Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.0% → 99.8%

Time: 4.2s

Alternatives: 15

Speedup: 0.7×

• 64.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\cosh x \cdot \frac{y}{x}}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* (cosh x) (/ y x)) z))

C

double code(double x, double y, double z) {
	return (cosh(x) * (y / x)) / z;
}

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 = (cosh(x) * (y / x)) / z
end function

Java

public static double code(double x, double y, double z) {
	return (Math.cosh(x) * (y / x)) / z;
}

Python

def code(x, y, z):
	return (math.cosh(x) * (y / x)) / z

Julia

function code(x, y, z)
	return Float64(Float64(cosh(x) * Float64(y / x)) / z)
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Initial Program: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\cosh x \cdot \frac{y}{x}}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* (cosh x) (/ y x)) z))

C

double code(double x, double y, double z) {
	return (cosh(x) * (y / x)) / z;
}

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 = (cosh(x) * (y / x)) / z
end function

Java

public static double code(double x, double y, double z) {
	return (Math.cosh(x) * (y / x)) / z;
}

Python

def code(x, y, z):
	return (math.cosh(x) * (y / x)) / z

Julia

function code(x, y, z)
	return Float64(Float64(cosh(x) * Float64(y / x)) / z)
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\cosh x\_m}{z\_m \cdot x\_m} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x\_m \cdot y\_m}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 5e-16)
      (* (/ (cosh x_m) (* z_m x_m)) y_m)
      (/ (/ (* (cosh x_m) y_m) z_m) x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5e-16) {
		tmp = (cosh(x_m) / (z_m * x_m)) * y_m;
	} else {
		tmp = ((cosh(x_m) * y_m) / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
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(z_s, y_s, x_s, x_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5d-16) then
        tmp = (cosh(x_m) / (z_m * x_m)) * y_m
    else
        tmp = ((cosh(x_m) * y_m) / z_m) / x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((Math.cosh(x_m) * (y_m / x_m)) / z_m) <= 5e-16) {
		tmp = (Math.cosh(x_m) / (z_m * x_m)) * y_m;
	} else {
		tmp = ((Math.cosh(x_m) * y_m) / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if ((math.cosh(x_m) * (y_m / x_m)) / z_m) <= 5e-16:
		tmp = (math.cosh(x_m) / (z_m * x_m)) * y_m
	else:
		tmp = ((math.cosh(x_m) * y_m) / z_m) / x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 5e-16)
		tmp = Float64(Float64(cosh(x_m) / Float64(z_m * x_m)) * y_m);
	else
		tmp = Float64(Float64(Float64(cosh(x_m) * y_m) / z_m) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5e-16)
		tmp = (cosh(x_m) / (z_m * x_m)) * y_m;
	else
		tmp = ((cosh(x_m) * y_m) / z_m) / x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 5e-16], N[(N[(N[Cosh[x$95$m], $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(N[Cosh[x$95$m], $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 5.0000000000000004e-16

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x \cdot \frac{y}{x}}}{z} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      5. lift-/.f64 N/A

         \[\leadsto \cosh x \cdot \frac{\color{blue}{\frac{y}{x}}}{z} \]

      6. associate-/r* N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

      7. div-flip N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{1}{\frac{x \cdot z}{y}}} \]

      8. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{\frac{x \cdot z}{y}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{\frac{x \cdot z}{y}}} \]

      10. lift-cosh.f64 N/A

          \[\leadsto \frac{\color{blue}{\cosh x}}{\frac{x \cdot z}{y}} \]

      11. associate-/l* N/A

          \[\leadsto \frac{\cosh x}{\color{blue}{x \cdot \frac{z}{y}}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\cosh x}{\color{blue}{x \cdot \frac{z}{y}}} \]

      13. lower-/.f64 81.0

          \[\leadsto \frac{\cosh x}{x \cdot \color{blue}{\frac{z}{y}}} \]

   3. Applied rewrites 81.0%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot \frac{z}{y}}} \]

      2. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x}}{x \cdot \frac{z}{y}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cosh x}{\color{blue}{x \cdot \frac{z}{y}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\cosh x}{x \cdot \color{blue}{\frac{z}{y}}} \]

      5. associate-/l* N/A

         \[\leadsto \frac{\cosh x}{\color{blue}{\frac{x \cdot z}{y}}} \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot z} \cdot y} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot z} \cdot y} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot z}} \cdot y \]

      9. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x}}{x \cdot z} \cdot y \]

      10. *-commutative N/A

          \[\leadsto \frac{\cosh x}{\color{blue}{z \cdot x}} \cdot y \]

      11. lift-*.f64 82.7

          \[\leadsto \frac{\cosh x}{\color{blue}{z \cdot x}} \cdot y \]

   5. Applied rewrites 82.7%

      \[\leadsto \color{blue}{\frac{\cosh x}{z \cdot x} \cdot y} \]

   if 5.0000000000000004e-16 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x \cdot \frac{y}{x}}}{z} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{y}{x}}}{z} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]

      6. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]

      7. cosh-def N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}} \cdot y}{x \cdot z} \]

      8. rec-exp N/A

         \[\leadsto \frac{\frac{e^{x} + \color{blue}{\frac{1}{e^{x}}}}{2} \cdot y}{x \cdot z} \]

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

   3. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\frac{\frac{\cosh x \cdot y}{z}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 94.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m}\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)}{z\_m}}{x\_m} \cdot y\_m\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (/ (* (cosh x_m) (/ y_m x_m)) z_m)))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= t_0 INFINITY)
        t_0
        (* (/ (/ (fma (* x_m x_m) 0.5 1.0) z_m) x_m) y_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = (cosh(x_m) * (y_m / x_m)) / z_m;
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = ((fma((x_m * x_m), 0.5, 1.0) / z_m) / x_m) * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	t_0 = Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m)
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) / z_m) / x_m) * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, Infinity], t$95$0, N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < +inf.0

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   if +inf.0 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x \cdot \frac{y}{x}}}{z} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      5. lift-/.f64 N/A

         \[\leadsto \cosh x \cdot \frac{\color{blue}{\frac{y}{x}}}{z} \]

