Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, B

Percentage Accurate: 99.9% → 99.9%

Time: 4.3s

Alternatives: 16

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(x + \cos y\right) - z \cdot \sin y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ x (cos y)) (* z (sin y))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y, z):
	return (x + math.cos(y)) - (z * math.sin(y))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + \cos y\right) - z \cdot \sin y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ x (cos y)) (* z (sin y))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y, z):
	return (x + math.cos(y)) - (z * math.sin(y))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (fma (sin y) (- z) (+ (cos y) x)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(x + \cos y\right) - z \cdot \sin y \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. *-lft-identity N/A

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

   7. metadata-eval N/A

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

   8. fp-cancel-sign-sub-inv N/A

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

   9. associate-+r+ N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. associate-+l+ N/A

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

   13. mul-1-neg N/A

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

   14. *-commutative N/A

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

   15. distribute-rgt-neg-in N/A

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

   16. +-commutative N/A

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

   17. lower-fma.f64 N/A

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

   18. lift-sin.f64 N/A

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

   19. lower-neg.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 N/A

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

   22. lift-cos.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 74.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \cos y\right) - z \cdot \sin y\\ \mathbf{if}\;t\_0 \leq -500:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x - -1\right)\\ \mathbf{elif}\;t\_0 \leq 0.95:\\ \;\;\;\;\cos y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+209}:\\ \;\;\;\;\left(x - z \cdot y\right) - -1\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (cos y)) (* z (sin y)))))
   (if (<= t_0 -500.0)
     (fma (- z) y (- x -1.0))
     (if (<= t_0 0.95)
       (cos y)
       (if (<= t_0 2e+209) (- (- x (* z y)) -1.0) (- x -1.0))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + cos(y)) - (z * sin(y));
	double tmp;
	if (t_0 <= -500.0) {
		tmp = fma(-z, y, (x - -1.0));
	} else if (t_0 <= 0.95) {
		tmp = cos(y);
	} else if (t_0 <= 2e+209) {
		tmp = (x - (z * y)) - -1.0;
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + cos(y)) - Float64(z * sin(y)))
	tmp = 0.0
	if (t_0 <= -500.0)
		tmp = fma(Float64(-z), y, Float64(x - -1.0));
	elseif (t_0 <= 0.95)
		tmp = cos(y);
	elseif (t_0 <= 2e+209)
		tmp = Float64(Float64(x - Float64(z * y)) - -1.0);
	else
		tmp = Float64(x - -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -500.0], N[((-z) * y + N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.95], N[Cos[y], $MachinePrecision], If[LessEqual[t$95$0, 2e+209], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision], N[(x - -1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -500

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

         \[\leadsto \left(y \cdot \left(\frac{-1}{2} \cdot y - z\right) + x\right) + 1 \]

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot y - z, y, x - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot y - z, y, x - -1 \cdot 1\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot y - z, y, x - -1\right) \]

      14. lower--.f64 57.2

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot y - z, y, x - -1\right) \]

   5. Applied rewrites 57.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot y - z, y, x - -1\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot z, y, x - -1\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(z\right), y, x - -1\right) \]

      2. lift-neg.f64 67.3

         \[\leadsto \mathsf{fma}\left(-z, y, x - -1\right) \]

   8. Applied rewrites 67.3%

      \[\leadsto \mathsf{fma}\left(-z, y, x - -1\right) \]

   if -500 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.94999999999999996

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \cos y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos y + \color{blue}{x} \]

      2. lower-+.f64 N/A

         \[\leadsto \cos y + \color{blue}{x} \]

      3. lift-cos.f64 96.8

         \[\leadsto \cos y + x \]

   5. Applied rewrites 96.8%

      \[\leadsto \color{blue}{\cos y + x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \cos y \]
   7. Step-by-step derivation

      1. lift-cos.f64 93.7

         \[\leadsto \cos y \]

   8. Applied rewrites 93.7%

      \[\leadsto \cos y \]

   if 0.94999999999999996 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 2.0000000000000001e209

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(x + \left(\mathsf{neg}\left(1\right)\right) \cdot \left(y \cdot z\right)\right) - -1 \]

