Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.9% → 97.3%

Time: 3.9s

Alternatives: 16

Speedup: 0.9×

• 64.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (cosh(x) * (y / x)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.9% 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: 97.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 4.9 \cdot 10^{+70}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y\_m, 0.5, y\_m\right)}{z}}{x}\\ \end{array} \end{array} \]

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 4.9e+70)
    (/ (/ (* (cosh x) y_m) x) z)
    (/ (/ (fma (* (* x x) y_m) 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 <= 4.9e+70) {
		tmp = ((cosh(x) * y_m) / x) / z;
	} else {
		tmp = (fma(((x * x) * y_m), 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 <= 4.9e+70)
		tmp = Float64(Float64(Float64(cosh(x) * y_m) / x) / z);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x * x) * y_m), 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, 4.9e+70], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y$95$m), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5 + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.90000000000000028e70

   1. Initial program 81.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 99.4

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

   3. Applied rewrites 99.4%

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

   if 4.90000000000000028e70 < y

   1. Initial program 90.9%

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

   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 94.1

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

   4. Applied rewrites 94.1%

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

Alternative 2: 88.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 20:\\ \;\;\;\;\frac{\cosh x \cdot y\_m}{z \cdot x}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x \cdot x}{z}}{x} \cdot y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

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 20.0)
    (/ (* (cosh x) y_m) (* z x))
    (if (<= x 1.5e+154)
      (/ (* (cosh x) (/ y_m x)) z)
      (* (* (/ (/ (* 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 <= 20.0) {
		tmp = (cosh(x) * y_m) / (z * x);
	} else if (x <= 1.5e+154) {
		tmp = (cosh(x) * (y_m / x)) / z;
	} 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 <= 20.0d0) then
        tmp = (cosh(x) * y_m) / (z * x)
    else if (x <= 1.5d+154) then
        tmp = (cosh(x) * (y_m / x)) / z
    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 <= 20.0) {
		tmp = (Math.cosh(x) * y_m) / (z * x);
	} else if (x <= 1.5e+154) {
		tmp = (Math.cosh(x) * (y_m / x)) / z;
	} 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 <= 20.0:
		tmp = (math.cosh(x) * y_m) / (z * x)
	elif x <= 1.5e+154:
		tmp = (math.cosh(x) * (y_m / x)) / z
	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 <= 20.0)
		tmp = Float64(Float64(cosh(x) * y_m) / Float64(z * x));
	elseif (x <= 1.5e+154)
		tmp = Float64(Float64(cosh(x) * Float64(y_m / x)) / z);
	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 <= 20.0)
		tmp = (cosh(x) * y_m) / (z * x);
	elseif (x <= 1.5e+154)
		tmp = (cosh(x) * (y_m / x)) / z;
	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, 20.0], N[(N[(N[Cosh[x], $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.5e+154], N[(N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $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 3 regimes

2. if x < 20

   1. Initial program 87.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 86.7

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

   3. Applied rewrites 86.7%

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

   if 20 < x < 1.50000000000000013e154

   1. Initial program 89.9%

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

   if 1.50000000000000013e154 < x

   1. Initial program 64.8%

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

   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 62.8

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

   4. Applied rewrites 62.8%

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

   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 100.0

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

   7. Applied rewrites 100.0%

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

   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. lift-*.f64 100.0

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

   10. Applied rewrites 100.0%

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

Alternative 3: 88.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{+140}:\\ \;\;\;\;\frac{\cosh x \cdot y\_m}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y\_m, 0.5, y\_m\right)}{z}}{x}\\ \end{array} \end{array} \]

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 (<= x 2.65e+140)
    (/ (* (cosh x) y_m) (* z x))
    (/ (/ (fma (* (* x x) y_m) 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 (x <= 2.65e+140) {
		tmp = (cosh(x) * y_m) / (z * x);
	} else {
		tmp = (fma(((x * x) * y_m), 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 (x <= 2.65e+140)
		tmp = Float64(Float64(cosh(x) * y_m) / Float64(z * x));
	else
		tmp = Float64(Float64(fma(Float64(Float64(x * x) * y_m), 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[x, 2.65e+140], N[(N[(N[Cosh[x], $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5 + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.64999999999999993e140

