/A

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

    16. lift-cos.f64 N/A

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

    17. lift-cos.f64 N/A

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

    18. *-commutative N/A

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

    19. lower-*.f64 99.5%

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

11. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\mathsf{min}\left(y, z\right)\right)\\ t_1 := \sin \left(\mathsf{max}\left(y, z\right)\right)\\ t_2 := \cos \left(\mathsf{min}\left(y, z\right)\right)\\ t_3 := \cos \left(\mathsf{max}\left(y, z\right)\right)\\ t_4 := t\_2 \cdot t\_3 - t\_0 \cdot t\_1\\ \left(\frac{t\_2}{t\_4} \cdot t\_1 + \left(x + \frac{t\_0}{t\_4} \cdot t\_3\right)\right) - \tan a \end{array} \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (let* ((t_0 (sin (fmin y z)))
       (t_1 (sin (fmax y z)))
       (t_2 (cos (fmin y z)))
       (t_3 (cos (fmax y z)))
       (t_4 (- (* t_2 t_3) (* t_0 t_1))))
  (- (+ (* (/ t_2 t_4) t_1) (+ x (* (/ t_0 t_4) t_3))) (tan a))))

C

double code(double x, double y, double z, double a) {
	double t_0 = sin(fmin(y, z));
	double t_1 = sin(fmax(y, z));
	double t_2 = cos(fmin(y, z));
	double t_3 = cos(fmax(y, z));
	double t_4 = (t_2 * t_3) - (t_0 * t_1);
	return (((t_2 / t_4) * t_1) + (x + ((t_0 / t_4) * t_3))) - 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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    t_0 = sin(fmin(y, z))
    t_1 = sin(fmax(y, z))
    t_2 = cos(fmin(y, z))
    t_3 = cos(fmax(y, z))
    t_4 = (t_2 * t_3) - (t_0 * t_1)
    code = (((t_2 / t_4) * t_1) + (x + ((t_0 / t_4) * t_3))) - tan(a)
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.sin(fmin(y, z));
	double t_1 = Math.sin(fmax(y, z));
	double t_2 = Math.cos(fmin(y, z));
	double t_3 = Math.cos(fmax(y, z));
	double t_4 = (t_2 * t_3) - (t_0 * t_1);
	return (((t_2 / t_4) * t_1) + (x + ((t_0 / t_4) * t_3))) - Math.tan(a);
}

Python

def code(x, y, z, a):
	t_0 = math.sin(fmin(y, z))
	t_1 = math.sin(fmax(y, z))
	t_2 = math.cos(fmin(y, z))
	t_3 = math.cos(fmax(y, z))
	t_4 = (t_2 * t_3) - (t_0 * t_1)
	return (((t_2 / t_4) * t_1) + (x + ((t_0 / t_4) * t_3))) - math.tan(a)

Julia

function code(x, y, z, a)
	t_0 = sin(fmin(y, z))
	t_1 = sin(fmax(y, z))
	t_2 = cos(fmin(y, z))
	t_3 = cos(fmax(y, z))
	t_4 = Float64(Float64(t_2 * t_3) - Float64(t_0 * t_1))
	return Float64(Float64(Float64(Float64(t_2 / t_4) * t_1) + Float64(x + Float64(Float64(t_0 / t_4) * t_3))) - tan(a))
end

MATLAB

function tmp = code(x, y, z, a)
	t_0 = sin(min(y, z));
	t_1 = sin(max(y, z));
	t_2 = cos(min(y, z));
	t_3 = cos(max(y, z));
	t_4 = (t_2 * t_3) - (t_0 * t_1);
	tmp = (((t_2 / t_4) * t_1) + (x + ((t_0 / t_4) * t_3))) - tan(a);
end

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. sub-to-fraction N/A

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

   7. mult-flip N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. lower-sin.f64 N/A

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

   18. lower-/.f64 N/A

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

   19. lower-cos.f64 79.6%

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

3. Applied rewrites 79.6%

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

5. Applied rewrites 80.3%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. lower--.f64 N/A

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

7. Applied rewrites 80.3%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. lower--.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 81.0%

