Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.9% → 93.1%

Time: 3.2s

Alternatives: 12

Speedup: 0.9×

• 63.4% 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: 93.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4.2 \cdot 10^{+142}:\\ \;\;\;\;\frac{\cosh x\_m \cdot y}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)}{z}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 4.2e+142)
    (/ (* (cosh x_m) y) (* z x_m))
    (/ (* y (/ (fma (* x_m x_m) 0.5 1.0) z)) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 4.2e+142) {
		tmp = (cosh(x_m) * y) / (z * x_m);
	} else {
		tmp = (y * (fma((x_m * x_m), 0.5, 1.0) / z)) / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 4.2e+142)
		tmp = Float64(Float64(cosh(x_m) * y) / Float64(z * x_m));
	else
		tmp = Float64(Float64(y * Float64(fma(Float64(x_m * x_m), 0.5, 1.0) / z)) / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 4.2e+142], N[(N[(N[Cosh[x$95$m], $MachinePrecision] * y), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.2e142

   1. Initial program 84.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 83.7

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

   3. Applied rewrites 83.7%

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

   if 4.2e142 < x

   1. Initial program 84.9%

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

   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 81.3

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

   4. Applied rewrites 81.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 77.3

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

   6. Applied rewrites 77.3%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. pow2 N/A

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

      9. div-add-rev N/A

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

      10. +-commutative N/A

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

      11. div-add N/A

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

      12. *-lft-identity N/A

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

      13. div-add N/A

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

      14. associate-*r* N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

   8. Applied rewrites 81.4%

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

Alternative 2: 92.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4.2 \cdot 10^{+142}:\\ \;\;\;\;y \cdot \frac{\cosh x\_m}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)}{z}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 4.2e+142)
    (* y (/ (cosh x_m) (* x_m z)))
    (/ (* y (/ (fma (* x_m x_m) 0.5 1.0) z)) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 4.2e+142) {
		tmp = y * (cosh(x_m) / (x_m * z));
	} else {
		tmp = (y * (fma((x_m * x_m), 0.5, 1.0) / z)) / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 4.2e+142)
		tmp = Float64(y * Float64(cosh(x_m) / Float64(x_m * z)));
	else
		tmp = Float64(Float64(y * Float64(fma(Float64(x_m * x_m), 0.5, 1.0) / z)) / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 4.2e+142], N[(y * N[(N[Cosh[x$95$m], $MachinePrecision] / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.2e142

   1. Initial program 84.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-cosh.f64 84.9

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

   3. Applied rewrites 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-cosh.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-cosh.f64 90.4

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

   5. Applied rewrites 90.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-cosh.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lift-cosh.f64 N/A

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

      15. lower-*.f64 83.3

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

   7. Applied rewrites 83.3%

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

   if 4.2e142 < x

   1. Initial program 84.9%

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

   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 81.3

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

   4. Applied rewrites 81.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 77.3

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

   6. Applied rewrites 77.3%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. pow2 N/A

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

      9. div-add-rev N/A

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

      10. +-commutative N/A

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

      11. div-add N/A

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

      12. *-lft-identity N/A

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

      13. div-add N/A

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

      14. associate-*r* N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

   8. Applied rewrites 81.4%

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

Alternative 3: 80.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y}{x\_m}}{z} \leq 5 \cdot 10^{+149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, \left(x\_m \cdot y\right) \cdot 0.5, y\right)}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)}{z}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* (cosh x_m) (/ y x_m)) z) 5e+149)
    (/ (fma x_m (* (* x_m y) 0.5) y) (* z x_m))
    (/ (* y (/ (fma (* x_m x_m) 0.5 1.0) z)) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((cosh(x_m) * (y / x_m)) / z) <= 5e+149) {
		tmp = fma(x_m, ((x_m * y) * 0.5), y) / (z * x_m);
	} else {
		tmp = (y * (fma((x_m * x_m), 0.5, 1.0) / z)) / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y / x_m)) / z) <= 5e+149)
		tmp = Float64(fma(x_m, Float64(Float64(x_m * y) * 0.5), y) / Float64(z * x_m));
	else
		tmp = Float64(Float64(y * Float64(fma(Float64(x_m * x_m), 0.5, 1.0) / z)) / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 5e+149], N[(N[(x$95$m * N[(N[(x$95$m * y), $MachinePrecision] * 0.5), $MachinePrecision] + y), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 4.9999999999999999e149

