Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.9% → 99.8%

Time: 8.7s

Alternatives: 18

Speedup: 2.3×

• 64.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \cosh x\_m \cdot \frac{y\_m}{x\_m}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+241}:\\ \;\;\;\;\frac{t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2 \cdot \cosh x\_m}{x\_m}}{z} \cdot y\_m}{2}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (cosh x_m) (/ y_m x_m))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 2e+241)
       (/ t_0 z)
       (/ (* (/ (/ (* 2.0 (cosh x_m)) x_m) z) y_m) 2.0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = cosh(x_m) * (y_m / x_m);
	double tmp;
	if (t_0 <= 2e+241) {
		tmp = t_0 / z;
	} else {
		tmp = ((((2.0 * cosh(x_m)) / x_m) / z) * y_m) / 2.0;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cosh(x_m) * (y_m / x_m)
    if (t_0 <= 2d+241) then
        tmp = t_0 / z
    else
        tmp = ((((2.0d0 * cosh(x_m)) / x_m) / z) * y_m) / 2.0d0
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = Math.cosh(x_m) * (y_m / x_m);
	double tmp;
	if (t_0 <= 2e+241) {
		tmp = t_0 / z;
	} else {
		tmp = ((((2.0 * Math.cosh(x_m)) / x_m) / z) * y_m) / 2.0;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x_s, x_m, y_m, z):
	t_0 = math.cosh(x_m) * (y_m / x_m)
	tmp = 0
	if t_0 <= 2e+241:
		tmp = t_0 / z
	else:
		tmp = ((((2.0 * math.cosh(x_m)) / x_m) / z) * y_m) / 2.0
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(cosh(x_m) * Float64(y_m / x_m))
	tmp = 0.0
	if (t_0 <= 2e+241)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(2.0 * cosh(x_m)) / x_m) / z) * y_m) / 2.0);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = cosh(x_m) * (y_m / x_m);
	tmp = 0.0;
	if (t_0 <= 2e+241)
		tmp = t_0 / z;
	else
		tmp = ((((2.0 * cosh(x_m)) / x_m) / z) * y_m) / 2.0;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 2e+241], N[(t$95$0 / z), $MachinePrecision], N[(N[(N[(N[(N[(2.0 * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.0000000000000001e241

   1. Initial program 99.7%

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

   if 2.0000000000000001e241 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 77.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cosh.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l* N/A

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

      6. cosh-def N/A

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

      7. rec-exp N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

Alternative 2: 94.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 2 \cdot 10^{+241}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot \frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x\_m \cdot x\_m, 0.08333333333333333\right), x\_m \cdot x\_m, 1\right), x\_m \cdot x\_m, 2\right)}{x\_m}}{z} \cdot y\_m}{2}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* (cosh x_m) (/ y_m x_m)) 2e+241)
     (/
      (*
       (fma
        (fma
         (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664)
         (* x_m x_m)
         0.5)
        (* x_m x_m)
        1.0)
       (/ y_m x_m))
      z)
     (/
      (*
       (/
        (/
         (fma
          (fma
           (fma 0.002777777777777778 (* x_m x_m) 0.08333333333333333)
           (* x_m x_m)
           1.0)
          (* x_m x_m)
          2.0)
         x_m)
        z)
       y_m)
      2.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 2e+241) {
		tmp = (fma(fma(fma(0.001388888888888889, (x_m * x_m), 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0) * (y_m / x_m)) / z;
	} else {
		tmp = (((fma(fma(fma(0.002777777777777778, (x_m * x_m), 0.08333333333333333), (x_m * x_m), 1.0), (x_m * x_m), 2.0) / x_m) / z) * y_m) / 2.0;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 2e+241)
		tmp = Float64(Float64(fma(fma(fma(0.001388888888888889, Float64(x_m * x_m), 0.041666666666666664), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) * Float64(y_m / x_m)) / z);
	else
		tmp = Float64(Float64(Float64(Float64(fma(fma(fma(0.002777777777777778, Float64(x_m * x_m), 0.08333333333333333), Float64(x_m * x_m), 1.0), Float64(x_m * x_m), 2.0) / x_m) / z) * y_m) / 2.0);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 2e+241], N[(N[(N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(0.002777777777777778 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.0000000000000001e241

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \frac{y}{x}}{z} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + \color{blue}{1}\right) \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), \color{blue}{{x}^{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}, {\color{blue}{x}}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} + \frac{1}{2}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right), {\color{blue}{x}}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      13. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot \color{blue}{x}, 1\right) \cdot \frac{y}{x}}{z} \]

