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

Percentage Accurate: 99.9% → 99.9%

Time: 3.1s

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: 73.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \cos y\right) - z \cdot \sin y\\ t_1 := x - \mathsf{fma}\left(z, y, -1\right)\\ \mathbf{if}\;t\_0 \leq -2700000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.99999:\\ \;\;\;\;\cos y\\ \mathbf{elif}\;t\_0 \leq 1.2 \cdot 10^{+188}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (cos y)) (* z (sin y)))) (t_1 (- x (fma z y -1.0))))
   (if (<= t_0 -2700000000000.0)
     t_1
     (if (<= t_0 0.99999) (cos y) (if (<= t_0 1.2e+188) t_1 (- x -1.0))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + cos(y)) - (z * sin(y));
	double t_1 = x - fma(z, y, -1.0);
	double tmp;
	if (t_0 <= -2700000000000.0) {
		tmp = t_1;
	} else if (t_0 <= 0.99999) {
		tmp = cos(y);
	} else if (t_0 <= 1.2e+188) {
		tmp = t_1;
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + cos(y)) - Float64(z * sin(y)))
	t_1 = Float64(x - fma(z, y, -1.0))
	tmp = 0.0
	if (t_0 <= -2700000000000.0)
		tmp = t_1;
	elseif (t_0 <= 0.99999)
		tmp = cos(y);
	elseif (t_0 <= 1.2e+188)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(x - N[(z * y + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2700000000000.0], t$95$1, If[LessEqual[t$95$0, 0.99999], N[Cos[y], $MachinePrecision], If[LessEqual[t$95$0, 1.2e+188], t$95$1, N[(x - -1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -2.7e12 or 0.999990000000000046 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 1.2e188

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(x + -1 \cdot \left(y \cdot z\right)\right)} \]
   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 77.3

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

   5. Applied rewrites 77.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 77.3

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

   7. Applied rewrites 77.3%

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

   if -2.7e12 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.999990000000000046

   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 94.7

         \[\leadsto \cos y + x \]

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in x around 0

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

      1. lift-cos.f64 94.7

         \[\leadsto \cos y \]

   8. Applied rewrites 94.7%

      \[\leadsto \cos y \]

   if 1.2e188 < (-.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 61.6

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

   5. Applied rewrites 61.6%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -2700000000000:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \mathbf{elif}\;\left(x + \cos y\right) - z \cdot \sin y \leq 0.99999:\\ \;\;\;\;\cos y\\ \mathbf{elif}\;\left(x + \cos y\right) - z \cdot \sin y \leq 1.2 \cdot 10^{+188}:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ t_1 := \left(x + \cos y\right) - t\_0\\ \mathbf{if}\;t\_1 \leq -2000000000000 \lor \neg \left(t\_1 \leq 10^{+14}\right):\\ \;\;\;\;x - t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))) (t_1 (- (+ x (cos y)) t_0)))
   (if (or (<= t_1 -2000000000000.0) (not (<= t_1 1e+14)))
     (- 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 <= -2000000000000.0) || !(t_1 <= 1e+14)) {
		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 <= (-2000000000000.0d0)) .or. (.not. (t_1 <= 1d+14))) 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 <= -2000000000000.0) || !(t_1 <= 1e+14)) {
		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 <= -2000000000000.0) or not (t_1 <= 1e+14):
		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 <= -2000000000000.0) || !(t_1 <= 1e+14))
		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 <= -2000000000000.0) || ~((t_1 <= 1e+14)))
		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, -2000000000000.0], N[Not[LessEqual[t$95$1, 1e+14]], $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))) < -2e12 or 1e14 < (-.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 99.6%

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

   if -2e12 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 1e14

   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 98.2

         \[\leadsto \cos y + x \]