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

9. Applied rewrites 81.0%

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

    1. lift-cos.f64 N/A

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

    2. lift-+.f64 N/A

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

    3. cos-sum N/A

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

    4. lower--.f64 N/A

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

    5. lift-cos.f64 N/A

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

    6. lift-cos.f64 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 N/A

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

    9. lift-sin.f64 N/A

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

    10. lift-sin.f64 N/A

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

    11. *-commutative N/A

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

    12. lower-*.f64 99.5%

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

11. Applied rewrites 99.5%

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

Alternative 3: 89.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \cos y \cdot \cos z - \sin y \cdot \sin z\\ t_1 := \cos \left(z + y\right)\\ t_2 := \frac{\cos z \cdot \sin y}{t\_1}\\ \mathbf{if}\;\tan a \leq \frac{-1152921504606847}{576460752303423488}:\\ \;\;\;\;x + \left(\left(\frac{\sin z \cdot \sin \left(\left(-y\right) + \pi \cdot \frac{1}{2}\right)}{t\_1} + t\_2\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq \frac{8920298079412249}{178405961588244985132285746181186892047843328}:\\ \;\;\;\;\left(x + \frac{\cos y}{t\_0} \cdot \sin z\right) + \frac{\sin y}{t\_0} \cdot \cos z\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(\frac{\sin z \cdot \cos y}{t\_1} + t\_2\right) - \tan a\right)\\ \end{array} \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (let* ((t_0 (- (* (cos y) (cos z)) (* (sin y) (sin z))))
       (t_1 (cos (+ z y)))
       (t_2 (/ (* (cos z) (sin y)) t_1)))
  (if (<= (tan a) -1152921504606847/576460752303423488)
    (+
     x
     (-
      (+ (/ (* (sin z) (sin (+ (- y) (* PI 1/2)))) t_1) t_2)
      (tan a)))
    (if (<=
         (tan a)
         8920298079412249/178405961588244985132285746181186892047843328)
      (+
       (+ x (* (/ (cos y) t_0) (sin z)))
       (* (/ (sin y) t_0) (cos z)))
      (+ x (- (+ (/ (* (sin z) (cos y)) t_1) t_2) (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = (cos(y) * cos(z)) - (sin(y) * sin(z));
	double t_1 = cos((z + y));
	double t_2 = (cos(z) * sin(y)) / t_1;
	double tmp;
	if (tan(a) <= -0.002) {
		tmp = x + ((((sin(z) * sin((-y + (((double) M_PI) * 0.5)))) / t_1) + t_2) - tan(a));
	} else if (tan(a) <= 5e-29) {
		tmp = (x + ((cos(y) / t_0) * sin(z))) + ((sin(y) / t_0) * cos(z));
	} else {
		tmp = x + ((((sin(z) * cos(y)) / t_1) + t_2) - tan(a));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = (Math.cos(y) * Math.cos(z)) - (Math.sin(y) * Math.sin(z));
	double t_1 = Math.cos((z + y));
	double t_2 = (Math.cos(z) * Math.sin(y)) / t_1;
	double tmp;
	if (Math.tan(a) <= -0.002) {
		tmp = x + ((((Math.sin(z) * Math.sin((-y + (Math.PI * 0.5)))) / t_1) + t_2) - Math.tan(a));
	} else if (Math.tan(a) <= 5e-29) {
		tmp = (x + ((Math.cos(y) / t_0) * Math.sin(z))) + ((Math.sin(y) / t_0) * Math.cos(z));
	} else {
		tmp = x + ((((Math.sin(z) * Math.cos(y)) / t_1) + t_2) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = (math.cos(y) * math.cos(z)) - (math.sin(y) * math.sin(z))
	t_1 = math.cos((z + y))
	t_2 = (math.cos(z) * math.sin(y)) / t_1
	tmp = 0
	if math.tan(a) <= -0.002:
		tmp = x + ((((math.sin(z) * math.sin((-y + (math.pi * 0.5)))) / t_1) + t_2) - math.tan(a))
	elif math.tan(a) <= 5e-29:
		tmp = (x + ((math.cos(y) / t_0) * math.sin(z))) + ((math.sin(y) / t_0) * math.cos(z))
	else:
		tmp = x + ((((math.sin(z) * math.cos(y)) / t_1) + t_2) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(Float64(cos(y) * cos(z)) - Float64(sin(y) * sin(z)))
	t_1 = cos(Float64(z + y))
	t_2 = Float64(Float64(cos(z) * sin(y)) / t_1)
	tmp = 0.0
	if (tan(a) <= -0.002)
		tmp = Float64(x + Float64(Float64(Float64(Float64(sin(z) * sin(Float64(Float64(-y) + Float64(pi * 0.5)))) / t_1) + t_2) - tan(a)));
	elseif (tan(a) <= 5e-29)
		tmp = Float64(Float64(x + Float64(Float64(cos(y) / t_0) * sin(z))) + Float64(Float64(sin(y) / t_0) * cos(z)));
	else
		tmp = Float64(x + Float64(Float64(Float64(Float64(sin(z) * cos(y)) / t_1) + t_2) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = (cos(y) * cos(z)) - (sin(y) * sin(z));
	t_1 = cos((z + y));
	t_2 = (cos(z) * sin(y)) / t_1;
	tmp = 0.0;
	if (tan(a) <= -0.002)
		tmp = x + ((((sin(z) * sin((-y + (pi * 0.5)))) / t_1) + t_2) - tan(a));
	elseif (tan(a) <= 5e-29)
		tmp = (x + ((cos(y) / t_0) * sin(z))) + ((sin(y) / t_0) * cos(z));
	else
		tmp = x + ((((sin(z) * cos(y)) / t_1) + t_2) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[(N[Cos[y], $MachinePrecision] * N[Cos[z], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * N[Sin[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(z + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Cos[z], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -1152921504606847/576460752303423488], N[(x + N[(N[(N[(N[(N[Sin[z], $MachinePrecision] * N[Sin[N[((-y) + N[(Pi * 1/2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 8920298079412249/178405961588244985132285746181186892047843328], N[(N[(x + N[(N[(N[Cos[y], $MachinePrecision] / t$95$0), $MachinePrecision] * N[Sin[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[y], $MachinePrecision] / t$95$0), $MachinePrecision] * N[Cos[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(N[(N[Sin[z], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -2e-3