      6. associate-/r* N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

      7. div-flip N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{1}{\frac{x \cdot z}{y}}} \]

      8. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{\frac{x \cdot z}{y}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{\frac{x \cdot z}{y}}} \]

      10. lift-cosh.f64 N/A

          \[\leadsto \frac{\color{blue}{\cosh x}}{\frac{x \cdot z}{y}} \]

      11. associate-/l* N/A

          \[\leadsto \frac{\cosh x}{\color{blue}{x \cdot \frac{z}{y}}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\cosh x}{\color{blue}{x \cdot \frac{z}{y}}} \]

      13. lower-/.f64 81.0

          \[\leadsto \frac{\cosh x}{x \cdot \color{blue}{\frac{z}{y}}} \]

   3. Applied rewrites 81.0%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot \frac{z}{y}}} \]

      2. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x}}{x \cdot \frac{z}{y}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cosh x}{\color{blue}{x \cdot \frac{z}{y}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\cosh x}{x \cdot \color{blue}{\frac{z}{y}}} \]

      5. associate-/l* N/A

         \[\leadsto \frac{\cosh x}{\color{blue}{\frac{x \cdot z}{y}}} \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot z} \cdot y} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot z} \cdot y} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot z}} \cdot y \]

      9. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x}}{x \cdot z} \cdot y \]

      10. *-commutative N/A

          \[\leadsto \frac{\cosh x}{\color{blue}{z \cdot x}} \cdot y \]

      11. lift-*.f64 82.7

          \[\leadsto \frac{\cosh x}{\color{blue}{z \cdot x}} \cdot y \]

   5. Applied rewrites 82.7%

      \[\leadsto \color{blue}{\frac{\cosh x}{z \cdot x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{{x}^{2}}{z} + \frac{1}{z}}{x}} \cdot y \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \frac{{x}^{2}}{z} + \frac{1}{z}}{\color{blue}{x}} \cdot y \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{1}{2} \cdot {x}^{2}}{z} + \frac{1}{z}}{x} \cdot y \]

      3. div-add-rev N/A

         \[\leadsto \frac{\frac{\frac{1}{2} \cdot {x}^{2} + 1}{z}}{x} \cdot y \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \frac{1}{2} \cdot {x}^{2}}{z}}{x} \cdot y \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1 + \frac{1}{2} \cdot {x}^{2}}{z}}{x} \cdot y \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{1}{2} \cdot {x}^{2} + 1}{z}}{x} \cdot y \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{{x}^{2} \cdot \frac{1}{2} + 1}{z}}{x} \cdot y \]

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 79.4

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

   8. Applied rewrites 79.4%

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

Alternative 3: 92.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.25 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m}}{x\_m}\\ \mathbf{elif}\;x\_m \leq 10^{+116}:\\ \;\;\;\;\frac{\cosh x\_m \cdot y\_m}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(x\_m \cdot x\_m\right) \cdot y\_m\right) \cdot 0.5}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 2.25e-72)
      (/ (/ y_m z_m) x_m)
      (if (<= x_m 1e+116)
        (/ (* (cosh x_m) y_m) (* z_m x_m))
        (/ (/ (* (* (* x_m x_m) y_m) 0.5) z_m) x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 2.25e-72) {
		tmp = (y_m / z_m) / x_m;
	} else if (x_m <= 1e+116) {
		tmp = (cosh(x_m) * y_m) / (z_m * x_m);
	} else {
		tmp = ((((x_m * x_m) * y_m) * 0.5) / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
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(z_s, y_s, x_s, x_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 2.25d-72) then
        tmp = (y_m / z_m) / x_m
    else if (x_m <= 1d+116) then
        tmp = (cosh(x_m) * y_m) / (z_m * x_m)
    else
        tmp = ((((x_m * x_m) * y_m) * 0.5d0) / z_m) / x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 2.25e-72) {
		tmp = (y_m / z_m) / x_m;
	} else if (x_m <= 1e+116) {
		tmp = (Math.cosh(x_m) * y_m) / (z_m * x_m);
	} else {
		tmp = ((((x_m * x_m) * y_m) * 0.5) / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 2.25e-72:
		tmp = (y_m / z_m) / x_m
	elif x_m <= 1e+116:
		tmp = (math.cosh(x_m) * y_m) / (z_m * x_m)
	else:
		tmp = ((((x_m * x_m) * y_m) * 0.5) / z_m) / x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 2.25e-72)
		tmp = Float64(Float64(y_m / z_m) / x_m);
	elseif (x_m <= 1e+116)
		tmp = Float64(Float64(cosh(x_m) * y_m) / Float64(z_m * x_m));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x_m * x_m) * y_m) * 0.5) / z_m) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 2.25e-72)
		tmp = (y_m / z_m) / x_m;
	elseif (x_m <= 1e+116)
		tmp = (cosh(x_m) * y_m) / (z_m * x_m);
	else
		tmp = ((((x_m * x_m) * y_m) * 0.5) / z_m) / x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 2.25e-72], N[(N[(y$95$m / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], If[LessEqual[x$95$m, 1e+116], N[(N[(N[Cosh[x$95$m], $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 2.25e-72

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x \cdot \frac{y}{x}}}{z} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{y}{x}}}{z} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]

      6. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]

      7. cosh-def N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}} \cdot y}{x \cdot z} \]

      8. rec-exp N/A

         \[\leadsto \frac{\frac{e^{x} + \color{blue}{\frac{1}{e^{x}}}}{2} \cdot y}{x \cdot z} \]

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

   3. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\frac{\frac{\cosh x \cdot y}{z}}{x}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\color{blue}{y}}{z}}{x} \]
   5. Step-by-step derivation

   6. Applied rewrites 52.8%

      \[\leadsto \frac{\frac{\color{blue}{y}}{z}}{x} \]

   if 2.25e-72 < x < 1.00000000000000002e116

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x \cdot \frac{y}{x}}}{z} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{y}{x}}}{z} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]

      6. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x \cdot y}}{x \cdot z} \]

      9. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot y}{x \cdot z} \]

