Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.4% → 97.6%

Time: 3.9s

Alternatives: 12

Speedup: 0.8×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.1 \cdot 10^{+188}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y\_m, 0.5, y\_m\right)}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.1e+188)
    (/ (/ (* (cosh x) y_m) x) z)
    (/ (/ (fma (* (* x x) y_m) 0.5 y_m) z) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.1e+188) {
		tmp = ((cosh(x) * y_m) / x) / z;
	} else {
		tmp = (fma(((x * x) * y_m), 0.5, y_m) / z) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.1e+188)
		tmp = Float64(Float64(Float64(cosh(x) * y_m) / x) / z);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x * x) * y_m), 0.5, y_m) / z) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.1e+188], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y$95$m), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5 + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.09999999999999999e188

   1. Initial program 83.6%

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

      1. lift-*.f64 N/A

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

      2. lift-cosh.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cosh.f64 97.7

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

   3. Applied rewrites 97.7%

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

   if 1.09999999999999999e188 < y

   1. Initial program 87.6%

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

   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 97.3

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

   4. Applied rewrites 97.3%

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

Alternative 2: 94.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.1%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 79.8

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

   4. Applied rewrites 79.8%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.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. lower-/.f64 N/A

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

      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. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

   6. Applied rewrites 52.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. +-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 87.0

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

   8. Applied rewrites 87.0%

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

Alternative 3: 77.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{-222}:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\cosh x \cdot y\_m}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot x\right) \cdot y\_m\right) \cdot 0.5}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 1.4e-222)
    (/ (/ y_m x) z)
    (if (<= x 3.5e+153)
      (/ (* (cosh x) y_m) (* z x))
      (/ (/ (* (* (* x x) y_m) 0.5) z) x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.4e-222) {
		tmp = (y_m / x) / z;
	} else if (x <= 3.5e+153) {
		tmp = (cosh(x) * y_m) / (z * x);
	} else {
		tmp = ((((x * x) * y_m) * 0.5) / z) / x;
	}
	return y_s * tmp;
}

Fortran

y\_m =     private
y\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.4d-222) then
        tmp = (y_m / x) / z
    else if (x <= 3.5d+153) then
        tmp = (cosh(x) * y_m) / (z * x)
    else
        tmp = ((((x * x) * y_m) * 0.5d0) / z) / x
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.4e-222) {
		tmp = (y_m / x) / z;
	} else if (x <= 3.5e+153) {
		tmp = (Math.cosh(x) * y_m) / (z * x);
	} else {
		tmp = ((((x * x) * y_m) * 0.5) / z) / x;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 1.4e-222:
		tmp = (y_m / x) / z
	elif x <= 3.5e+153:
		tmp = (math.cosh(x) * y_m) / (z * x)
	else:
		tmp = ((((x * x) * y_m) * 0.5) / z) / x
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 1.4e-222)
		tmp = Float64(Float64(y_m / x) / z);
	elseif (x <= 3.5e+153)
		tmp = Float64(Float64(cosh(x) * y_m) / Float64(z * x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * y_m) * 0.5) / z) / x);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 1.4e-222)
		tmp = (y_m / x) / z;
	elseif (x <= 3.5e+153)
		tmp = (cosh(x) * y_m) / (z * x);
	else
		tmp = ((((x * x) * y_m) * 0.5) / z) / x;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 1.4e-222], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[x, 3.5e+153], N[(N[(N[Cosh[x], $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.40000000000000004e-222

   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 54.3

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

   4. Applied rewrites 54.3%

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

   if 1.40000000000000004e-222 < x < 3.4999999999999999e153

   1. Initial program 91.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cosh.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 91.1

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

   3. Applied rewrites 91.1%

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

   if 3.4999999999999999e153 < x

   1. Initial program 65.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

Alternative 4: 71.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y\_m}{x}}{z} \leq 10^{+57}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}{z \cdot x} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y\_m, 0.5, y\_m\right)}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (* (cosh x) (/ y_m x)) z) 1e+57)
    (* (/ (fma (* x x) 0.5 1.0) (* z x)) y_m)
    (/ (/ (fma (* (* x x) y_m) 0.5 y_m) z) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((cosh(x) * (y_m / x)) / z) <= 1e+57) {
		tmp = (fma((x * x), 0.5, 1.0) / (z * x)) * y_m;
	} else {
		tmp = (fma(((x * x) * y_m), 0.5, y_m) / z) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x) * Float64(y_m / x)) / z) <= 1e+57)
		tmp = Float64(Float64(fma(Float64(x * x), 0.5, 1.0) / Float64(z * x)) * y_m);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x * x) * y_m), 0.5, y_m) / z) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e+57], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5 + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.00000000000000005e57