      14. lower-*.f64 98.8

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot \color{blue}{x}, 1\right) \cdot \frac{y}{x}}{z} \]

   4. Applied rewrites 98.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]

   if 2.0000000000000001e241 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 77.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cosh.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l* N/A

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

      6. cosh-def N/A

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

      7. rec-exp N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2 \cdot \cosh x}{x}}{z} \cdot y}{2}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\frac{\color{blue}{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}}{x}}{z} \cdot y}{2} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) + \color{blue}{2}}{x}}{z} \cdot y}{2} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 2}{x}}{z} \cdot y}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right), \color{blue}{{x}^{2}}, 2\right)}{x}}{z} \cdot y}{2} \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 1, {\color{blue}{x}}^{2}, 2\right)}{x}}{z} \cdot y}{2} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot {x}^{2} + 1, {x}^{2}, 2\right)}{x}}{z} \cdot y}{2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, {x}^{2}, 1\right), {\color{blue}{x}}^{2}, 2\right)}{x}}{z} \cdot y}{2} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {x}^{2} + \frac{1}{12}, {x}^{2}, 1\right), {x}^{2}, 2\right)}{x}}{z} \cdot y}{2} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {x}^{2}, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)}{x}}{z} \cdot y}{2} \]

      9. pow2 N/A

         \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)}{x}}{z} \cdot y}{2} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)}{x}}{z} \cdot y}{2} \]

      11. pow2 N/A

          \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), {x}^{2}, 2\right)}{x}}{z} \cdot y}{2} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), {x}^{2}, 2\right)}{x}}{z} \cdot y}{2} \]

      13. pow2 N/A

          \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)}{x}}{z} \cdot y}{2} \]

      14. lift-*.f64 92.1

          \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)}{x}}{z} \cdot y}{2} \]

   6. Applied rewrites 92.1%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}}{x}}{z} \cdot y}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664, x\_m \cdot x\_m, 1\right) \cdot \frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(x\_m, x\_m, 2\right)}{z}}{x\_m} \cdot y\_m}{2}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) INFINITY)
     (/
      (*
       (fma (* (* x_m x_m) 0.041666666666666664) (* x_m x_m) 1.0)
       (/ y_m x_m))
      z)
     (/ (* (/ (/ (fma x_m x_m 2.0) z) x_m) y_m) 2.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= ((double) INFINITY)) {
		tmp = (fma(((x_m * x_m) * 0.041666666666666664), (x_m * x_m), 1.0) * (y_m / x_m)) / z;
	} else {
		tmp = (((fma(x_m, x_m, 2.0) / z) / x_m) * y_m) / 2.0;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= Inf)
		tmp = Float64(Float64(fma(Float64(Float64(x_m * x_m) * 0.041666666666666664), Float64(x_m * x_m), 1.0) * Float64(y_m / x_m)) / z);
	else
		tmp = Float64(Float64(Float64(Float64(fma(x_m, x_m, 2.0) / z) / x_m) * y_m) / 2.0);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], Infinity], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m + 2.0), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / 2.0), $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 95.6%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + \color{blue}{1}\right) \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, \color{blue}{{x}^{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}, {\color{blue}{x}}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right), {\color{blue}{x}}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot \color{blue}{x}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-*.f64 86.8

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot \color{blue}{x}, 1\right) \cdot \frac{y}{x}}{z} \]

   4. Applied rewrites 86.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2}, \color{blue}{x} \cdot x, 1\right) \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({x}^{2} \cdot \frac{1}{24}, x \cdot x, 1\right) \cdot \frac{y}{x}}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({x}^{2} \cdot \frac{1}{24}, x \cdot x, 1\right) \cdot \frac{y}{x}}{z} \]

      3. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{24}, x \cdot x, 1\right) \cdot \frac{y}{x}}{z} \]

      4. lift-*.f64 86.4

         \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x \cdot x, 1\right) \cdot \frac{y}{x}}{z} \]

   7. Applied rewrites 86.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, \color{blue}{x} \cdot x, 1\right) \cdot \frac{y}{x}}{z} \]

   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. 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-/l* N/A

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

      6. cosh-def N/A

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

      7. rec-exp N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2 \cdot \cosh x}{x}}{z} \cdot y}{2}} \]