   1. Initial program 87.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 86.9

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

   3. Applied rewrites 86.9%

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

   if 2.64999999999999993e140 < x

   1. Initial program 65.9%

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

   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 96.2

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

   4. Applied rewrites 96.2%

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

Alternative 4: 77.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y\_m}{x}}{z} \leq 10^{+214}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.5, y\_m, y\_m\right)}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y\_m, 0.5, y\_m\right)}{z}}{x}\\ \end{array} \end{array} \]

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 (<= (/ (* (cosh x) (/ y_m x)) z) 1e+214)
    (/ (fma (* (* x x) 0.5) y_m y_m) (* z x))
    (/ (/ (fma (* (* x x) y_m) 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 (((cosh(x) * (y_m / x)) / z) <= 1e+214) {
		tmp = fma(((x * x) * 0.5), y_m, y_m) / (z * x);
	} else {
		tmp = (fma(((x * x) * y_m), 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 (Float64(Float64(cosh(x) * Float64(y_m / x)) / z) <= 1e+214)
		tmp = Float64(fma(Float64(Float64(x * x) * 0.5), y_m, y_m) / Float64(z * x));
	else
		tmp = Float64(Float64(fma(Float64(Float64(x * x) * y_m), 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[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e+214], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * y$95$m + y$95$m), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5 + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 9.9999999999999995e213

   1. Initial program 96.4%

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

   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 82.1

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

   4. Applied rewrites 82.1%

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

      1. lift-/.f64 N/A

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

      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. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 76.4

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

   6. Applied rewrites 76.4%

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

   if 9.9999999999999995e213 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 68.5%

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

   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 78.6

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

   4. Applied rewrites 78.6%

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

Alternative 5: 77.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y\_m}{x}}{z} \leq 10^{+214}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.5, y\_m, y\_m\right)}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}{z}}{x}\\ \end{array} \end{array} \]

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 (<= (/ (* (cosh x) (/ y_m x)) z) 1e+214)
    (/ (fma (* (* x x) 0.5) y_m y_m) (* z x))
    (/ (* y_m (/ (fma (* x x) 0.5 1.0) 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) <= 1e+214) {
		tmp = fma(((x * x) * 0.5), y_m, y_m) / (z * x);
	} else {
		tmp = (y_m * (fma((x * x), 0.5, 1.0) / 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) <= 1e+214)
		tmp = Float64(fma(Float64(Float64(x * x) * 0.5), y_m, y_m) / Float64(z * x));
	else
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x * x), 0.5, 1.0) / 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[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e+214], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * y$95$m + y$95$m), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 9.9999999999999995e213

   1. Initial program 96.4%

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

   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 82.1

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

   4. Applied rewrites 82.1%

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

      1. lift-/.f64 N/A

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

      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. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 76.4

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

   6. Applied rewrites 76.4%

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

   if 9.9999999999999995e213 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 68.5%

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

   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 78.6

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

   4. Applied rewrites 78.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 35.4%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-lft-identity N/A

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

      4. div-add N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 79.8

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

   10. Applied rewrites 79.8%

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

Alternative 6: 74.5% accurate, 1.0× speedup

Math

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

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 (<= x 1e+75)
    (/ (fma (* (* x x) 0.5) y_m y_m) (* z x))
    (/ (/ (* (* (* x x) y_m) 0.5) 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 (x <= 1e+75) {
		tmp = fma(((x * x) * 0.5), y_m, y_m) / (z * x);
	} else {
		tmp = ((((x * x) * y_m) * 0.5) / 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 (x <= 1e+75)
		tmp = Float64(fma(Float64(Float64(x * x) * 0.5), y_m, y_m) / Float64(z * x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * y_m) * 0.5) / 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[x, 1e+75], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * y$95$m + y$95$m), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999927e74

   1. Initial program 88.1%

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

   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 80.4

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

   4. Applied rewrites 80.4%

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

      1. lift-/.f64 N/A

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

      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. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 72.9