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 81.4

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

   5. Applied rewrites 81.4%

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

   if 2.0000000000000001e209 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y)))

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \color{blue}{1} \]

      2. metadata-eval N/A

         \[\leadsto x + 1 \cdot \color{blue}{1} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1} \]

      4. metadata-eval N/A

         \[\leadsto x - -1 \cdot 1 \]

      5. metadata-eval N/A

         \[\leadsto x - -1 \]

      6. lower--.f64 63.7

         \[\leadsto x - \color{blue}{-1} \]

   5. Applied rewrites 63.7%

      \[\leadsto \color{blue}{x - -1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -500:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x - -1\right)\\ \mathbf{elif}\;\left(x + \cos y\right) - z \cdot \sin y \leq 0.95:\\ \;\;\;\;\cos y\\ \mathbf{elif}\;\left(x + \cos y\right) - z \cdot \sin y \leq 2 \cdot 10^{+209}:\\ \;\;\;\;\left(x - z \cdot y\right) - -1\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ t_1 := \left(x + \cos y\right) - t\_0\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+22} \lor \neg \left(t\_1 \leq 2\right):\\ \;\;\;\;x - t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))) (t_1 (- (+ x (cos y)) t_0)))
   (if (or (<= t_1 -5e+22) (not (<= t_1 2.0))) (- x t_0) (+ (cos y) x))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double t_1 = (x + cos(y)) - t_0;
	double tmp;
	if ((t_1 <= -5e+22) || !(t_1 <= 2.0)) {
		tmp = x - t_0;
	} else {
		tmp = cos(y) + x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z * sin(y)
    t_1 = (x + cos(y)) - t_0
    if ((t_1 <= (-5d+22)) .or. (.not. (t_1 <= 2.0d0))) then
        tmp = x - t_0
    else
        tmp = cos(y) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.sin(y);
	double t_1 = (x + Math.cos(y)) - t_0;
	double tmp;
	if ((t_1 <= -5e+22) || !(t_1 <= 2.0)) {
		tmp = x - t_0;
	} else {
		tmp = Math.cos(y) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.sin(y)
	t_1 = (x + math.cos(y)) - t_0
	tmp = 0
	if (t_1 <= -5e+22) or not (t_1 <= 2.0):
		tmp = x - t_0
	else:
		tmp = math.cos(y) + x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	t_1 = Float64(Float64(x + cos(y)) - t_0)
	tmp = 0.0
	if ((t_1 <= -5e+22) || !(t_1 <= 2.0))
		tmp = Float64(x - t_0);
	else
		tmp = Float64(cos(y) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * sin(y);
	t_1 = (x + cos(y)) - t_0;
	tmp = 0.0;
	if ((t_1 <= -5e+22) || ~((t_1 <= 2.0)))
		tmp = x - t_0;
	else
		tmp = cos(y) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+22], N[Not[LessEqual[t$95$1, 2.0]], $MachinePrecision]], N[(x - t$95$0), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -4.9999999999999996e22 or 2 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y)))

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} - z \cdot \sin y \]
   4. Step-by-step derivation

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{x} - z \cdot \sin y \]

   if -4.9999999999999996e22 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 2

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \cos y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos y + \color{blue}{x} \]

      2. lower-+.f64 N/A

         \[\leadsto \cos y + \color{blue}{x} \]

      3. lift-cos.f64 95.7

         \[\leadsto \cos y + x \]

   5. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\cos y + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -5 \cdot 10^{+22} \lor \neg \left(\left(x + \cos y\right) - z \cdot \sin y \leq 2\right):\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ t_1 := \left(x + \cos y\right) - t\_0\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, x\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;x - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))) (t_1 (- (+ x (cos y)) t_0)))
   (if (<= t_1 -5e+22)
     (fma (sin y) (- z) x)
     (if (<= t_1 2.0) (+ (cos y) x) (- x t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double t_1 = (x + cos(y)) - t_0;
	double tmp;
	if (t_1 <= -5e+22) {
		tmp = fma(sin(y), -z, x);
	} else if (t_1 <= 2.0) {
		tmp = cos(y) + x;
	} else {
		tmp = x - t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	t_1 = Float64(Float64(x + cos(y)) - t_0)
	tmp = 0.0
	if (t_1 <= -5e+22)
		tmp = fma(sin(y), Float64(-z), x);
	elseif (t_1 <= 2.0)
		tmp = Float64(cos(y) + x);
	else
		tmp = Float64(x - t_0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+22], N[(N[Sin[y], $MachinePrecision] * (-z) + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision], N[(x - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -4.9999999999999996e22