   5. Applied rewrites 98.2%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -2000000000000 \lor \neg \left(\left(x + \cos y\right) - z \cdot \sin y \leq 10^{+14}\right):\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ t_1 := \left(x + \cos y\right) - t\_0\\ \mathbf{if}\;t\_1 \leq -2000000000000:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+14}:\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;x - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))) (t_1 (- (+ x (cos y)) t_0)))
   (if (<= t_1 -2000000000000.0)
     (fma (sin y) (- z) x)
     (if (<= t_1 1e+14) (+ (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 <= -2000000000000.0) {
		tmp = fma(sin(y), -z, x);
	} else if (t_1 <= 1e+14) {
		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 <= -2000000000000.0)
		tmp = fma(sin(y), Float64(-z), x);
	elseif (t_1 <= 1e+14)
		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, -2000000000000.0], N[(N[Sin[y], $MachinePrecision] * (-z) + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+14], 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))) < -2e12

   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. 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 99.3%

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

   if -2e12 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 1e14

   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 98.2

         \[\leadsto \cos y + x \]

   5. Applied rewrites 98.2%

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

   if 1e14 < (-.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 99.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -2000000000000:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, x\right)\\ \mathbf{elif}\;\left(x + \cos y\right) - z \cdot \sin y \leq 10^{+14}:\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;x - t\_0\\ \mathbf{elif}\;x \leq 0.122:\\ \;\;\;\;\cos y - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))))
   (if (<= x -1.0)
     (- x t_0)
     (if (<= x 0.122) (- (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 <= -1.0) {
		tmp = x - t_0;
	} else if (x <= 0.122) {
		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 <= -1.0)
		tmp = Float64(x - t_0);
	elseif (x <= 0.122)
		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, -1.0], N[(x - t$95$0), $MachinePrecision], If[LessEqual[x, 0.122], 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 < -1

   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 99.7%

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

   if -1 < x < 0.122

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

   if 0.122 < x

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   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 98.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{elif}\;x \leq 0.122:\\ \;\;\;\;\cos y - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, x\right)\\ \end{array} \]
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: 80.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+206} \lor \neg \left(z \leq 1.45 \cdot 10^{+66}\right):\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.5e+206) (not (<= z 1.45e+66)))
   (* (- z) (sin y))
   (+ (cos y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.5e+206) || !(z <= 1.45e+66)) {
		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 <= (-5.5d+206)) .or. (.not. (z <= 1.45d+66))) 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 <= -5.5e+206) || !(z <= 1.45e+66)) {
		tmp = -z * Math.sin(y);
	} else {
		tmp = Math.cos(y) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -5.5e+206) or not (z <= 1.45e+66):
		tmp = -z * math.sin(y)
	else:
		tmp = math.cos(y) + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.5e+206) || !(z <= 1.45e+66))
		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 <= -5.5e+206) || ~((z <= 1.45e+66)))
		tmp = -z * sin(y);
	else
		tmp = cos(y) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.5e+206], N[Not[LessEqual[z, 1.45e+66]], $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 < -5.50000000000000021e206 or 1.44999999999999993e66 < 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 66.9

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

   5. Applied rewrites 66.9%

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

   if -5.50000000000000021e206 < z < 1.44999999999999993e66

   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 90.6

         \[\leadsto \cos y + x \]

   5. Applied rewrites 90.6%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+206} \lor \neg \left(z \leq 1.45 \cdot 10^{+66}\right):\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+30} \lor \neg \left(y \leq 0.086\right):\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.16666666666666666 \cdot \left(z \cdot y\right) - 0.5\right) \cdot y - z, y, x - -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5.8e+30) (not (<= y 0.086)))
   (+ (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 <= -5.8e+30) || !(y <= 0.086)) {
		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 <= -5.8e+30) || !(y <= 0.086))
		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, -5.8e+30], N[Not[LessEqual[y, 0.086]], $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 < -5.7999999999999996e30 or 0.085999999999999993 < y

   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 59.1

         \[\leadsto \cos y + x \]

