Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 85.3% → 97.2%

Time: 3.9s

Alternatives: 13

Speedup: 0.9×

• 64.3% 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: 85.3% 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: 97.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3.9 \cdot 10^{+75}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot \left(y\_m \cdot x\right), 0.5, y\_m\right)}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 3.9e+75)
    (/ (/ (* (cosh x) y_m) x) z)
    (/ (/ (fma (* x (* y_m x)) 0.5 y_m) z) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 3.9e+75) {
		tmp = ((cosh(x) * y_m) / x) / z;
	} else {
		tmp = (fma((x * (y_m * x)), 0.5, y_m) / z) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 3.9e+75)
		tmp = Float64(Float64(Float64(cosh(x) * y_m) / x) / z);
	else
		tmp = Float64(Float64(fma(Float64(x * Float64(y_m * x)), 0.5, y_m) / z) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 3.9e+75], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y$95$m), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(x * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision] * 0.5 + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.90000000000000038e75

   1. Initial program 85.3%

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

      1. lift-*.f64 N/A

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

      2. lift-cosh.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cosh.f64 95.9

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

   3. Applied rewrites 95.9%

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

   if 3.90000000000000038e75 < y

   1. Initial program 85.3%

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

   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 80.8

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

   4. Applied rewrites 80.8%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y, 0.5, y\right)}{z}}{x}} \]
   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 76.7

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

   6. Applied rewrites 76.7%

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

Alternative 2: 88.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\cosh x \cdot y\_m}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x \cdot x}{z}}{x} \cdot y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 1.35e+154)
    (/ (* (cosh x) y_m) (* z x))
    (* (* (/ (/ (* x x) z) x) y_m) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (cosh(x) * y_m) / (z * x);
	} else {
		tmp = ((((x * x) / z) / x) * y_m) * 0.5;
	}
	return y_s * tmp;
}

Fortran

y\_m =     private
y\_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(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.35d+154) then
        tmp = (cosh(x) * y_m) / (z * x)
    else
        tmp = ((((x * x) / z) / x) * y_m) * 0.5d0
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (Math.cosh(x) * y_m) / (z * x);
	} else {
		tmp = ((((x * x) / z) / x) * y_m) * 0.5;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 1.35e+154:
		tmp = (math.cosh(x) * y_m) / (z * x)
	else:
		tmp = ((((x * x) / z) / x) * y_m) * 0.5
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(Float64(cosh(x) * y_m) / Float64(z * x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) / z) / x) * y_m) * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 1.35e+154)
		tmp = (cosh(x) * y_m) / (z * x);
	else
		tmp = ((((x * x) / z) / x) * y_m) * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 1.35e+154], N[(N[(N[Cosh[x], $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 85.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 84.0

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

   3. Applied rewrites 84.0%

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

   if 1.35000000000000003e154 < x

   1. Initial program 85.3%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.6

          \[\leadsto \left(\frac{2 \cdot \cosh x}{z \cdot x} \cdot y\right) \cdot 0.5 \]

   4. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\left(\frac{2 \cdot \cosh x}{z \cdot x} \cdot y\right) \cdot 0.5} \]

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 79.4

         \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(x, x, 2\right)}{z}}{x} \cdot y\right) \cdot 0.5 \]

   7. Applied rewrites 79.4%

      \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(x, x, 2\right)}{z}}{x} \cdot y\right) \cdot 0.5 \]

   8. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 38.9

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

   10. Applied rewrites 38.9%

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

Alternative 3: 81.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y\_m}{x} \leq 10^{+291}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y\_m, 0.5, y\_m\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\mathsf{fma}\left(x, x, 2\right)}{z}}{x} \cdot y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* (cosh x) (/ y_m x)) 1e+291)
    (/ (/ (fma (* (* x x) y_m) 0.5 y_m) x) z)
    (* (* (/ (/ (fma x x 2.0) z) x) y_m) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((cosh(x) * (y_m / x)) <= 1e+291) {
		tmp = (fma(((x * x) * y_m), 0.5, y_m) / x) / z;
	} else {
		tmp = (((fma(x, x, 2.0) / z) / x) * y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(y_m / x)) <= 1e+291)
		tmp = Float64(Float64(fma(Float64(Float64(x * x) * y_m), 0.5, y_m) / x) / z);
	else
		tmp = Float64(Float64(Float64(Float64(fma(x, x, 2.0) / z) / x) * y_m) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], 1e+291], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5 + y$95$m), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x * x + 2.0), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 9.9999999999999996e290