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-neg-in N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-sin.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lift-cos.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.8%

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

   if -4.9999999999999996e22 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 2

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \cos y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos y + \color{blue}{x} \]

      2. lower-+.f64 N/A

         \[\leadsto \cos y + \color{blue}{x} \]

      3. lift-cos.f64 95.7

         \[\leadsto \cos y + x \]

   5. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\cos y + x} \]

   if 2 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y)))

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} - z \cdot \sin y \]
   4. Step-by-step derivation

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{x} - z \cdot \sin y \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, x\right)\\ \mathbf{elif}\;\left(x + \cos y\right) - z \cdot \sin y \leq 2:\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ \mathbf{if}\;x \leq -25:\\ \;\;\;\;x - t\_0\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-9}:\\ \;\;\;\;\cos y - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))))
   (if (<= x -25.0)
     (- x t_0)
     (if (<= x 4.6e-9) (- (cos y) t_0) (fma (sin y) (- z) x)))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double tmp;
	if (x <= -25.0) {
		tmp = x - t_0;
	} else if (x <= 4.6e-9) {
		tmp = cos(y) - t_0;
	} else {
		tmp = fma(sin(y), -z, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	tmp = 0.0
	if (x <= -25.0)
		tmp = Float64(x - t_0);
	elseif (x <= 4.6e-9)
		tmp = Float64(cos(y) - t_0);
	else
		tmp = fma(sin(y), Float64(-z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -25.0], N[(x - t$95$0), $MachinePrecision], If[LessEqual[x, 4.6e-9], N[(N[Cos[y], $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * (-z) + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -25

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} - z \cdot \sin y \]
   4. Step-by-step derivation

   5. Applied rewrites 97.8%

      \[\leadsto \color{blue}{x} - z \cdot \sin y \]

   if -25 < x < 4.5999999999999998e-9

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\cos y} - z \cdot \sin y \]
   4. Step-by-step derivation

      1. lift-cos.f64 98.5

         \[\leadsto \cos y - z \cdot \sin y \]

   5. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\cos y} - z \cdot \sin y \]

   if 4.5999999999999998e-9 < x

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-neg-in N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-sin.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lift-cos.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.6%

      \[\leadsto \mathsf{fma}\left(\sin y, -z, \color{blue}{x}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -25:\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-9}:\\ \;\;\;\;\cos y - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + \cos y\right) - z \cdot \sin y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ x (cos y)) (* z (sin y))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y, z):
	return (x + math.cos(y)) - (z * math.sin(y))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(x + \cos y\right) - z \cdot \sin y \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 7: 81.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+86} \lor \neg \left(z \leq 2.95 \cdot 10^{+72}\right):\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.95e+86) (not (<= z 2.95e+72)))
   (* (- z) (sin y))
   (+ (cos y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.95e+86) || !(z <= 2.95e+72)) {
		tmp = -z * sin(y);
	} else {
		tmp = cos(y) + x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.95d+86)) .or. (.not. (z <= 2.95d+72))) then
        tmp = -z * sin(y)
    else
        tmp = cos(y) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.95e+86) || !(z <= 2.95e+72)) {
		tmp = -z * Math.sin(y);
	} else {
		tmp = Math.cos(y) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.95e+86) or not (z <= 2.95e+72):
		tmp = -z * math.sin(y)
	else:
		tmp = math.cos(y) + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.95e+86) || !(z <= 2.95e+72))
		tmp = Float64(Float64(-z) * sin(y));
	else
		tmp = Float64(cos(y) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.95e+86) || ~((z <= 2.95e+72)))
		tmp = -z * sin(y);
	else
		tmp = cos(y) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.95e+86], N[Not[LessEqual[z, 2.95e+72]], $MachinePrecision]], N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.9500000000000001e86 or 2.9500000000000001e72 < z