   5. Applied rewrites 59.1%

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

   if -5.7999999999999996e30 < y < 0.085999999999999993

   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 99.7%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+30} \lor \neg \left(y \leq 0.086\right):\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.16666666666666666 \cdot \left(z \cdot y\right) - 0.5\right) \cdot y - z, y, x - -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+57} \lor \neg \left(y \leq 4.6\right):\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.16666666666666666 \cdot \left(z \cdot y\right) - 0.5\right) \cdot y - z, y, x - -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.7e+57) (not (<= y 4.6)))
   (- 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 <= -1.7e+57) || !(y <= 4.6)) {
		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 <= -1.7e+57) || !(y <= 4.6))
		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, -1.7e+57], N[Not[LessEqual[y, 4.6]], $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 < -1.69999999999999996e57 or 4.5999999999999996 < y

   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 39.1

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

   5. Applied rewrites 39.1%

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

   if -1.69999999999999996e57 < y < 4.5999999999999996

   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 94.8%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+57} \lor \neg \left(y \leq 4.6\right):\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.16666666666666666 \cdot \left(z \cdot y\right) - 0.5\right) \cdot y - z, y, x - -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+55} \lor \neg \left(y \leq 2.9 \cdot 10^{+36}\right):\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(0.16666666666666666 \cdot y\right) \cdot z\right) \cdot y - z, y, x - -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -7.8e+55) (not (<= y 2.9e+36)))
   (- x -1.0)
   (fma (- (* (* (* 0.16666666666666666 y) z) y) z) y (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7.8e+55) || !(y <= 2.9e+36)) {
		tmp = x - -1.0;
	} else {
		tmp = fma(((((0.16666666666666666 * y) * z) * y) - z), y, (x - -1.0));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -7.8e+55) || !(y <= 2.9e+36))
		tmp = Float64(x - -1.0);
	else
		tmp = fma(Float64(Float64(Float64(Float64(0.16666666666666666 * y) * z) * y) - z), y, Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -7.8e+55], N[Not[LessEqual[y, 2.9e+36]], $MachinePrecision]], N[(x - -1.0), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * y), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.80000000000000054e55 or 2.9e36 < y

   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 39.6

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

   5. Applied rewrites 39.6%

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

   if -7.80000000000000054e55 < y < 2.9e36

   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 91.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.16666666666666666 \cdot \left(z \cdot y\right) - 0.5\right) \cdot y - z, y, x - -1\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right)\right) \cdot y - z, y, x - -1\right) \]
   7. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 91.9

         \[\leadsto \mathsf{fma}\left(\left(\left(0.16666666666666666 \cdot y\right) \cdot z\right) \cdot y - z, y, x - -1\right) \]

   8. Applied rewrites 91.9%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+55} \lor \neg \left(y \leq 2.9 \cdot 10^{+36}\right):\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(0.16666666666666666 \cdot y\right) \cdot z\right) \cdot y - z, y, x - -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+32} \lor \neg \left(y \leq 4.6\right):\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - z, y, x - -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.25e+32) (not (<= y 4.6)))
   (- x -1.0)
   (fma (- (* -0.5 y) z) y (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.25e+32) || !(y <= 4.6)) {
		tmp = x - -1.0;
	} else {
		tmp = fma(((-0.5 * y) - z), y, (x - -1.0));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.25e+32) || !(y <= 4.6))
		tmp = Float64(x - -1.0);
	else
		tmp = fma(Float64(Float64(-0.5 * y) - z), y, Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.25e+32], N[Not[LessEqual[y, 4.6]], $MachinePrecision]], N[(x - -1.0), $MachinePrecision], N[(N[(N[(-0.5 * y), $MachinePrecision] - z), $MachinePrecision] * y + N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2499999999999999e32 or 4.5999999999999996 < y