   1. Initial program 84.9%

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

   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 81.3

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

   4. Applied rewrites 81.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y, 0.5, y\right)}{z}}{x}} \]
   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 69.4

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

   6. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 65.9

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

   8. Applied rewrites 65.9%

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

   if 4.9999999999999999e149 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 84.9%

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

   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 81.3

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

   4. Applied rewrites 81.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 77.3

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

   6. Applied rewrites 77.3%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. pow2 N/A

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

      9. div-add-rev N/A

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

      10. +-commutative N/A

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

      11. div-add N/A

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

      12. *-lft-identity N/A

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

      13. div-add N/A

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

      14. associate-*r* N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

   8. Applied rewrites 81.4%

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

Alternative 4: 80.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y}{x\_m} \leq 10^{+265}:\\ \;\;\;\;\frac{\frac{y}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\mathsf{fma}\left(x\_m, x\_m, 2\right)}{z}}{x\_m} \cdot y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* (cosh x_m) (/ y x_m)) 1e+265)
    (/ (/ y x_m) z)
    (* (* (/ (/ (fma x_m x_m 2.0) z) x_m) y) 0.5))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((cosh(x_m) * (y / x_m)) <= 1e+265) {
		tmp = (y / x_m) / z;
	} else {
		tmp = (((fma(x_m, x_m, 2.0) / z) / x_m) * y) * 0.5;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y / x_m)) <= 1e+265)
		tmp = Float64(Float64(y / x_m) / z);
	else
		tmp = Float64(Float64(Float64(Float64(fma(x_m, x_m, 2.0) / z) / x_m) * y) * 0.5);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision], 1e+265], N[(N[(y / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m + 2.0), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision] * y), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.00000000000000007e265

   1. Initial program 84.9%

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

   2. Taylor expanded in x around 0

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

      1. lift-/.f64 49.9

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

   4. Applied rewrites 49.9%

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

   if 1.00000000000000007e265 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 84.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.3

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

   4. Applied rewrites 83.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 79.8

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

   7. Applied rewrites 79.8%

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

Alternative 5: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.55 \cdot 10^{+76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, \left(x\_m \cdot y\right) \cdot 0.5, y\right)}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x\_m \cdot x\_m}{z}}{x\_m} \cdot y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.55e+76)
    (/ (fma x_m (* (* x_m y) 0.5) y) (* z x_m))
    (* (* (/ (/ (* x_m x_m) z) x_m) y) 0.5))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1.55e+76) {
		tmp = fma(x_m, ((x_m * y) * 0.5), y) / (z * x_m);
	} else {
		tmp = ((((x_m * x_m) / z) / x_m) * y) * 0.5;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 1.55e+76)
		tmp = Float64(fma(x_m, Float64(Float64(x_m * y) * 0.5), y) / Float64(z * x_m));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x_m * x_m) / z) / x_m) * y) * 0.5);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 1.55e+76], N[(N[(x$95$m * N[(N[(x$95$m * y), $MachinePrecision] * 0.5), $MachinePrecision] + y), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision] * y), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000006e76

   1. Initial program 84.9%

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

   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 81.3

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

   4. Applied rewrites 81.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y, 0.5, y\right)}{z}}{x}} \]
   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 69.4