   1. Initial program 96.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 80.4

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

   4. Applied rewrites 80.4%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.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. lower-/.f64 N/A

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

      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. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

   6. Applied rewrites 73.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 73.7

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

   8. Applied rewrites 73.7%

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

   if 1.00000000000000005e57 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 71.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 82.0

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

   4. Applied rewrites 82.0%

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

Alternative 5: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 500000:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \frac{1}{x}\right) \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot x\right) \cdot y\_m\right) \cdot 0.5}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 500000.0)
    (* (fma 0.5 x (/ 1.0 x)) (/ y_m z))
    (/ (/ (* (* (* x x) y_m) 0.5) z) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 500000.0) {
		tmp = fma(0.5, x, (1.0 / x)) * (y_m / z);
	} else {
		tmp = ((((x * x) * y_m) * 0.5) / z) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 500000.0)
		tmp = Float64(fma(0.5, x, Float64(1.0 / x)) * Float64(y_m / z));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * y_m) * 0.5) / z) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 500000.0], N[(N[(0.5 * x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5e5

   1. Initial program 86.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 84.3

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

   4. Applied rewrites 84.3%

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

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

      3. lift-*.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. lower-/.f64 N/A

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

      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. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 79.7%

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

   7. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. pow-flip N/A

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

      3. pow-plus N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. inv-pow N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 71.2

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

   9. Applied rewrites 71.2%

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

   if 5e5 < x

   1. Initial program 77.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 71.5

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

   4. Applied rewrites 71.5%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 71.5

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

   7. Applied rewrites 71.5%

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

Alternative 6: 65.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 500000:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \frac{1}{x}\right) \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot y\_m}{z \cdot x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 500000.0)
    (* (fma 0.5 x (/ 1.0 x)) (/ y_m z))
    (/ (* (* (* x x) 0.5) y_m) (* z x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 500000.0) {
		tmp = fma(0.5, x, (1.0 / x)) * (y_m / z);
	} else {
		tmp = (((x * x) * 0.5) * y_m) / (z * x);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 500000.0)
		tmp = Float64(fma(0.5, x, Float64(1.0 / x)) * Float64(y_m / z));
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * 0.5) * y_m) / Float64(z * x));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 500000.0], N[(N[(0.5 * x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5e5

   1. Initial program 86.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 84.3

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

   4. Applied rewrites 84.3%

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

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

      3. lift-*.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. lower-/.f64 N/A

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

      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. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 79.7%

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

   7. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. pow-flip N/A

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

      3. pow-plus N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. inv-pow N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 71.2

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

   9. Applied rewrites 71.2%

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

   if 5e5 < x

   1. Initial program 77.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 47.8

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

   4. Applied rewrites 47.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 47.8

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

   7. Applied rewrites 47.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 48.2

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

   9. Applied rewrites 48.2%

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

Alternative 7: 64.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \frac{1}{x}\right) \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\left(x \cdot x\right) \cdot 0.5}{z \cdot x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 2.2e+117)
    (* (fma 0.5 x (/ 1.0 x)) (/ y_m z))
    (* y_m (/ (* (* x x) 0.5) (* z x))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.2e+117) {
		tmp = fma(0.5, x, (1.0 / x)) * (y_m / z);
	} else {
		tmp = y_m * (((x * x) * 0.5) / (z * x));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 2.2e+117)
		tmp = Float64(fma(0.5, x, Float64(1.0 / x)) * Float64(y_m / z));
	else
		tmp = Float64(y_m * Float64(Float64(Float64(x * x) * 0.5) / Float64(z * x)));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 2.2e+117], N[(N[(0.5 * x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.20000000000000014e117

   1. Initial program 87.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 79.3

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

   4. Applied rewrites 79.3%

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

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

      3. lift-*.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. lower-/.f64 N/A

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

      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. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 73.6%

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

   7. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. pow-flip N/A

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

      3. pow-plus N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. inv-pow N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 66.0