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \sin y\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(z \cdot \sin y\right) \]

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-sin.f64 64.6

         \[\leadsto \left(-z\right) \cdot \sin y \]

   5. Applied rewrites 64.6%

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

   if -1.9500000000000001e86 < z < 2.9500000000000001e72

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \cos y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos y + \color{blue}{x} \]

      2. lower-+.f64 N/A

         \[\leadsto \cos y + \color{blue}{x} \]

      3. lift-cos.f64 93.9

         \[\leadsto \cos y + x \]

   5. Applied rewrites 93.9%

      \[\leadsto \color{blue}{\cos y + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+86} \lor \neg \left(z \leq 2.95 \cdot 10^{+72}\right):\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -16500 \lor \neg \left(y \leq 2400000\right):\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.16666666666666666 \cdot \left(z \cdot y\right) - 0.5\right) \cdot y - z, y, x - -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -16500.0) (not (<= y 2400000.0)))
   (+ (cos y) x)
   (fma (- (* (- (* 0.16666666666666666 (* z y)) 0.5) y) z) y (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -16500.0) || !(y <= 2400000.0)) {
		tmp = cos(y) + x;
	} else {
		tmp = fma(((((0.16666666666666666 * (z * y)) - 0.5) * y) - z), y, (x - -1.0));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -16500.0) || !(y <= 2400000.0))
		tmp = Float64(cos(y) + x);
	else
		tmp = fma(Float64(Float64(Float64(Float64(0.16666666666666666 * Float64(z * y)) - 0.5) * y) - z), y, Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -16500.0], N[Not[LessEqual[y, 2400000.0]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * N[(z * y), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -16500 or 2.4e6 < y

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \cos y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos y + \color{blue}{x} \]

      2. lower-+.f64 N/A

         \[\leadsto \cos y + \color{blue}{x} \]

      3. lift-cos.f64 59.6

         \[\leadsto \cos y + x \]

   5. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\cos y + x} \]

   if -16500 < y < 2.4e6

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(x + y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x + y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right)\right) + \color{blue}{1} \]

      2. +-commutative N/A

         \[\leadsto \left(y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right) + x\right) + 1 \]

      3. associate-+l+ N/A

         \[\leadsto y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right) + \color{blue}{\left(x + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right) \cdot y + \left(\color{blue}{x} + 1\right) \]

      5. +-commutative N/A

         \[\leadsto \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right) \cdot y + \left(1 + \color{blue}{x}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z, \color{blue}{y}, 1 + x\right) \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z, y, 1 + x\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x + 1\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x + 1 \cdot 1\right) \]

      16. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x - -1 \cdot 1\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x - -1\right) \]

   5. Applied rewrites 97.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.16666666666666666 \cdot \left(z \cdot y\right) - 0.5\right) \cdot y - z, y, x - -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -16500 \lor \neg \left(y \leq 2400000\right):\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.16666666666666666 \cdot \left(z \cdot y\right) - 0.5\right) \cdot y - z, y, x - -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3100000 \lor \neg \left(y \leq 3000000\right):\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.16666666666666666 \cdot \left(z \cdot y\right) - 0.5\right) \cdot y - z, y, x - -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3100000.0) (not (<= y 3000000.0)))
   (- x -1.0)
   (fma (- (* (- (* 0.16666666666666666 (* z y)) 0.5) y) z) y (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3100000.0) || !(y <= 3000000.0)) {
		tmp = x - -1.0;
	} else {
		tmp = fma(((((0.16666666666666666 * (z * y)) - 0.5) * y) - z), y, (x - -1.0));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3100000.0) || !(y <= 3000000.0))
		tmp = Float64(x - -1.0);
	else
		tmp = fma(Float64(Float64(Float64(Float64(0.16666666666666666 * Float64(z * y)) - 0.5) * y) - z), y, Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3100000.0], N[Not[LessEqual[y, 3000000.0]], $MachinePrecision]], N[(x - -1.0), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * N[(z * y), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.1e6 or 3e6 < y