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

   6. Applied rewrites 72.9%

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

   if 9.99999999999999927e74 < x

   1. Initial program 71.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 81.6

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

   4. Applied rewrites 81.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 81.6

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

   7. Applied rewrites 81.6%

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

Alternative 7: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, 2\right) \cdot y\_m}{z \cdot x} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot x\right) \cdot y\_m\right) \cdot 0.5}{z}}{x}\\ \end{array} \end{array} \]

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 (<= x 1e+75)
    (* (/ (* (fma x x 2.0) y_m) (* z x)) 0.5)
    (/ (/ (* (* (* x x) y_m) 0.5) 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 (x <= 1e+75) {
		tmp = ((fma(x, x, 2.0) * y_m) / (z * x)) * 0.5;
	} else {
		tmp = ((((x * x) * y_m) * 0.5) / 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 (x <= 1e+75)
		tmp = Float64(Float64(Float64(fma(x, x, 2.0) * y_m) / Float64(z * x)) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * y_m) * 0.5) / 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[x, 1e+75], N[(N[(N[(N[(x * x + 2.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999927e74

   1. Initial program 88.1%

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

   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 86.9

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

   4. Applied rewrites 86.9%

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

   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 71.6

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

   7. Applied rewrites 71.6%

      \[\leadsto \left(\frac{\mathsf{fma}\left(x, x, 2\right)}{z \cdot x} \cdot y\right) \cdot 0.5 \]
   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. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 72.9

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

   9. Applied rewrites 72.9%

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

   if 9.99999999999999927e74 < x

   1. Initial program 71.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 81.6

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

   4. Applied rewrites 81.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 81.6

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

   7. Applied rewrites 81.6%

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

Alternative 8: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 10^{+15}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(x, x, 2\right)}{z \cdot x} \cdot y\_m\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot x\right) \cdot y\_m\right) \cdot 0.5}{z}}{x}\\ \end{array} \end{array} \]

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 (<= x 1e+15)
    (* (* (/ (fma x x 2.0) (* z x)) y_m) 0.5)
    (/ (/ (* (* (* x x) y_m) 0.5) 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 (x <= 1e+15) {
		tmp = ((fma(x, x, 2.0) / (z * x)) * y_m) * 0.5;
	} else {
		tmp = ((((x * x) * y_m) * 0.5) / 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 (x <= 1e+15)
		tmp = Float64(Float64(Float64(fma(x, x, 2.0) / Float64(z * x)) * y_m) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * y_m) * 0.5) / 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[x, 1e+15], N[(N[(N[(N[(x * x + 2.0), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1e15

   1. Initial program 87.6%

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

   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 86.5

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

   4. Applied rewrites 86.5%

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

   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 74.9

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

   7. Applied rewrites 74.9%

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

   if 1e15 < x

   1. Initial program 76.5%

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

   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 71.2

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

   4. Applied rewrites 71.2%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 71.2

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

   7. Applied rewrites 71.2%

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

Alternative 9: 64.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\frac{y\_m}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot x\right) \cdot y\_m\right) \cdot 0.5}{z}}{x}\\ \end{array} \end{array} \]

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 (<= x 1.45) (/ y_m (* z x)) (/ (/ (* (* (* x x) y_m) 0.5) 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 (x <= 1.45) {
		tmp = y_m / (z * x);
	} else {
		tmp = ((((x * x) * y_m) * 0.5) / 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 (x <= 1.45d0) then
        tmp = y_m / (z * x)
    else
        tmp = ((((x * x) * y_m) * 0.5d0) / 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 (x <= 1.45) {
		tmp = y_m / (z * x);
	} else {
		tmp = ((((x * x) * y_m) * 0.5) / 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 x <= 1.45:
		tmp = y_m / (z * x)
	else:
		tmp = ((((x * x) * y_m) * 0.5) / 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 (x <= 1.45)
		tmp = Float64(y_m / Float64(z * x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * y_m) * 0.5) / 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 (x <= 1.45)
		tmp = y_m / (z * x);
	else
		tmp = ((((x * x) * y_m) * 0.5) / 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[x, 1.45], N[(y$95$m / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 87.4%