      10. *-commutative N/A

          \[\leadsto \frac{\cosh x \cdot y}{\color{blue}{z \cdot x}} \]

      11. lower-*.f64 83.0

          \[\leadsto \frac{\cosh x \cdot y}{\color{blue}{z \cdot x}} \]

   3. Applied rewrites 83.0%

      \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}} \]

   if 1.00000000000000002e116 < x

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{\color{blue}{x}} \]

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 81.4

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

   4. Applied rewrites 81.4%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 41.4

         \[\leadsto \frac{\frac{\left(\left(x \cdot x\right) \cdot y\right) \cdot 0.5}{z}}{x} \]

   7. Applied rewrites 41.4%

      \[\leadsto \frac{\frac{\left(\left(x \cdot x\right) \cdot y\right) \cdot 0.5}{z}}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 92.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.25 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m}}{x\_m}\\ \mathbf{elif}\;x\_m \leq 10^{+116}:\\ \;\;\;\;\frac{\cosh x\_m}{z\_m \cdot x\_m} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(x\_m \cdot x\_m\right) \cdot y\_m\right) \cdot 0.5}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 2.25e-72)
      (/ (/ y_m z_m) x_m)
      (if (<= x_m 1e+116)
        (* (/ (cosh x_m) (* z_m x_m)) y_m)
        (/ (/ (* (* (* x_m x_m) y_m) 0.5) z_m) x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 2.25e-72) {
		tmp = (y_m / z_m) / x_m;
	} else if (x_m <= 1e+116) {
		tmp = (cosh(x_m) / (z_m * x_m)) * y_m;
	} else {
		tmp = ((((x_m * x_m) * y_m) * 0.5) / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
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(z_s, y_s, x_s, x_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 2.25d-72) then
        tmp = (y_m / z_m) / x_m
    else if (x_m <= 1d+116) then
        tmp = (cosh(x_m) / (z_m * x_m)) * y_m
    else
        tmp = ((((x_m * x_m) * y_m) * 0.5d0) / z_m) / x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 2.25e-72) {
		tmp = (y_m / z_m) / x_m;
	} else if (x_m <= 1e+116) {
		tmp = (Math.cosh(x_m) / (z_m * x_m)) * y_m;
	} else {
		tmp = ((((x_m * x_m) * y_m) * 0.5) / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 2.25e-72:
		tmp = (y_m / z_m) / x_m
	elif x_m <= 1e+116:
		tmp = (math.cosh(x_m) / (z_m * x_m)) * y_m
	else:
		tmp = ((((x_m * x_m) * y_m) * 0.5) / z_m) / x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 2.25e-72)
		tmp = Float64(Float64(y_m / z_m) / x_m);
	elseif (x_m <= 1e+116)
		tmp = Float64(Float64(cosh(x_m) / Float64(z_m * x_m)) * y_m);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x_m * x_m) * y_m) * 0.5) / z_m) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 2.25e-72)
		tmp = (y_m / z_m) / x_m;
	elseif (x_m <= 1e+116)
		tmp = (cosh(x_m) / (z_m * x_m)) * y_m;
	else
		tmp = ((((x_m * x_m) * y_m) * 0.5) / z_m) / x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 2.25e-72], N[(N[(y$95$m / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], If[LessEqual[x$95$m, 1e+116], N[(N[(N[Cosh[x$95$m], $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 2.25e-72

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x \cdot \frac{y}{x}}}{z} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{y}{x}}}{z} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]

      6. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]

      7. cosh-def N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}} \cdot y}{x \cdot z} \]

      8. rec-exp N/A

         \[\leadsto \frac{\frac{e^{x} + \color{blue}{\frac{1}{e^{x}}}}{2} \cdot y}{x \cdot z} \]

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

   3. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\frac{\frac{\cosh x \cdot y}{z}}{x}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\color{blue}{y}}{z}}{x} \]
   5. Step-by-step derivation

   6. Applied rewrites 52.8%

      \[\leadsto \frac{\frac{\color{blue}{y}}{z}}{x} \]

   if 2.25e-72 < x < 1.00000000000000002e116

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x \cdot \frac{y}{x}}}{z} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      5. lift-/.f64 N/A

         \[\leadsto \cosh x \cdot \frac{\color{blue}{\frac{y}{x}}}{z} \]

      6. associate-/r* N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

      7. div-flip N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{1}{\frac{x \cdot z}{y}}} \]

      8. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{\frac{x \cdot z}{y}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{\frac{x \cdot z}{y}}} \]

      10. lift-cosh.f64 N/A

          \[\leadsto \frac{\color{blue}{\cosh x}}{\frac{x \cdot z}{y}} \]

      11. associate-/l* N/A

          \[\leadsto \frac{\cosh x}{\color{blue}{x \cdot \frac{z}{y}}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\cosh x}{\color{blue}{x \cdot \frac{z}{y}}} \]

      13. lower-/.f64 81.0

          \[\leadsto \frac{\cosh x}{x \cdot \color{blue}{\frac{z}{y}}} \]

   3. Applied rewrites 81.0%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot \frac{z}{y}}} \]

      2. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x}}{x \cdot \frac{z}{y}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cosh x}{\color{blue}{x \cdot \frac{z}{y}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\cosh x}{x \cdot \color{blue}{\frac{z}{y}}} \]

      5. associate-/l* N/A

         \[\leadsto \frac{\cosh x}{\color{blue}{\frac{x \cdot z}{y}}} \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot z} \cdot y} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot z} \cdot y} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot z}} \cdot y \]

      9. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x}}{x \cdot z} \cdot y \]

      10. *-commutative N/A

          \[\leadsto \frac{\cosh x}{\color{blue}{z \cdot x}} \cdot y \]

      11. lift-*.f64 82.7

          \[\leadsto \frac{\cosh x}{\color{blue}{z \cdot x}} \cdot y \]

   5. Applied rewrites 82.7%

      \[\leadsto \color{blue}{\frac{\cosh x}{z \cdot x} \cdot y} \]

   if 1.00000000000000002e116 < x

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{\color{blue}{x}} \]