   1. Initial program 79.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 80.3%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 79.9%

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

   5. Applied rewrites 79.9%

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

   if -2e-3 < (tan.f64 a) < 4.9999999999999999e-29

   1. Initial program 79.8%

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

   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 50.7%

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

   4. Applied rewrites 50.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. quot-tan N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-+.f64 N/A

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

      11. sin-sum-rev N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-cos.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. div-add-rev N/A

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

      20. lift-/.f64 N/A

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

   6. Applied rewrites 51.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. lower--.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 51.6%

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

   8. Applied rewrites 51.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. lower--.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 61.2%

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

   10. Applied rewrites 61.2%

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

   if 4.9999999999999999e-29 < (tan.f64 a)

   1. Initial program 79.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 80.3%

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

Alternative 4: 80.3% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z a)
  :precision binary64
  (let* ((t_0 (cos (+ z 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 = cos((z + y));
	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 + y))
    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 + y));
	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 + y))
	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 = cos(Float64(z + 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

MATLAB

function tmp = code(x, y, z, a)
	t_0 = cos((z + y));
	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[Cos[N[(z + y), $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 79.8%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 80.3%

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

Alternative 5: 79.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(z + y\right)\\ x + \left(\left(\frac{\sin z \cdot 1}{t\_0} + \frac{\cos z \cdot \sin y}{t\_0}\right) - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (let* ((t_0 (cos (+ z y))))
  (+
   x
   (- (+ (/ (* (sin z) 1) 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 + y));
	return x + ((((sin(z) * 1.0) / 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 + y))
    code = x + ((((sin(z) * 1.0d0) / 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 + y));
	return x + ((((Math.sin(z) * 1.0) / 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 + y))
	return x + ((((math.sin(z) * 1.0) / t_0) + ((math.cos(z) * math.sin(y)) / t_0)) - math.tan(a))

Julia

function code(x, y, z, a)
	t_0 = cos(Float64(z + y))
	return Float64(x + Float64(Float64(Float64(Float64(sin(z) * 1.0) / 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 + y));
	tmp = x + ((((sin(z) * 1.0) / t_0) + ((cos(z) * sin(y)) / t_0)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Cos[N[(z + y), $MachinePrecision]], $MachinePrecision]}, N[(x + N[(N[(N[(N[(N[Sin[z], $MachinePrecision] * 1), $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 79.8%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 80.3%