   1. Initial program 85.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 79.2

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

   4. Applied rewrites 79.2%

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

   if 9.9999999999999996e290 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 85.3%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.6

          \[\leadsto \left(\frac{2 \cdot \cosh x}{z \cdot x} \cdot y\right) \cdot 0.5 \]

   4. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\left(\frac{2 \cdot \cosh x}{z \cdot x} \cdot y\right) \cdot 0.5} \]

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 79.4

         \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(x, x, 2\right)}{z}}{x} \cdot y\right) \cdot 0.5 \]

   7. Applied rewrites 79.4%

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

Alternative 4: 80.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y\_m}{x} \leq 10^{+291}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, 2\right) \cdot \frac{y\_m}{x}}{z} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\mathsf{fma}\left(x, x, 2\right)}{z}}{x} \cdot y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* (cosh x) (/ y_m x)) 1e+291)
    (* (/ (* (fma x x 2.0) (/ y_m x)) z) 0.5)
    (* (* (/ (/ (fma x x 2.0) z) x) y_m) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((cosh(x) * (y_m / x)) <= 1e+291) {
		tmp = ((fma(x, x, 2.0) * (y_m / x)) / z) * 0.5;
	} else {
		tmp = (((fma(x, x, 2.0) / z) / x) * y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(y_m / x)) <= 1e+291)
		tmp = Float64(Float64(Float64(fma(x, x, 2.0) * Float64(y_m / x)) / z) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(fma(x, x, 2.0) / z) / x) * y_m) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], 1e+291], N[(N[(N[(N[(x * x + 2.0), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(x * x + 2.0), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 9.9999999999999996e290

   1. Initial program 85.3%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.6

          \[\leadsto \left(\frac{2 \cdot \cosh x}{z \cdot x} \cdot y\right) \cdot 0.5 \]

   4. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\left(\frac{2 \cdot \cosh x}{z \cdot x} \cdot y\right) \cdot 0.5} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 68.5

         \[\leadsto \left(\frac{\mathsf{fma}\left(x, x, 2\right)}{z \cdot x} \cdot y\right) \cdot 0.5 \]

   7. Applied rewrites 68.5%

      \[\leadsto \left(\frac{\mathsf{fma}\left(x, x, 2\right)}{z \cdot x} \cdot y\right) \cdot 0.5 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 69.5

         \[\leadsto \frac{\mathsf{fma}\left(x, x, 2\right) \cdot y}{z \cdot x} \cdot 0.5 \]

   9. Applied rewrites 69.5%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, 2\right) \cdot y}{z \cdot x} \cdot 0.5 \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. frac-times N/A

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

       5. lift-/.f64 N/A

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

       6. associate-*l/ N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 69.6

          \[\leadsto \frac{\mathsf{fma}\left(x, x, 2\right) \cdot \frac{y}{x}}{z} \cdot 0.5 \]

   11. Applied rewrites 69.6%

       \[\leadsto \frac{\mathsf{fma}\left(x, x, 2\right) \cdot \frac{y}{x}}{z} \cdot 0.5 \]

   if 9.9999999999999996e290 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 85.3%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.6

          \[\leadsto \left(\frac{2 \cdot \cosh x}{z \cdot x} \cdot y\right) \cdot 0.5 \]

   4. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\left(\frac{2 \cdot \cosh x}{z \cdot x} \cdot y\right) \cdot 0.5} \]

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 79.4

         \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(x, x, 2\right)}{z}}{x} \cdot y\right) \cdot 0.5 \]

   7. Applied rewrites 79.4%

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

Alternative 5: 70.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y\_m}{x} \leq 2 \cdot 10^{+200}:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\mathsf{fma}\left(x, x, 2\right)}{z}}{x} \cdot y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* (cosh x) (/ y_m x)) 2e+200)
    (/ (/ y_m x) z)
    (* (* (/ (/ (fma x x 2.0) z) x) y_m) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((cosh(x) * (y_m / x)) <= 2e+200) {
		tmp = (y_m / x) / z;
	} else {
		tmp = (((fma(x, x, 2.0) / z) / x) * y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(y_m / x)) <= 2e+200)
		tmp = Float64(Float64(y_m / x) / z);
	else
		tmp = Float64(Float64(Float64(Float64(fma(x, x, 2.0) / z) / x) * y_m) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], 2e+200], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x * x + 2.0), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.9999999999999999e200