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 81.4

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

   4. Applied rewrites 81.4%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 41.4

         \[\leadsto \frac{\frac{\left(\left(x \cdot x\right) \cdot y\right) \cdot 0.5}{z}}{x} \]

   7. Applied rewrites 41.4%

      \[\leadsto \frac{\frac{\left(\left(x \cdot x\right) \cdot y\right) \cdot 0.5}{z}}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 90.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.25 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m}}{x\_m}\\ \mathbf{elif}\;x\_m \leq 10^{+116}:\\ \;\;\;\;\frac{y\_m}{z\_m \cdot x\_m} \cdot \cosh x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(x\_m \cdot x\_m\right) \cdot y\_m\right) \cdot 0.5}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 2.25e-72)
      (/ (/ y_m z_m) x_m)
      (if (<= x_m 1e+116)
        (* (/ y_m (* z_m x_m)) (cosh x_m))
        (/ (/ (* (* (* x_m x_m) y_m) 0.5) z_m) x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 2.25e-72) {
		tmp = (y_m / z_m) / x_m;
	} else if (x_m <= 1e+116) {
		tmp = (y_m / (z_m * x_m)) * cosh(x_m);
	} else {
		tmp = ((((x_m * x_m) * y_m) * 0.5) / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
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(z_s, y_s, x_s, x_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 2.25d-72) then
        tmp = (y_m / z_m) / x_m
    else if (x_m <= 1d+116) then
        tmp = (y_m / (z_m * x_m)) * cosh(x_m)
    else
        tmp = ((((x_m * x_m) * y_m) * 0.5d0) / z_m) / x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 2.25e-72) {
		tmp = (y_m / z_m) / x_m;
	} else if (x_m <= 1e+116) {
		tmp = (y_m / (z_m * x_m)) * Math.cosh(x_m);
	} else {
		tmp = ((((x_m * x_m) * y_m) * 0.5) / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 2.25e-72:
		tmp = (y_m / z_m) / x_m
	elif x_m <= 1e+116:
		tmp = (y_m / (z_m * x_m)) * math.cosh(x_m)
	else:
		tmp = ((((x_m * x_m) * y_m) * 0.5) / z_m) / x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 2.25e-72)
		tmp = Float64(Float64(y_m / z_m) / x_m);
	elseif (x_m <= 1e+116)
		tmp = Float64(Float64(y_m / Float64(z_m * x_m)) * cosh(x_m));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x_m * x_m) * y_m) * 0.5) / z_m) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 2.25e-72)
		tmp = (y_m / z_m) / x_m;
	elseif (x_m <= 1e+116)
		tmp = (y_m / (z_m * x_m)) * cosh(x_m);
	else
		tmp = ((((x_m * x_m) * y_m) * 0.5) / z_m) / x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 2.25e-72], N[(N[(y$95$m / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], If[LessEqual[x$95$m, 1e+116], N[(N[(y$95$m / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 2.25e-72

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x \cdot \frac{y}{x}}}{z} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{y}{x}}}{z} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]

      6. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]

      7. cosh-def N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}} \cdot y}{x \cdot z} \]

      8. rec-exp N/A

         \[\leadsto \frac{\frac{e^{x} + \color{blue}{\frac{1}{e^{x}}}}{2} \cdot y}{x \cdot z} \]

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

   3. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\frac{\frac{\cosh x \cdot y}{z}}{x}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\color{blue}{y}}{z}}{x} \]
   5. Step-by-step derivation

   6. Applied rewrites 52.8%

      \[\leadsto \frac{\frac{\color{blue}{y}}{z}}{x} \]

   if 2.25e-72 < x < 1.00000000000000002e116

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x \cdot \frac{y}{x}}}{z} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      5. lift-/.f64 N/A

         \[\leadsto \cosh x \cdot \frac{\color{blue}{\frac{y}{x}}}{z} \]

      6. associate-/r* N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

      7. *-commutative N/A

         \[\leadsto \cosh x \cdot \frac{y}{\color{blue}{z \cdot x}} \]

      8. associate-/l/ N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{\frac{y}{z}}{x}} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x} \cdot \cosh x} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x} \cdot \cosh x} \]

      11. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{y}{z \cdot x}} \cdot \cosh x \]

      12. *-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \cdot \cosh x \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \cdot \cosh x \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \cdot \cosh x \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \cdot \cosh x \]

      16. lift-cosh.f64 77.6

          \[\leadsto \frac{y}{z \cdot x} \cdot \color{blue}{\cosh x} \]

   3. Applied rewrites 77.6%

      \[\leadsto \color{blue}{\frac{y}{z \cdot x} \cdot \cosh x} \]

   if 1.00000000000000002e116 < x

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{\color{blue}{x}} \]

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 81.4

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

   4. Applied rewrites 81.4%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 41.4

         \[\leadsto \frac{\frac{\left(\left(x \cdot x\right) \cdot y\right) \cdot 0.5}{z}}{x} \]

   7. Applied rewrites 41.4%

      \[\leadsto \frac{\frac{\left(\left(x \cdot x\right) \cdot y\right) \cdot 0.5}{z}}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 10^{-127}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)}{z\_m}}{x\_m} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot y\_m, 0.5, y\_m\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 1e-127)
      (* (/ (/ (fma (* x_m x_m) 0.5 1.0) z_m) x_m) y_m)
      (/ (/ (fma (* (* x_m x_m) y_m) 0.5 y_m) z_m) x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 1e-127) {
		tmp = ((fma((x_m * x_m), 0.5, 1.0) / z_m) / x_m) * y_m;
	} else {
		tmp = (fma(((x_m * x_m) * y_m), 0.5, y_m) / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 1e-127)
		tmp = Float64(Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) / z_m) / x_m) * y_m);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x_m * x_m) * y_m), 0.5, y_m) / z_m) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 1e-127], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5 + y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1e-127

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x \cdot \frac{y}{x}}}{z} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      5. lift-/.f64 N/A

         \[\leadsto \cosh x \cdot \frac{\color{blue}{\frac{y}{x}}}{z} \]

      6. associate-/r* N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

      7. div-flip N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{1}{\frac{x \cdot z}{y}}} \]