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

4. Taylor expanded in y around 0

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

6. Applied rewrites 79.9%

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

Alternative 6: 79.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. sub-to-fraction N/A

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

   7. mult-flip N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. lower-sin.f64 N/A

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

   18. lower-/.f64 N/A

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

   19. lower-cos.f64 79.6%

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

3. Applied rewrites 79.6%

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

5. Applied rewrites 80.3%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. lower--.f64 N/A

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

7. Applied rewrites 80.3%

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

8. Taylor expanded in z around 0

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

   1. lower-+.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-cos.f64 79.8%

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

10. Applied rewrites 79.8%

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

Alternative 7: 69.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(y, z\right) + \mathsf{max}\left(y, z\right) \leq -500:\\ \;\;\;\;\tan \left(\mathsf{max}\left(y, z\right) + \mathsf{min}\left(y, z\right)\right) + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(\mathsf{max}\left(y, z\right)\right) - \tan a\right)\\ \end{array} \]

FPCore

(FPCore (x y z a)
  :precision binary64
  (if (<= (+ (fmin y z) (fmax y z)) -500)
  (+ (tan (+ (fmax y z) (fmin y z))) x)
  (+ x (- (tan (fmax y z)) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((fmin(y, z) + fmax(y, z)) <= -500.0) {
		tmp = tan((fmax(y, z) + fmin(y, z))) + x;
	} else {
		tmp = x + (tan(fmax(y, z)) - tan(a));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((fmin(y, z) + fmax(y, z)) <= (-500.0d0)) then
        tmp = tan((fmax(y, z) + fmin(y, z))) + x
    else
        tmp = x + (tan(fmax(y, z)) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((fmin(y, z) + fmax(y, z)) <= -500.0) {
		tmp = Math.tan((fmax(y, z) + fmin(y, z))) + x;
	} else {
		tmp = x + (Math.tan(fmax(y, z)) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (fmin(y, z) + fmax(y, z)) <= -500.0:
		tmp = math.tan((fmax(y, z) + fmin(y, z))) + x
	else:
		tmp = x + (math.tan(fmax(y, z)) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(fmin(y, z) + fmax(y, z)) <= -500.0)
		tmp = Float64(tan(Float64(fmax(y, z) + fmin(y, z))) + x);
	else
		tmp = Float64(x + Float64(tan(fmax(y, z)) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((min(y, z) + max(y, z)) <= -500.0)
		tmp = tan((max(y, z) + min(y, z))) + x;
	else
		tmp = x + (tan(max(y, z)) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(N[Min[y, z], $MachinePrecision] + N[Max[y, z], $MachinePrecision]), $MachinePrecision], -500], N[(N[Tan[N[(N[Max[y, z], $MachinePrecision] + N[Min[y, z], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision], N[(x + N[(N[Tan[N[Max[y, z], $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -500

   1. Initial program 79.8%

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

   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 50.7%

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

   4. Applied rewrites 50.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. quot-tan N/A

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

      7. lower-tan.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 50.7%

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

   6. Applied rewrites 50.7%

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

   if -500 < (+.f64 y z)

   1. Initial program 79.8%

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

   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 60.6%

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

Alternative 8: 50.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

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 50.7%

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

4. Applied rewrites 50.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. quot-tan N/A

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

   7. lower-tan.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-+.f64 N/A

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

   11. lift-+.f64 50.7%

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

6. Applied rewrites 50.7%

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

Reproduce

herbie shell --seed 2025285 -o generate:evaluate
(FPCore (x y z a)
  :name "tan-example (used to crash)"
  :precision binary64
  :pre (and (and (and (or (== x 0) (and (<= 2942071/5000000 x) (<= x 5055909/10000))) (or (and (<= -179665800000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 y) (<= y -1885117/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)) (and (<= 642469/500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 y) (<= y 175122400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)))) (or (and (<= -177670700000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 z) (<= z -2149949/2500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)) (and (<= 658629/20000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 z) (<= z 172515400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)))) (or (and (<= -179665800000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 a) (<= a -1885117/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)) (and (<= 642469/500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 a) (<= a 175122400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000))))
  (+ x (- (tan (+ y z)) (tan a))))