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

   9. Applied rewrites 66.0%

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

   if 2.20000000000000014e117 < x

   1. Initial program 68.6%

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

   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 91.5

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

   4. Applied rewrites 91.5%

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

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

      3. lift-*.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. lower-/.f64 N/A

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

      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. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

   6. Applied rewrites 58.8%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 58.8

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

   9. Applied rewrites 58.8%

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

Alternative 8: 63.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \frac{1}{x}\right) \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(\frac{x}{z} \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 4e+88)
    (* (fma 0.5 x (/ 1.0 x)) (/ y_m z))
    (* y_m (* (/ x z) 0.5)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 4e+88) {
		tmp = fma(0.5, x, (1.0 / x)) * (y_m / z);
	} else {
		tmp = y_m * ((x / z) * 0.5);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 4e+88)
		tmp = Float64(fma(0.5, x, Float64(1.0 / x)) * Float64(y_m / z));
	else
		tmp = Float64(y_m * Float64(Float64(x / z) * 0.5));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 4e+88], N[(N[(0.5 * x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(x / z), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.99999999999999984e88

   1. Initial program 87.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 80.3

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

   4. Applied rewrites 80.3%

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

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

      3. lift-*.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. lower-/.f64 N/A

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

      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. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 75.2%

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

   7. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. pow-flip N/A

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

      3. pow-plus N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. inv-pow N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 67.4

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

   9. Applied rewrites 67.4%

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

   if 3.99999999999999984e88 < x

   1. Initial program 70.9%

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

   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 85.0

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

   4. Applied rewrites 85.0%

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

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

      3. lift-*.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. lower-/.f64 N/A

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

      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. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

   6. Applied rewrites 54.6%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 46.7

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

   9. Applied rewrites 46.7%

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

Alternative 9: 59.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

y\_m =     private
y\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 86.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 84.7

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

   4. Applied rewrites 84.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 65.7%

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

   if 1.3999999999999999 < x

   1. Initial program 77.6%

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

   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 70.5

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

   4. Applied rewrites 70.5%

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

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

      3. lift-*.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. lower-/.f64 N/A

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

      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. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

   6. Applied rewrites 45.9%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 40.1

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

   9. Applied rewrites 40.1%

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

Alternative 10: 57.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

y\_m =     private
y\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 86.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 84.7

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

   4. Applied rewrites 84.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 65.7%

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

   if 1.3999999999999999 < x

   1. Initial program 77.6%

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

   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 70.5

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

   4. Applied rewrites 70.5%

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

   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 32.9

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

   7. Applied rewrites 32.9%

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

Alternative 11: 53.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y\_m}{x}}{z} \leq 10^{+57}:\\ \;\;\;\;\frac{y\_m}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (* (cosh x) (/ y_m x)) z) 1e+57)
    (/ y_m (* z x))
    (/ (/ y_m z) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((cosh(x) * (y_m / x)) / z) <= 1e+57) {
		tmp = y_m / (z * x);
	} else {
		tmp = (y_m / z) / x;
	}
	return y_s * tmp;
}

Fortran

y\_m =     private
y\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((cosh(x) * (y_m / x)) / z) <= 1d+57) then
        tmp = y_m / (z * x)
    else
        tmp = (y_m / z) / x
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((Math.cosh(x) * (y_m / x)) / z) <= 1e+57) {
		tmp = y_m / (z * x);
	} else {
		tmp = (y_m / z) / x;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if ((math.cosh(x) * (y_m / x)) / z) <= 1e+57:
		tmp = y_m / (z * x)
	else:
		tmp = (y_m / z) / x
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x) * Float64(y_m / x)) / z) <= 1e+57)
		tmp = Float64(y_m / Float64(z * x));
	else
		tmp = Float64(Float64(y_m / z) / x);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (((cosh(x) * (y_m / x)) / z) <= 1e+57)
		tmp = y_m / (z * x);
	else
		tmp = (y_m / z) / x;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e+57], N[(y$95$m / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.00000000000000005e57

   1. Initial program 96.1%

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

   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 62.0

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

   4. Applied rewrites 62.0%

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

   if 1.00000000000000005e57 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 71.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 82.0

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

   4. Applied rewrites 82.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 43.1%

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

Alternative 12: 49.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

y\_m =     private
y\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

2. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 49.3

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

4. Applied rewrites 49.3%

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

Reproduce

herbie shell --seed 2025112 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64
  (/ (* (cosh x) (/ y x)) z))