      8. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{\frac{x \cdot z}{y}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{\frac{x \cdot z}{y}}} \]

      10. lift-cosh.f64 N/A

          \[\leadsto \frac{\color{blue}{\cosh x}}{\frac{x \cdot z}{y}} \]

      11. associate-/l* N/A

          \[\leadsto \frac{\cosh x}{\color{blue}{x \cdot \frac{z}{y}}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\cosh x}{\color{blue}{x \cdot \frac{z}{y}}} \]

      13. lower-/.f64 81.0

          \[\leadsto \frac{\cosh x}{x \cdot \color{blue}{\frac{z}{y}}} \]

   3. Applied rewrites 81.0%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot \frac{z}{y}}} \]

      2. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x}}{x \cdot \frac{z}{y}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cosh x}{\color{blue}{x \cdot \frac{z}{y}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\cosh x}{x \cdot \color{blue}{\frac{z}{y}}} \]

      5. associate-/l* N/A

         \[\leadsto \frac{\cosh x}{\color{blue}{\frac{x \cdot z}{y}}} \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot z} \cdot y} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot z} \cdot y} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot z}} \cdot y \]

      9. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x}}{x \cdot z} \cdot y \]

      10. *-commutative N/A

          \[\leadsto \frac{\cosh x}{\color{blue}{z \cdot x}} \cdot y \]

      11. lift-*.f64 82.7

          \[\leadsto \frac{\cosh x}{\color{blue}{z \cdot x}} \cdot y \]

   5. Applied rewrites 82.7%

      \[\leadsto \color{blue}{\frac{\cosh x}{z \cdot x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{{x}^{2}}{z} + \frac{1}{z}}{x}} \cdot y \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \frac{{x}^{2}}{z} + \frac{1}{z}}{\color{blue}{x}} \cdot y \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{1}{2} \cdot {x}^{2}}{z} + \frac{1}{z}}{x} \cdot y \]

      3. div-add-rev N/A

         \[\leadsto \frac{\frac{\frac{1}{2} \cdot {x}^{2} + 1}{z}}{x} \cdot y \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \frac{1}{2} \cdot {x}^{2}}{z}}{x} \cdot y \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1 + \frac{1}{2} \cdot {x}^{2}}{z}}{x} \cdot y \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{1}{2} \cdot {x}^{2} + 1}{z}}{x} \cdot y \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{{x}^{2} \cdot \frac{1}{2} + 1}{z}}{x} \cdot y \]

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 79.4

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

   8. Applied rewrites 79.4%

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

   if 1e-127 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{\color{blue}{x}} \]

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 81.4

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

   4. Applied rewrites 81.4%

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

Alternative 7: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)}{z\_m \cdot x\_m} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot y\_m, 0.5, y\_m\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 5e-16)
      (* (/ (fma (* x_m x_m) 0.5 1.0) (* z_m x_m)) y_m)
      (/ (/ (fma (* (* x_m x_m) y_m) 0.5 y_m) z_m) x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5e-16) {
		tmp = (fma((x_m * x_m), 0.5, 1.0) / (z_m * x_m)) * y_m;
	} else {
		tmp = (fma(((x_m * x_m) * y_m), 0.5, y_m) / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 5e-16)
		tmp = Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) / Float64(z_m * x_m)) * y_m);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x_m * x_m) * y_m), 0.5, y_m) / z_m) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 5e-16], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5 + y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 5.0000000000000004e-16

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x \cdot \frac{y}{x}}}{z} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      5. lift-/.f64 N/A

         \[\leadsto \cosh x \cdot \frac{\color{blue}{\frac{y}{x}}}{z} \]

      6. associate-/r* N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

      7. div-flip N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{1}{\frac{x \cdot z}{y}}} \]

      8. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{\frac{x \cdot z}{y}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{\frac{x \cdot z}{y}}} \]

      10. lift-cosh.f64 N/A

          \[\leadsto \frac{\color{blue}{\cosh x}}{\frac{x \cdot z}{y}} \]

      11. associate-/l* N/A

          \[\leadsto \frac{\cosh x}{\color{blue}{x \cdot \frac{z}{y}}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\cosh x}{\color{blue}{x \cdot \frac{z}{y}}} \]

      13. lower-/.f64 81.0

          \[\leadsto \frac{\cosh x}{x \cdot \color{blue}{\frac{z}{y}}} \]

   3. Applied rewrites 81.0%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot \frac{z}{y}}} \]

      2. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x}}{x \cdot \frac{z}{y}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cosh x}{\color{blue}{x \cdot \frac{z}{y}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\cosh x}{x \cdot \color{blue}{\frac{z}{y}}} \]

      5. associate-/l* N/A

         \[\leadsto \frac{\cosh x}{\color{blue}{\frac{x \cdot z}{y}}} \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot z} \cdot y} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot z} \cdot y} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot z}} \cdot y \]

      9. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x}}{x \cdot z} \cdot y \]

      10. *-commutative N/A

          \[\leadsto \frac{\cosh x}{\color{blue}{z \cdot x}} \cdot y \]

      11. lift-*.f64 82.7

          \[\leadsto \frac{\cosh x}{\color{blue}{z \cdot x}} \cdot y \]

   5. Applied rewrites 82.7%

      \[\leadsto \color{blue}{\frac{\cosh x}{z \cdot x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + \frac{1}{2} \cdot {x}^{2}}}{z \cdot x} \cdot y \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{1}{2} \cdot {x}^{2} + \color{blue}{1}}{z \cdot x} \cdot y \]

      2. *-commutative N/A

         \[\leadsto \frac{{x}^{2} \cdot \frac{1}{2} + 1}{z \cdot x} \cdot y \]

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 67.9

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

   8. Applied rewrites 67.9%

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

   if 5.0000000000000004e-16 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{\color{blue}{x}} \]