   1. Initial program 85.3%

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

   2. Taylor expanded in x around 0

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

      1. lift-/.f64 49.3

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

   4. Applied rewrites 49.3%

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

   if 1.9999999999999999e200 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 85.3%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.6

          \[\leadsto \left(\frac{2 \cdot \cosh x}{z \cdot x} \cdot y\right) \cdot 0.5 \]

   4. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\left(\frac{2 \cdot \cosh x}{z \cdot x} \cdot y\right) \cdot 0.5} \]

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 79.4

         \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(x, x, 2\right)}{z}}{x} \cdot y\right) \cdot 0.5 \]

   7. Applied rewrites 79.4%

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

Alternative 6: 69.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y\_m}{x} \leq 2 \cdot 10^{+200}:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\mathsf{fma}\left(x, x, 2\right)}{x}}{z} \cdot y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* (cosh x) (/ y_m x)) 2e+200)
    (/ (/ y_m x) z)
    (* (* (/ (/ (fma x x 2.0) x) z) y_m) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((cosh(x) * (y_m / x)) <= 2e+200) {
		tmp = (y_m / x) / z;
	} else {
		tmp = (((fma(x, x, 2.0) / x) / z) * y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(y_m / x)) <= 2e+200)
		tmp = Float64(Float64(y_m / x) / z);
	else
		tmp = Float64(Float64(Float64(Float64(fma(x, x, 2.0) / x) / z) * y_m) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], 2e+200], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x * x + 2.0), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.9999999999999999e200

   1. Initial program 85.3%

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

   2. Taylor expanded in x around 0

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

      1. lift-/.f64 49.3

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

   4. Applied rewrites 49.3%

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

   if 1.9999999999999999e200 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 85.3%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.6

          \[\leadsto \left(\frac{2 \cdot \cosh x}{z \cdot x} \cdot y\right) \cdot 0.5 \]

   4. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\left(\frac{2 \cdot \cosh x}{z \cdot x} \cdot y\right) \cdot 0.5} \]

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 79.4

         \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(x, x, 2\right)}{z}}{x} \cdot y\right) \cdot 0.5 \]

   7. Applied rewrites 79.4%

      \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(x, x, 2\right)}{z}}{x} \cdot y\right) \cdot 0.5 \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-fma.f64 N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. lift-fma.f64 N/A

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

      13. lower-/.f64 77.7

          \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(x, x, 2\right)}{x}}{z} \cdot y\right) \cdot 0.5 \]

   9. Applied rewrites 77.7%

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

Alternative 7: 66.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 85.3%

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

   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 49.8

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

   4. Applied rewrites 49.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 53.7

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

   6. Applied rewrites 53.7%

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

   if 1.3999999999999999 < x

   1. Initial program 85.3%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.6

          \[\leadsto \left(\frac{2 \cdot \cosh x}{z \cdot x} \cdot y\right) \cdot 0.5 \]

   4. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\left(\frac{2 \cdot \cosh x}{z \cdot x} \cdot y\right) \cdot 0.5} \]

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 79.4

         \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(x, x, 2\right)}{z}}{x} \cdot y\right) \cdot 0.5 \]

   7. Applied rewrites 79.4%

      \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(x, x, 2\right)}{z}}{x} \cdot y\right) \cdot 0.5 \]

   8. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 38.9

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

   10. Applied rewrites 38.9%

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

Alternative 8: 60.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 85.3%

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

   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 49.8

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

   4. Applied rewrites 49.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 53.7

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

   6. Applied rewrites 53.7%

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

   if 1.3999999999999999 < x

   1. Initial program 85.3%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.6

          \[\leadsto \left(\frac{2 \cdot \cosh x}{z \cdot x} \cdot y\right) \cdot 0.5 \]

   4. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\left(\frac{2 \cdot \cosh x}{z \cdot x} \cdot y\right) \cdot 0.5} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 68.5

         \[\leadsto \left(\frac{\mathsf{fma}\left(x, x, 2\right)}{z \cdot x} \cdot y\right) \cdot 0.5 \]

   7. Applied rewrites 68.5%

      \[\leadsto \left(\frac{\mathsf{fma}\left(x, x, 2\right)}{z \cdot x} \cdot y\right) \cdot 0.5 \]

   8. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 29.0

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

   10. Applied rewrites 29.0%

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

Alternative 9: 59.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 85.3%

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

   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 49.8

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

   4. Applied rewrites 49.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 53.7

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

   6. Applied rewrites 53.7%

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

   if 1.3999999999999999 < x

   1. Initial program 85.3%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.6

          \[\leadsto \left(\frac{2 \cdot \cosh x}{z \cdot x} \cdot y\right) \cdot 0.5 \]