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

   6. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 65.9

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

   8. Applied rewrites 65.9%

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

   if 1.55000000000000006e76 < x

   1. Initial program 84.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.3

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

   4. Applied rewrites 83.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 79.8

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

   7. Applied rewrites 79.8%

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

   8. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 39.1

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

   10. Applied rewrites 39.1%

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

Alternative 6: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.55 \cdot 10^{+76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, x\_m, 2\right) \cdot y}{z \cdot x\_m} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x\_m \cdot x\_m}{z}}{x\_m} \cdot y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.55e+76)
    (* (/ (* (fma x_m x_m 2.0) y) (* z x_m)) 0.5)
    (* (* (/ (/ (* x_m x_m) z) x_m) y) 0.5))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1.55e+76) {
		tmp = ((fma(x_m, x_m, 2.0) * y) / (z * x_m)) * 0.5;
	} else {
		tmp = ((((x_m * x_m) / z) / x_m) * y) * 0.5;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 1.55e+76)
		tmp = Float64(Float64(Float64(fma(x_m, x_m, 2.0) * y) / Float64(z * x_m)) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x_m * x_m) / z) / x_m) * y) * 0.5);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 1.55e+76], N[(N[(N[(N[(x$95$m * x$95$m + 2.0), $MachinePrecision] * y), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision] * y), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000006e76

   1. Initial program 84.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.3

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

   4. Applied rewrites 83.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 68.3

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

   7. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 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 69.4

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

   9. Applied rewrites 69.4%

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

   if 1.55000000000000006e76 < x

   1. Initial program 84.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.3

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

   4. Applied rewrites 83.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 79.8

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

   7. Applied rewrites 79.8%

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

   8. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 39.1

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

   10. Applied rewrites 39.1%

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

Alternative 7: 71.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   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 50.0

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

   4. Applied rewrites 50.0%

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

   if 1.3999999999999999 < x

   1. Initial program 84.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.3

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

   4. Applied rewrites 83.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 79.8

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

   7. Applied rewrites 79.8%

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

   8. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 39.1

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

   10. Applied rewrites 39.1%

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

Alternative 8: 68.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   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 50.0

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

   4. Applied rewrites 50.0%

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

   if 1.3999999999999999 < x

   1. Initial program 84.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.3

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

   4. Applied rewrites 83.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 68.3

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

   7. Applied rewrites 68.3%

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

   8. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 28.6

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

   10. Applied rewrites 28.6%

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

Alternative 9: 66.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   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 50.0

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

   4. Applied rewrites 50.0%

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

   if 1.3999999999999999 < x

   1. Initial program 84.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.3

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

   4. Applied rewrites 83.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 79.8

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

   7. Applied rewrites 79.8%

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

   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 25.7

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

   10. Applied rewrites 25.7%

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

Alternative 10: 62.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   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 50.0

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

   4. Applied rewrites 50.0%

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

   if 1.3999999999999999 < x

   1. Initial program 84.9%

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

   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 81.3

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

   4. Applied rewrites 81.3%

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

   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 25.9

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

   7. Applied rewrites 25.9%

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

Alternative 11: 53.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y}{x\_m}}{z} \leq 10^{+30}:\\ \;\;\;\;\frac{y}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* (cosh x_m) (/ y x_m)) z) 1e+30)
    (/ y (* z x_m))
    (/ (/ y z) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((cosh(x_m) * (y / x_m)) / z) <= 1e+30) {
		tmp = y / (z * x_m);
	} else {
		tmp = (y / z) / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_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(x_s, x_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((cosh(x_m) * (y / x_m)) / z) <= 1d+30) then
        tmp = y / (z * x_m)
    else
        tmp = (y / z) / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((Math.cosh(x_m) * (y / x_m)) / z) <= 1e+30) {
		tmp = y / (z * x_m);
	} else {
		tmp = (y / z) / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if ((math.cosh(x_m) * (y / x_m)) / z) <= 1e+30:
		tmp = y / (z * x_m)
	else:
		tmp = (y / z) / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y / x_m)) / z) <= 1e+30)
		tmp = Float64(y / Float64(z * x_m));
	else
		tmp = Float64(Float64(y / z) / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (((cosh(x_m) * (y / x_m)) / z) <= 1e+30)
		tmp = y / (z * x_m);
	else
		tmp = (y / z) / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e+30], N[(y / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1e30

   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 50.0

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

   4. Applied rewrites 50.0%

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

   if 1e30 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 84.9%

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

   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 81.3

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

   4. Applied rewrites 81.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 54.3%

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

Alternative 12: 50.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[(y / N[(z * x$95$m), $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 50.0

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

4. Applied rewrites 50.0%

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

Reproduce

herbie shell --seed 2025139 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64
  (/ (* (cosh x) (/ y x)) z))