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

   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 63.8

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

   4. Applied rewrites 63.8%

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

   if 1.44999999999999996 < x

   1. Initial program 77.7%

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

   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 68.3

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

   4. Applied rewrites 68.3%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 68.3

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

   7. Applied rewrites 68.3%

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

Alternative 10: 64.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\frac{y\_m}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x \cdot x}{z}}{x} \cdot y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

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 1.45) (/ 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 <= 1.45) {
		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 <= 1.45d0) 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 <= 1.45) {
		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 <= 1.45:
		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 <= 1.45)
		tmp = Float64(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 <= 1.45)
		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, 1.45], N[(y$95$m / 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 < 1.44999999999999996

   1. Initial program 87.4%

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

   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 63.8

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

   4. Applied rewrites 63.8%

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

   if 1.44999999999999996 < x

   1. Initial program 77.7%

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

   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 75.6

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

   4. Applied rewrites 75.6%

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

   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 66.7

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

   7. Applied rewrites 66.7%

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

   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. lift-*.f64 66.7

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

   10. Applied rewrites 66.7%

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

Alternative 11: 59.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\frac{y\_m}{z \cdot x}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+259}:\\ \;\;\;\;\frac{\left(\left(x \cdot x\right) \cdot y\_m\right) \cdot 0.5}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(y\_m \cdot x\right)}{z}\\ \end{array} \end{array} \]

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 1.45)
    (/ y_m (* z x))
    (if (<= x 3.5e+259)
      (/ (* (* (* x x) y_m) 0.5) (* z x))
      (/ (* 0.5 (* y_m x)) z)))))

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 <= 1.45) {
		tmp = y_m / (z * x);
	} else if (x <= 3.5e+259) {
		tmp = (((x * x) * y_m) * 0.5) / (z * x);
	} else {
		tmp = (0.5 * (y_m * x)) / z;
	}
	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 <= 1.45d0) then
        tmp = y_m / (z * x)
    else if (x <= 3.5d+259) then
        tmp = (((x * x) * y_m) * 0.5d0) / (z * x)
    else
        tmp = (0.5d0 * (y_m * x)) / z
    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 <= 1.45) {
		tmp = y_m / (z * x);
	} else if (x <= 3.5e+259) {
		tmp = (((x * x) * y_m) * 0.5) / (z * x);
	} else {
		tmp = (0.5 * (y_m * x)) / z;
	}
	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 <= 1.45:
		tmp = y_m / (z * x)
	elif x <= 3.5e+259:
		tmp = (((x * x) * y_m) * 0.5) / (z * x)
	else:
		tmp = (0.5 * (y_m * x)) / z
	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 <= 1.45)
		tmp = Float64(y_m / Float64(z * x));
	elseif (x <= 3.5e+259)
		tmp = Float64(Float64(Float64(Float64(x * x) * y_m) * 0.5) / Float64(z * x));
	else
		tmp = Float64(Float64(0.5 * Float64(y_m * x)) / z);
	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 <= 1.45)
		tmp = y_m / (z * x);
	elseif (x <= 3.5e+259)
		tmp = (((x * x) * y_m) * 0.5) / (z * x);
	else
		tmp = (0.5 * (y_m * x)) / z;
	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, 1.45], N[(y$95$m / N[(z * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.5e+259], N[(N[(N[(N[(x * x), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.44999999999999996

   1. Initial program 87.4%

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

   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 63.8

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

   4. Applied rewrites 63.8%

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

   if 1.44999999999999996 < x < 3.4999999999999998e259

   1. Initial program 81.9%

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

   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 62.4

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

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 62.4

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

   7. Applied rewrites 62.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 44.3

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

   9. Applied rewrites 44.3%

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

   if 3.4999999999999998e259 < x

   1. Initial program 55.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   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 47.5

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

   7. Applied rewrites 47.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 57.9

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

   9. Applied rewrites 57.9%

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

Alternative 12: 57.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\frac{y\_m}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} \cdot y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

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 1.45) (/ 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 <= 1.45) {
		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 <= 1.45d0) 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 <= 1.45) {
		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 <= 1.45:
		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 <= 1.45)
		tmp = Float64(y_m / Float64(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 <= 1.45)
		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, 1.45], N[(y$95$m / N[(z * x), $MachinePrecision]), $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 < 1.44999999999999996