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 81.4

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

   4. Applied rewrites 81.4%

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

Alternative 8: 84.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m}\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)}{z\_m \cdot x\_m} \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m \cdot \left(y\_m \cdot x\_m\right), 0.5, y\_m\right)}{z\_m}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(x\_m \cdot x\_m\right) \cdot y\_m\right) \cdot 0.5}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (/ (* (cosh x_m) (/ y_m x_m)) z_m)))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= t_0 5e-16)
        (* (/ (fma (* x_m x_m) 0.5 1.0) (* z_m x_m)) y_m)
        (if (<= t_0 INFINITY)
          (/ (/ (fma (* x_m (* y_m x_m)) 0.5 y_m) z_m) x_m)
          (/ (/ (* (* (* x_m x_m) y_m) 0.5) z_m) x_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = (cosh(x_m) * (y_m / x_m)) / z_m;
	double tmp;
	if (t_0 <= 5e-16) {
		tmp = (fma((x_m * x_m), 0.5, 1.0) / (z_m * x_m)) * y_m;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (fma((x_m * (y_m * x_m)), 0.5, y_m) / z_m) / x_m;
	} else {
		tmp = ((((x_m * x_m) * y_m) * 0.5) / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	t_0 = Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m)
	tmp = 0.0
	if (t_0 <= 5e-16)
		tmp = Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) / Float64(z_m * x_m)) * y_m);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(fma(Float64(x_m * Float64(y_m * x_m)), 0.5, y_m) / z_m) / x_m);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x_m * x_m) * y_m) * 0.5) / z_m) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 5e-16], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[(x$95$m * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 5.0000000000000004e-16

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x \cdot \frac{y}{x}}}{z} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      5. lift-/.f64 N/A

         \[\leadsto \cosh x \cdot \frac{\color{blue}{\frac{y}{x}}}{z} \]

      6. associate-/r* N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

      7. div-flip N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{1}{\frac{x \cdot z}{y}}} \]

      8. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{\frac{x \cdot z}{y}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{\frac{x \cdot z}{y}}} \]

      10. lift-cosh.f64 N/A

          \[\leadsto \frac{\color{blue}{\cosh x}}{\frac{x \cdot z}{y}} \]

      11. associate-/l* N/A

          \[\leadsto \frac{\cosh x}{\color{blue}{x \cdot \frac{z}{y}}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\cosh x}{\color{blue}{x \cdot \frac{z}{y}}} \]

      13. lower-/.f64 81.0

          \[\leadsto \frac{\cosh x}{x \cdot \color{blue}{\frac{z}{y}}} \]

   3. Applied rewrites 81.0%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot \frac{z}{y}}} \]

      2. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x}}{x \cdot \frac{z}{y}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cosh x}{\color{blue}{x \cdot \frac{z}{y}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\cosh x}{x \cdot \color{blue}{\frac{z}{y}}} \]

      5. associate-/l* N/A

         \[\leadsto \frac{\cosh x}{\color{blue}{\frac{x \cdot z}{y}}} \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot z} \cdot y} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot z} \cdot y} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{x \cdot z}} \cdot y \]

      9. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x}}{x \cdot z} \cdot y \]

      10. *-commutative N/A

          \[\leadsto \frac{\cosh x}{\color{blue}{z \cdot x}} \cdot y \]

      11. lift-*.f64 82.7

          \[\leadsto \frac{\cosh x}{\color{blue}{z \cdot x}} \cdot y \]

   5. Applied rewrites 82.7%

      \[\leadsto \color{blue}{\frac{\cosh x}{z \cdot x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + \frac{1}{2} \cdot {x}^{2}}}{z \cdot x} \cdot y \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{1}{2} \cdot {x}^{2} + \color{blue}{1}}{z \cdot x} \cdot y \]

      2. *-commutative N/A

         \[\leadsto \frac{{x}^{2} \cdot \frac{1}{2} + 1}{z \cdot x} \cdot y \]

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 67.9

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

   8. Applied rewrites 67.9%

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

   if 5.0000000000000004e-16 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < +inf.0

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{\color{blue}{x}} \]

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 81.4

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

   4. Applied rewrites 81.4%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y, 0.5, y\right)}{z}}{x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 77.3

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

   6. Applied rewrites 77.3%

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

   if +inf.0 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{\color{blue}{x}} \]

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 81.4

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

   4. Applied rewrites 81.4%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 41.4

         \[\leadsto \frac{\frac{\left(\left(x \cdot x\right) \cdot y\right) \cdot 0.5}{z}}{x} \]

   7. Applied rewrites 41.4%

      \[\leadsto \frac{\frac{\left(\left(x \cdot x\right) \cdot y\right) \cdot 0.5}{z}}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.4:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(x\_m \cdot x\_m\right) \cdot y\_m\right) \cdot 0.5}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 1.4)
      (/ (/ y_m z_m) x_m)
      (/ (/ (* (* (* x_m x_m) y_m) 0.5) z_m) x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = (y_m / z_m) / x_m;
	} else {
		tmp = ((((x_m * x_m) * y_m) * 0.5) / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
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(z_s, y_s, x_s, x_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 1.4d0) then
        tmp = (y_m / z_m) / x_m
    else
        tmp = ((((x_m * x_m) * y_m) * 0.5d0) / z_m) / x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = (y_m / z_m) / x_m;
	} else {
		tmp = ((((x_m * x_m) * y_m) * 0.5) / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 1.4:
		tmp = (y_m / z_m) / x_m
	else:
		tmp = ((((x_m * x_m) * y_m) * 0.5) / z_m) / x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 1.4)
		tmp = Float64(Float64(y_m / z_m) / x_m);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x_m * x_m) * y_m) * 0.5) / z_m) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 1.4)
		tmp = (y_m / z_m) / x_m;
	else
		tmp = ((((x_m * x_m) * y_m) * 0.5) / z_m) / x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 1.4], N[(N[(y$95$m / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x \cdot \frac{y}{x}}}{z} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{y}{x}}}{z} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]

      6. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]

      7. cosh-def N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}} \cdot y}{x \cdot z} \]

      8. rec-exp N/A

         \[\leadsto \frac{\frac{e^{x} + \color{blue}{\frac{1}{e^{x}}}}{2} \cdot y}{x \cdot z} \]

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

   3. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\frac{\frac{\cosh x \cdot y}{z}}{x}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\color{blue}{y}}{z}}{x} \]
   5. Step-by-step derivation