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \color{blue}{1} \]

      2. metadata-eval N/A

         \[\leadsto x + 1 \cdot \color{blue}{1} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1} \]

      4. metadata-eval N/A

         \[\leadsto x - -1 \cdot 1 \]

      5. metadata-eval N/A

         \[\leadsto x - -1 \]

      6. lower--.f64 40.5

         \[\leadsto x - \color{blue}{-1} \]

   5. Applied rewrites 40.5%

      \[\leadsto \color{blue}{x - -1} \]

   if -3.1e6 < y < 3e6

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(x + y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x + y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right)\right) + \color{blue}{1} \]

      2. +-commutative N/A

         \[\leadsto \left(y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right) + x\right) + 1 \]

      3. associate-+l+ N/A

         \[\leadsto y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right) + \color{blue}{\left(x + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right) \cdot y + \left(\color{blue}{x} + 1\right) \]

      5. +-commutative N/A

         \[\leadsto \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right) \cdot y + \left(1 + \color{blue}{x}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z, \color{blue}{y}, 1 + x\right) \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z, y, 1 + x\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x + 1\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x + 1 \cdot 1\right) \]

      16. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x - -1 \cdot 1\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x - -1\right) \]

   5. Applied rewrites 97.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.16666666666666666 \cdot \left(z \cdot y\right) - 0.5\right) \cdot y - z, y, x - -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3100000 \lor \neg \left(y \leq 3000000\right):\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.16666666666666666 \cdot \left(z \cdot y\right) - 0.5\right) \cdot y - z, y, x - -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.3% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -40000000 \lor \neg \left(y \leq 1.55 \cdot 10^{+63}\right):\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x - -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -40000000.0) (not (<= y 1.55e+63)))
   (- x -1.0)
   (fma (- z) y (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -40000000.0) || !(y <= 1.55e+63)) {
		tmp = x - -1.0;
	} else {
		tmp = fma(-z, y, (x - -1.0));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -40000000.0) || !(y <= 1.55e+63))
		tmp = Float64(x - -1.0);
	else
		tmp = fma(Float64(-z), y, Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -40000000.0], N[Not[LessEqual[y, 1.55e+63]], $MachinePrecision]], N[(x - -1.0), $MachinePrecision], N[((-z) * y + N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4e7 or 1.55e63 < y

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \color{blue}{1} \]

      2. metadata-eval N/A

         \[\leadsto x + 1 \cdot \color{blue}{1} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1} \]

      4. metadata-eval N/A

         \[\leadsto x - -1 \cdot 1 \]

      5. metadata-eval N/A

         \[\leadsto x - -1 \]

      6. lower--.f64 41.1

         \[\leadsto x - \color{blue}{-1} \]

   5. Applied rewrites 41.1%

      \[\leadsto \color{blue}{x - -1} \]

   if -4e7 < y < 1.55e63

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

         \[\leadsto \left(y \cdot \left(\frac{-1}{2} \cdot y - z\right) + x\right) + 1 \]

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot y - z, y, x - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot y - z, y, x - -1 \cdot 1\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot y - z, y, x - -1\right) \]

      14. lower--.f64 92.5

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot y - z, y, x - -1\right) \]

   5. Applied rewrites 92.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot y - z, y, x - -1\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot z, y, x - -1\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(z\right), y, x - -1\right) \]

      2. lift-neg.f64 93.3

         \[\leadsto \mathsf{fma}\left(-z, y, x - -1\right) \]