   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 37.8

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

   5. Applied rewrites 37.8%

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

   if -1.2499999999999999e32 < y < 4.5999999999999996

   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 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+32} \lor \neg \left(y \leq 4.6\right):\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - z, y, x - -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.6% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+51} \lor \neg \left(y \leq 1.7 \cdot 10^{+26}\right):\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2e+51) (not (<= y 1.7e+26))) (- x -1.0) (- x (fma z y -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2e+51) || !(y <= 1.7e+26)) {
		tmp = x - -1.0;
	} else {
		tmp = x - fma(z, y, -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2e+51) || !(y <= 1.7e+26))
		tmp = Float64(x - -1.0);
	else
		tmp = Float64(x - fma(z, y, -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2e+51], N[Not[LessEqual[y, 1.7e+26]], $MachinePrecision]], N[(x - -1.0), $MachinePrecision], N[(x - N[(z * y + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2e51 or 1.7000000000000001e26 < y

   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 38.7

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

   5. Applied rewrites 38.7%

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

   if -2e51 < y < 1.7000000000000001e26

   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 94.0

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

   5. Applied rewrites 94.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 94.0

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

   7. Applied rewrites 94.0%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+51} \lor \neg \left(y \leq 1.7 \cdot 10^{+26}\right):\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.2% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9500000:\\ \;\;\;\;1 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.5e+16) x (if (<= x 9500000.0) (- 1.0 (* z y)) (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.5e+16) {
		tmp = x;
	} else if (x <= 9500000.0) {
		tmp = 1.0 - (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 (x <= (-5.5d+16)) then
        tmp = x
    else if (x <= 9500000.0d0) then
        tmp = 1.0d0 - (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 (x <= -5.5e+16) {
		tmp = x;
	} else if (x <= 9500000.0) {
		tmp = 1.0 - (z * y);
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.5e+16:
		tmp = x
	elif x <= 9500000.0:
		tmp = 1.0 - (z * y)
	else:
		tmp = x - -1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.5e+16)
		tmp = x;
	elseif (x <= 9500000.0)
		tmp = Float64(1.0 - 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 (x <= -5.5e+16)
		tmp = x;
	elseif (x <= 9500000.0)
		tmp = 1.0 - (z * y);
	else
		tmp = x - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.5e+16], x, If[LessEqual[x, 9500000.0], N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.5e16

   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 84.4%

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

   if -5.5e16 < x < 9.5e6

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(x + -1 \cdot \left(y \cdot z\right)\right)} \]
   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 57.8

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 - \color{blue}{y \cdot z} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 - y \cdot \color{blue}{z} \]

      2. *-commutative N/A

         \[\leadsto 1 - z \cdot y \]

      3. lift-*.f64 56.2

         \[\leadsto 1 - z \cdot y \]

   8. Applied rewrites 56.2%

      \[\leadsto 1 - \color{blue}{z \cdot y} \]

   if 9.5e6 < x

   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 83.1

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

   5. Applied rewrites 83.1%

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

Alternative 14: 61.9% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.6 \cdot 10^{+212}:\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2.6e+212) (- x -1.0) (* (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.6e+212) {
		tmp = x - -1.0;
	} else {
		tmp = -z * y;
	}
	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 <= 2.6d+212) then
        tmp = x - (-1.0d0)
    else
        tmp = -z * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.6e+212) {
		tmp = x - -1.0;
	} else {
		tmp = -z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.6e+212:
		tmp = x - -1.0
	else:
		tmp = -z * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.6e+212)
		tmp = Float64(x - -1.0);
	else
		tmp = Float64(Float64(-z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2.6e+212)
		tmp = x - -1.0;
	else
		tmp = -z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.6e+212], N[(x - -1.0), $MachinePrecision], N[((-z) * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.5999999999999998e212

   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 66.4

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

   5. Applied rewrites 66.4%

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

   if 2.5999999999999998e212 < 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 82.9

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

   5. Applied rewrites 82.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 38.0%

      \[\leadsto \left(-z\right) \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 61.4% 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 62.3

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

5. Applied rewrites 62.3%

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

Alternative 16: 42.2% 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 38.7%

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

Reproduce

herbie shell --seed 2025080 
(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))))