   6. Applied rewrites 52.8%

      \[\leadsto \frac{\frac{\color{blue}{y}}{z}}{x} \]

   if 1.3999999999999999 < x

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{\color{blue}{x}} \]

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 81.4

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

   4. Applied rewrites 81.4%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 41.4

         \[\leadsto \frac{\frac{\left(\left(x \cdot x\right) \cdot y\right) \cdot 0.5}{z}}{x} \]

   7. Applied rewrites 41.4%

      \[\leadsto \frac{\frac{\left(\left(x \cdot x\right) \cdot y\right) \cdot 0.5}{z}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 69.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.4:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x\_m \cdot x\_m\right) \cdot 0.5\right) \cdot y\_m}{z\_m \cdot x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 1.4)
      (/ (/ y_m z_m) x_m)
      (/ (* (* (* x_m x_m) 0.5) y_m) (* z_m x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = (y_m / z_m) / x_m;
	} else {
		tmp = (((x_m * x_m) * 0.5) * y_m) / (z_m * x_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
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(z_s, y_s, x_s, x_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 1.4d0) then
        tmp = (y_m / z_m) / x_m
    else
        tmp = (((x_m * x_m) * 0.5d0) * y_m) / (z_m * x_m)
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = (y_m / z_m) / x_m;
	} else {
		tmp = (((x_m * x_m) * 0.5) * y_m) / (z_m * x_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 1.4:
		tmp = (y_m / z_m) / x_m
	else:
		tmp = (((x_m * x_m) * 0.5) * y_m) / (z_m * x_m)
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 1.4)
		tmp = Float64(Float64(y_m / z_m) / x_m);
	else
		tmp = Float64(Float64(Float64(Float64(x_m * x_m) * 0.5) * y_m) / Float64(z_m * x_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 1.4)
		tmp = (y_m / z_m) / x_m;
	else
		tmp = (((x_m * x_m) * 0.5) * y_m) / (z_m * x_m);
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 1.4], N[(N[(y$95$m / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x \cdot \frac{y}{x}}}{z} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{y}{x}}}{z} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]

      6. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]

      7. cosh-def N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}} \cdot y}{x \cdot z} \]

      8. rec-exp N/A

         \[\leadsto \frac{\frac{e^{x} + \color{blue}{\frac{1}{e^{x}}}}{2} \cdot y}{x \cdot z} \]

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

   3. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\frac{\frac{\cosh x \cdot y}{z}}{x}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\color{blue}{y}}{z}}{x} \]
   5. Step-by-step derivation

   6. Applied rewrites 52.8%

      \[\leadsto \frac{\frac{\color{blue}{y}}{z}}{x} \]

   if 1.3999999999999999 < x

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 68.7

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

   4. Applied rewrites 68.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 64.3

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

   6. Applied rewrites 64.3%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 27.2

         \[\leadsto \left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \frac{y}{z \cdot x} \]

   9. Applied rewrites 27.2%

      \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{0.5}\right) \cdot \frac{y}{z \cdot x} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-/.f64 N/A

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

       8. *-commutative N/A

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

       9. lift-*.f64 29.7

          \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot y}{\color{blue}{z \cdot x}} \]

   11. Applied rewrites 29.7%

       \[\leadsto \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot y}{z \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 64.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.4:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x\_m \cdot x\_m\right) \cdot 0.5\right) \cdot \frac{y\_m}{z\_m \cdot x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 1.4)
      (/ (/ y_m z_m) x_m)
      (* (* (* x_m x_m) 0.5) (/ y_m (* z_m x_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = (y_m / z_m) / x_m;
	} else {
		tmp = ((x_m * x_m) * 0.5) * (y_m / (z_m * x_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
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(z_s, y_s, x_s, x_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 1.4d0) then
        tmp = (y_m / z_m) / x_m
    else
        tmp = ((x_m * x_m) * 0.5d0) * (y_m / (z_m * x_m))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = (y_m / z_m) / x_m;
	} else {
		tmp = ((x_m * x_m) * 0.5) * (y_m / (z_m * x_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 1.4:
		tmp = (y_m / z_m) / x_m
	else:
		tmp = ((x_m * x_m) * 0.5) * (y_m / (z_m * x_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 1.4)
		tmp = Float64(Float64(y_m / z_m) / x_m);
	else
		tmp = Float64(Float64(Float64(x_m * x_m) * 0.5) * Float64(y_m / Float64(z_m * x_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 1.4)
		tmp = (y_m / z_m) / x_m;
	else
		tmp = ((x_m * x_m) * 0.5) * (y_m / (z_m * x_m));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 1.4], N[(N[(y$95$m / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * N[(y$95$m / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x \cdot \frac{y}{x}}}{z} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{y}{x}}}{z} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]

      6. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]

      7. cosh-def N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}} \cdot y}{x \cdot z} \]

      8. rec-exp N/A

         \[\leadsto \frac{\frac{e^{x} + \color{blue}{\frac{1}{e^{x}}}}{2} \cdot y}{x \cdot z} \]

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

   3. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\frac{\frac{\cosh x \cdot y}{z}}{x}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\color{blue}{y}}{z}}{x} \]
   5. Step-by-step derivation

   6. Applied rewrites 52.8%

      \[\leadsto \frac{\frac{\color{blue}{y}}{z}}{x} \]

   if 1.3999999999999999 < x

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 68.7

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

   4. Applied rewrites 68.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 64.3

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

   6. Applied rewrites 64.3%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 27.2

         \[\leadsto \left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \frac{y}{z \cdot x} \]