   1. Initial program 87.4%

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

   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 63.8

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

   4. Applied rewrites 63.8%

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

   if 1.44999999999999996 < x

   1. Initial program 77.7%

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

   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 75.6

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

   4. Applied rewrites 75.6%

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

   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 66.7

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

   7. Applied rewrites 66.7%

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

   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 38.3

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

   10. Applied rewrites 38.3%

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

Alternative 13: 57.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\frac{y\_m}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(y\_m \cdot x\right)}{z}\\ \end{array} \end{array} \]

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 1.45) (/ y_m (* z x)) (/ (* 0.5 (* y_m x)) z))))

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 <= 1.45) {
		tmp = y_m / (z * x);
	} else {
		tmp = (0.5 * (y_m * x)) / z;
	}
	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 <= 1.45d0) then
        tmp = y_m / (z * x)
    else
        tmp = (0.5d0 * (y_m * x)) / z
    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 <= 1.45) {
		tmp = y_m / (z * x);
	} else {
		tmp = (0.5 * (y_m * x)) / z;
	}
	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 <= 1.45:
		tmp = y_m / (z * x)
	else:
		tmp = (0.5 * (y_m * x)) / z
	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 <= 1.45)
		tmp = Float64(y_m / Float64(z * x));
	else
		tmp = Float64(Float64(0.5 * Float64(y_m * x)) / z);
	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 <= 1.45)
		tmp = y_m / (z * x);
	else
		tmp = (0.5 * (y_m * x)) / z;
	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, 1.45], N[(y$95$m / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 87.4%

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

   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 63.8

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

   4. Applied rewrites 63.8%

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

   if 1.44999999999999996 < x

   1. Initial program 77.7%

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

   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 68.3

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

   4. Applied rewrites 68.3%

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

   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 30.9

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

   7. Applied rewrites 30.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 39.0

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

   9. Applied rewrites 39.0%

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

Alternative 14: 55.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\frac{y\_m}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{y\_m}{z}\right) \cdot 0.5\\ \end{array} \end{array} \]

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 1.45) (/ 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 <= 1.45) {
		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 <= 1.45d0) 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 <= 1.45) {
		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 <= 1.45:
		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 <= 1.45)
		tmp = Float64(y_m / Float64(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 <= 1.45)
		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, 1.45], N[(y$95$m / N[(z * x), $MachinePrecision]), $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 < 1.44999999999999996

   1. Initial program 87.4%

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

   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 63.8

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

   4. Applied rewrites 63.8%

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

   if 1.44999999999999996 < x

   1. Initial program 77.7%

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

   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 68.3

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

   4. Applied rewrites 68.3%

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

   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 30.9

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

   7. Applied rewrites 30.9%

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

Alternative 15: 51.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y\_m}{x}}{z} \leq 10^{+214}:\\ \;\;\;\;\frac{y\_m}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x}\\ \end{array} \end{array} \]

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) 1e+214)
    (/ 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 (((cosh(x) * (y_m / x)) / z) <= 1e+214) {
		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 (((cosh(x) * (y_m / x)) / z) <= 1d+214) 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 (((Math.cosh(x) * (y_m / x)) / z) <= 1e+214) {
		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 ((math.cosh(x) * (y_m / x)) / z) <= 1e+214:
		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 (Float64(Float64(cosh(x) * Float64(y_m / x)) / z) <= 1e+214)
		tmp = Float64(y_m / Float64(z * x));
	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) <= 1e+214)
		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[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e+214], N[(y$95$m / N[(z * x), $MachinePrecision]), $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) < 9.9999999999999995e213

   1. Initial program 96.4%

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

   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 63.1

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

   4. Applied rewrites 63.1%

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

   if 9.9999999999999995e213 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 68.5%

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

   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 78.6

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

   4. Applied rewrites 78.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 35.4%

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

Alternative 16: 48.9% 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 84.9%

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

2. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 48.9

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

4. Applied rewrites 48.9%

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

Reproduce

herbie shell --seed 2025110 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64
  (/ (* (cosh x) (/ y x)) z))