   8. Applied rewrites 93.3%

      \[\leadsto \mathsf{fma}\left(-z, y, x - -1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -40000000 \lor \neg \left(y \leq 1.55 \cdot 10^{+63}\right):\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x - -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.3% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -40000000 \lor \neg \left(y \leq 1.55 \cdot 10^{+63}\right):\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;\left(x - z \cdot y\right) - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -40000000.0) (not (<= y 1.55e+63)))
   (- x -1.0)
   (- (- x (* z y)) -1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -40000000.0) || !(y <= 1.55e+63)) {
		tmp = x - -1.0;
	} else {
		tmp = (x - (z * y)) - -1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-40000000.0d0)) .or. (.not. (y <= 1.55d+63))) then
        tmp = x - (-1.0d0)
    else
        tmp = (x - (z * y)) - (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -40000000.0) || !(y <= 1.55e+63)) {
		tmp = x - -1.0;
	} else {
		tmp = (x - (z * y)) - -1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -40000000.0) or not (y <= 1.55e+63):
		tmp = x - -1.0
	else:
		tmp = (x - (z * y)) - -1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -40000000.0) || !(y <= 1.55e+63))
		tmp = Float64(x - -1.0);
	else
		tmp = Float64(Float64(x - Float64(z * y)) - -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -40000000.0) || ~((y <= 1.55e+63)))
		tmp = x - -1.0;
	else
		tmp = (x - (z * y)) - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -40000000.0], N[Not[LessEqual[y, 1.55e+63]], $MachinePrecision]], N[(x - -1.0), $MachinePrecision], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4e7 or 1.55e63 < y

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \color{blue}{1} \]

      2. metadata-eval N/A

         \[\leadsto x + 1 \cdot \color{blue}{1} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1} \]

      4. metadata-eval N/A

         \[\leadsto x - -1 \cdot 1 \]

      5. metadata-eval N/A

         \[\leadsto x - -1 \]

      6. lower--.f64 41.1

         \[\leadsto x - \color{blue}{-1} \]

   5. Applied rewrites 41.1%

      \[\leadsto \color{blue}{x - -1} \]

   if -4e7 < y < 1.55e63

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(x + \left(\mathsf{neg}\left(1\right)\right) \cdot \left(y \cdot z\right)\right) - -1 \]

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 93.2

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

   5. Applied rewrites 93.2%

      \[\leadsto \color{blue}{\left(x - z \cdot y\right) - -1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -40000000 \lor \neg \left(y \leq 1.55 \cdot 10^{+63}\right):\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;\left(x - z \cdot y\right) - -1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.7% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-10} \lor \neg \left(x \leq 9 \cdot 10^{-10}\right):\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.6e-10) (not (<= x 9e-10))) (- x -1.0) (fma (- z) y 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.6e-10) || !(x <= 9e-10)) {
		tmp = x - -1.0;
	} else {
		tmp = fma(-z, y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.6e-10) || !(x <= 9e-10))
		tmp = Float64(x - -1.0);
	else
		tmp = fma(Float64(-z), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.6e-10], N[Not[LessEqual[x, 9e-10]], $MachinePrecision]], N[(x - -1.0), $MachinePrecision], N[((-z) * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.60000000000000014e-10 or 8.9999999999999999e-10 < x

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \color{blue}{1} \]

      2. metadata-eval N/A

         \[\leadsto x + 1 \cdot \color{blue}{1} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1} \]

      4. metadata-eval N/A

         \[\leadsto x - -1 \cdot 1 \]

      5. metadata-eval N/A

         \[\leadsto x - -1 \]

      6. lower--.f64 81.6

         \[\leadsto x - \color{blue}{-1} \]

   5. Applied rewrites 81.6%

      \[\leadsto \color{blue}{x - -1} \]

   if -4.60000000000000014e-10 < x < 8.9999999999999999e-10

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

         \[\leadsto \left(y \cdot \left(\frac{-1}{2} \cdot y - z\right) + x\right) + 1 \]

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot y - z, y, x - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot y - z, y, x - -1 \cdot 1\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot y - z, y, x - -1\right) \]

      14. lower--.f64 48.3

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot y - z, y, x - -1\right) \]

   5. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot y - z, y, x - -1\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot z, y, x - -1\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(z\right), y, x - -1\right) \]

      2. lift-neg.f64 50.0

         \[\leadsto \mathsf{fma}\left(-z, y, x - -1\right) \]