   4. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\left(\frac{2 \cdot \cosh x}{z \cdot x} \cdot y\right) \cdot 0.5} \]

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 79.4

         \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(x, x, 2\right)}{z}}{x} \cdot y\right) \cdot 0.5 \]

   7. Applied rewrites 79.4%

      \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(x, x, 2\right)}{z}}{x} \cdot y\right) \cdot 0.5 \]

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 26.0

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

   10. Applied rewrites 26.0%

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

Alternative 10: 57.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 85.3%

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

   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 49.8

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

   4. Applied rewrites 49.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 53.7

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

   6. Applied rewrites 53.7%

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

   if 1.3999999999999999 < x

   1. Initial program 85.3%

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

   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 80.8

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

   4. Applied rewrites 80.8%

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

   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 26.5

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

   7. Applied rewrites 26.5%

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

Alternative 11: 55.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y\_m}{x}}{z} \leq 5 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (* (cosh x) (/ y_m x)) z) 5e-33)
    (/ (/ y_m x) z)
    (/ (/ y_m z) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((cosh(x) * (y_m / x)) / z) <= 5e-33) {
		tmp = (y_m / x) / z;
	} else {
		tmp = (y_m / z) / x;
	}
	return y_s * tmp;
}

Fortran

y\_m =     private
y\_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(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((cosh(x) * (y_m / x)) / z) <= 5d-33) then
        tmp = (y_m / x) / z
    else
        tmp = (y_m / z) / x
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((Math.cosh(x) * (y_m / x)) / z) <= 5e-33) {
		tmp = (y_m / x) / z;
	} else {
		tmp = (y_m / z) / x;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if ((math.cosh(x) * (y_m / x)) / z) <= 5e-33:
		tmp = (y_m / x) / z
	else:
		tmp = (y_m / z) / x
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x) * Float64(y_m / x)) / z) <= 5e-33)
		tmp = Float64(Float64(y_m / x) / z);
	else
		tmp = Float64(Float64(y_m / z) / x);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (((cosh(x) * (y_m / x)) / z) <= 5e-33)
		tmp = (y_m / x) / z;
	else
		tmp = (y_m / z) / x;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 5e-33], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 5.00000000000000028e-33

   1. Initial program 85.3%

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

   2. Taylor expanded in x around 0

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

      1. lift-/.f64 49.3

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

   4. Applied rewrites 49.3%

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

   if 5.00000000000000028e-33 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 85.3%

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

   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 49.8

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

   4. Applied rewrites 49.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 53.7

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

   6. Applied rewrites 53.7%

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

Alternative 12: 53.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 1.75 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z \cdot x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= z 1.75e-28) (/ (/ y_m z) x) (/ y_m (* z x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 1.75e-28) {
		tmp = (y_m / z) / x;
	} else {
		tmp = y_m / (z * x);
	}
	return y_s * tmp;
}

Fortran

y\_m =     private
y\_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(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.75d-28) then
        tmp = (y_m / z) / x
    else
        tmp = y_m / (z * x)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 1.75e-28) {
		tmp = (y_m / z) / x;
	} else {
		tmp = y_m / (z * x);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= 1.75e-28:
		tmp = (y_m / z) / x
	else:
		tmp = y_m / (z * x)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= 1.75e-28)
		tmp = Float64(Float64(y_m / z) / x);
	else
		tmp = Float64(y_m / Float64(z * x));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= 1.75e-28)
		tmp = (y_m / z) / x;
	else
		tmp = y_m / (z * x);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 1.75e-28], N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision], N[(y$95$m / N[(z * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.75e-28

   1. Initial program 85.3%

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

   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 49.8

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

   4. Applied rewrites 49.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 53.7

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

   6. Applied rewrites 53.7%

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

   if 1.75e-28 < z

   1. Initial program 85.3%

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

   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 49.8

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

   4. Applied rewrites 49.8%

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

Alternative 13: 49.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \frac{y\_m}{z \cdot x} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z) :precision binary64 (* y_s (/ y_m (* z x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (y_m / (z * x));
}

Fortran

y\_m =     private
y\_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(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (y_m / (z * x))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * (y_m / (z * x));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	return y_s * (y_m / (z * x))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(y_m / Float64(z * x)))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * (y_m / (z * x));
end

Wolfram

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

Derivation

1. Initial program 85.3%

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

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 49.8

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

4. Applied rewrites 49.8%

   \[\leadsto \color{blue}{\frac{y}{z \cdot x}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025138 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64
  (/ (* (cosh x) (/ y x)) z))