   9. Applied rewrites 27.2%

      \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{0.5}\right) \cdot \frac{y}{z \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 61.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.4:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot \frac{y\_m}{z\_m}\right) \cdot 0.5\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (* x_s (if (<= x_m 1.4) (/ (/ y_m z_m) x_m) (* (* x_m (/ y_m z_m)) 0.5))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = (y_m / z_m) / x_m;
	} else {
		tmp = (x_m * (y_m / z_m)) * 0.5;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
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(z_s, y_s, x_s, x_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 1.4d0) then
        tmp = (y_m / z_m) / x_m
    else
        tmp = (x_m * (y_m / z_m)) * 0.5d0
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = (y_m / z_m) / x_m;
	} else {
		tmp = (x_m * (y_m / z_m)) * 0.5;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 1.4:
		tmp = (y_m / z_m) / x_m
	else:
		tmp = (x_m * (y_m / z_m)) * 0.5
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 1.4)
		tmp = Float64(Float64(y_m / z_m) / x_m);
	else
		tmp = Float64(Float64(x_m * Float64(y_m / z_m)) * 0.5);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 1.4)
		tmp = (y_m / z_m) / x_m;
	else
		tmp = (x_m * (y_m / z_m)) * 0.5;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 1.4], N[(N[(y$95$m / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(x$95$m * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x \cdot \frac{y}{x}}}{z} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{y}{x}}}{z} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]

      6. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]

      7. cosh-def N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}} \cdot y}{x \cdot z} \]

      8. rec-exp N/A

         \[\leadsto \frac{\frac{e^{x} + \color{blue}{\frac{1}{e^{x}}}}{2} \cdot y}{x \cdot z} \]

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

   3. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\frac{\frac{\cosh x \cdot y}{z}}{x}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\color{blue}{y}}{z}}{x} \]
   5. Step-by-step derivation

   6. Applied rewrites 52.8%

      \[\leadsto \frac{\frac{\color{blue}{y}}{z}}{x} \]

   if 1.3999999999999999 < x

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{\color{blue}{x}} \]

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 81.4

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

   4. Applied rewrites 81.4%

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

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{x \cdot y}{z}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot y}{z} \cdot \frac{1}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{z} \cdot \frac{1}{2} \]

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 26.2

         \[\leadsto \left(x \cdot \frac{y}{z}\right) \cdot 0.5 \]

   7. Applied rewrites 26.2%

      \[\leadsto \left(x \cdot \frac{y}{z}\right) \cdot \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 56.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 2 \cdot 10^{+66}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 2e+66)
      (/ (/ y_m x_m) z_m)
      (/ (/ y_m z_m) x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 2e+66) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = (y_m / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
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(z_s, y_s, x_s, x_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 2d+66) then
        tmp = (y_m / x_m) / z_m
    else
        tmp = (y_m / z_m) / x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((Math.cosh(x_m) * (y_m / x_m)) / z_m) <= 2e+66) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = (y_m / z_m) / x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if ((math.cosh(x_m) * (y_m / x_m)) / z_m) <= 2e+66:
		tmp = (y_m / x_m) / z_m
	else:
		tmp = (y_m / z_m) / x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 2e+66)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(Float64(y_m / z_m) / x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 2e+66)
		tmp = (y_m / x_m) / z_m;
	else
		tmp = (y_m / z_m) / x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 2e+66], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(y$95$m / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.99999999999999989e66

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]
   3. Step-by-step derivation

      1. lift-/.f64 48.4

         \[\leadsto \frac{\frac{y}{\color{blue}{x}}}{z} \]

   4. Applied rewrites 48.4%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   if 1.99999999999999989e66 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x \cdot \frac{y}{x}}}{z} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{y}{x}}}{z} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]

      6. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]

      7. cosh-def N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}} \cdot y}{x \cdot z} \]

      8. rec-exp N/A

         \[\leadsto \frac{\frac{e^{x} + \color{blue}{\frac{1}{e^{x}}}}{2} \cdot y}{x \cdot z} \]

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

   3. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\frac{\frac{\cosh x \cdot y}{z}}{x}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\color{blue}{y}}{z}}{x} \]
   5. Step-by-step derivation

   6. Applied rewrites 52.8%

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

Alternative 14: 52.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 10^{+145}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z\_m \cdot x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= (* (cosh x_m) (/ y_m x_m)) 1e+145)
      (/ (/ y_m x_m) z_m)
      (/ y_m (* z_m x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 1e+145) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = y_m / (z_m * x_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
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(z_s, y_s, x_s, x_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((cosh(x_m) * (y_m / x_m)) <= 1d+145) then
        tmp = (y_m / x_m) / z_m
    else
        tmp = y_m / (z_m * x_m)
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((Math.cosh(x_m) * (y_m / x_m)) <= 1e+145) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = y_m / (z_m * x_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if (math.cosh(x_m) * (y_m / x_m)) <= 1e+145:
		tmp = (y_m / x_m) / z_m
	else:
		tmp = y_m / (z_m * x_m)
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 1e+145)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(y_m / Float64(z_m * x_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if ((cosh(x_m) * (y_m / x_m)) <= 1e+145)
		tmp = (y_m / x_m) / z_m;
	else
		tmp = y_m / (z_m * x_m);
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 1e+145], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 9.9999999999999999e144

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]
   3. Step-by-step derivation

      1. lift-/.f64 48.4

         \[\leadsto \frac{\frac{y}{\color{blue}{x}}}{z} \]

   4. Applied rewrites 48.4%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   if 9.9999999999999999e144 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 84.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 48.8

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

   4. Applied rewrites 48.8%

      \[\leadsto \color{blue}{\frac{y}{z \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 48.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \frac{y\_m}{z\_m \cdot x\_m}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m)
 :precision binary64
 (* z_s (* y_s (* x_s (/ y_m (* z_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	return z_s * (y_s * (x_s * (y_m / (z_m * x_m))));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
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(z_s, y_s, x_s, x_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    code = z_s * (y_s * (x_s * (y_m / (z_m * x_m))))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m) {
	return z_s * (y_s * (x_s * (y_m / (z_m * x_m))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m):
	return z_s * (y_s * (x_s * (y_m / (z_m * x_m))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m)
	return Float64(z_s * Float64(y_s * Float64(x_s * Float64(y_m / Float64(z_m * x_m)))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, y_s, x_s, x_m, y_m, z_m)
	tmp = z_s * (y_s * (x_s * (y_m / (z_m * x_m))));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * N[(y$95$m / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.0%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

2. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 48.8

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

4. Applied rewrites 48.8%

   \[\leadsto \color{blue}{\frac{y}{z \cdot x}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64
  (/ (* (cosh x) (/ y x)) z))