   8. Applied rewrites 50.0%

      \[\leadsto \mathsf{fma}\left(-z, y, x - -1\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(-z, y, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 49.9%

       \[\leadsto \mathsf{fma}\left(-z, y, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-10} \lor \neg \left(x \leq 9 \cdot 10^{-10}\right):\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.1% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.38 \cdot 10^{+219}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.38e+219) (* (- z) y) (- x -1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.38e+219) {
		tmp = -z * y;
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.38d+219)) then
        tmp = -z * y
    else
        tmp = x - (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.38e+219) {
		tmp = -z * y;
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.38e+219:
		tmp = -z * y
	else:
		tmp = x - -1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.38e+219)
		tmp = Float64(Float64(-z) * y);
	else
		tmp = Float64(x - -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.38e+219)
		tmp = -z * y;
	else
		tmp = x - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.38e+219], N[((-z) * y), $MachinePrecision], N[(x - -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3800000000000001e219

   1. Initial program 99.6%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \sin y\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(z \cdot \sin y\right) \]

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-sin.f64 82.7

         \[\leadsto \left(-z\right) \cdot \sin y \]

   5. Applied rewrites 82.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(-z\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 50.1%

      \[\leadsto \left(-z\right) \cdot y \]

   if -1.3800000000000001e219 < z

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \color{blue}{1} \]

      2. metadata-eval N/A

         \[\leadsto x + 1 \cdot \color{blue}{1} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1} \]

      4. metadata-eval N/A

         \[\leadsto x - -1 \cdot 1 \]

      5. metadata-eval N/A

         \[\leadsto x - -1 \]

      6. lower--.f64 65.5

         \[\leadsto x - \color{blue}{-1} \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{x - -1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 60.7% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.05e-7) x (if (<= x 4.6e-9) 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.05e-7) {
		tmp = x;
	} else if (x <= 4.6e-9) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.05d-7)) then
        tmp = x
    else if (x <= 4.6d-9) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.05e-7) {
		tmp = x;
	} else if (x <= 4.6e-9) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.05e-7:
		tmp = x
	elif x <= 4.6e-9:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.05e-7)
		tmp = x;
	elseif (x <= 4.6e-9)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.05e-7)
		tmp = x;
	elseif (x <= 4.6e-9)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.05e-7], x, If[LessEqual[x, 4.6e-9], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05e-7 or 4.5999999999999998e-9 < x

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 80.6%

      \[\leadsto \color{blue}{x} \]

   if -1.05e-7 < x < 4.5999999999999998e-9

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \color{blue}{1} \]

      2. metadata-eval N/A

         \[\leadsto x + 1 \cdot \color{blue}{1} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1} \]

      4. metadata-eval N/A

         \[\leadsto x - -1 \cdot 1 \]

      5. metadata-eval N/A

         \[\leadsto x - -1 \]

      6. lower--.f64 36.4

         \[\leadsto x - \color{blue}{-1} \]

   5. Applied rewrites 36.4%

      \[\leadsto \color{blue}{x - -1} \]

   6. Taylor expanded in x around 0

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 35.7%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 61.6% accurate, 53.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- x -1.0))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x - (-1.0d0)
end function

Java

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

Python

def code(x, y, z):
	return x - -1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(x + \cos y\right) - z \cdot \sin y \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1 + x} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x + \color{blue}{1} \]

   2. metadata-eval N/A

      \[\leadsto x + 1 \cdot \color{blue}{1} \]

   3. fp-cancel-sign-sub-inv N/A

      \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1} \]

   4. metadata-eval N/A

      \[\leadsto x - -1 \cdot 1 \]

   5. metadata-eval N/A

      \[\leadsto x - -1 \]

   6. lower--.f64 61.2

      \[\leadsto x - \color{blue}{-1} \]

5. Applied rewrites 61.2%

   \[\leadsto \color{blue}{x - -1} \]
6. Add Preprocessing

Alternative 16: 43.1% accurate, 212.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 x)

C

double code(double x, double y, double z) {
	return 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

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

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.9%

   \[\left(x + \cos y\right) - z \cdot \sin y \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Applied rewrites 45.8%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025085 
(FPCore (x y z)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, B"
  :precision binary64
  (- (+ x (cos y)) (* z (sin y))))