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 85.7

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

   6. Applied rewrites 85.7%

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

Alternative 4: 80.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq 10^{+32}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right) \cdot \frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot y\_m, 0.5, y\_m\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) 1e+32)
     (/ (* (fma (* x_m x_m) 0.5 1.0) (/ y_m x_m)) z)
     (/ (/ (fma (* (* x_m x_m) y_m) 0.5 y_m) z) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 1e+32) {
		tmp = (fma((x_m * x_m), 0.5, 1.0) * (y_m / x_m)) / z;
	} else {
		tmp = (fma(((x_m * x_m) * y_m), 0.5, y_m) / z) / x_m;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 1e+32)
		tmp = Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) * Float64(y_m / x_m)) / z);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x_m * x_m) * y_m), 0.5, y_m) / z) / x_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e+32], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5 + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.00000000000000005e32

   1. Initial program 96.2%

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

   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 81.1

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

   4. Applied rewrites 81.1%

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

   if 1.00000000000000005e32 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 72.6%

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

   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 80.6

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

   4. Applied rewrites 80.6%

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

Alternative 5: 81.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot y\_m, 0.5, y\_m\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq 10^{+32}:\\ \;\;\;\;\frac{\frac{t\_0}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{z}}{x\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (fma (* (* x_m x_m) y_m) 0.5 y_m)))
   (*
    y_s
    (*
     x_s
     (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) 1e+32)
       (/ (/ t_0 x_m) z)
       (/ (/ t_0 z) x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = fma(((x_m * x_m) * y_m), 0.5, y_m);
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 1e+32) {
		tmp = (t_0 / x_m) / z;
	} else {
		tmp = (t_0 / z) / x_m;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	t_0 = fma(Float64(Float64(x_m * x_m) * y_m), 0.5, y_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 1e+32)
		tmp = Float64(Float64(t_0 / x_m) / z);
	else
		tmp = Float64(Float64(t_0 / z) / x_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5 + y$95$m), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e+32], N[(N[(t$95$0 / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(t$95$0 / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.00000000000000005e32

   1. Initial program 96.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 82.6

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

   4. Applied rewrites 82.6%

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

   if 1.00000000000000005e32 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 72.6%

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

   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 80.6

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

   4. Applied rewrites 80.6%

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

Alternative 6: 81.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq 10^{+32}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right) \cdot y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot y\_m, 0.5, y\_m\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) 1e+32)
     (/ (/ (* (fma (* x_m x_m) 0.5 1.0) y_m) x_m) z)
     (/ (/ (fma (* (* x_m x_m) y_m) 0.5 y_m) z) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 1e+32) {
		tmp = ((fma((x_m * x_m), 0.5, 1.0) * y_m) / x_m) / z;
	} else {
		tmp = (fma(((x_m * x_m) * y_m), 0.5, y_m) / z) / x_m;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 1e+32)
		tmp = Float64(Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) * y_m) / x_m) / z);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x_m * x_m) * y_m), 0.5, y_m) / z) / x_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e+32], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5 + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.00000000000000005e32

   1. Initial program 96.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 82.6

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

   4. Applied rewrites 82.6%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 82.6

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

   7. Applied rewrites 82.6%

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

   if 1.00000000000000005e32 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 72.6%

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

   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 80.6

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

   4. Applied rewrites 80.6%

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

Alternative 7: 80.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq 10^{+32}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot y\_m, 0.5, y\_m\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) 1e+32)
     (/ (* y_m (/ (fma 0.5 (* x_m x_m) 1.0) x_m)) z)
     (/ (/ (fma (* (* x_m x_m) y_m) 0.5 y_m) z) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 1e+32) {
		tmp = (y_m * (fma(0.5, (x_m * x_m), 1.0) / x_m)) / z;
	} else {
		tmp = (fma(((x_m * x_m) * y_m), 0.5, y_m) / z) / x_m;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 1e+32)
		tmp = Float64(Float64(y_m * Float64(fma(0.5, Float64(x_m * x_m), 1.0) / x_m)) / z);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x_m * x_m) * y_m), 0.5, y_m) / z) / x_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e+32], N[(N[(y$95$m * N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5 + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.00000000000000005e32

   1. Initial program 96.2%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{x}}}{z} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{\color{blue}{x}}}{z} \]

   4. Applied rewrites 90.4%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5 \cdot y\right), x \cdot x, y\right)}{x}}}{z} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}{\color{blue}{x}}}{z} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{y \cdot \frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}{\color{blue}{x}}}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}{\color{blue}{x}}}{z} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot \frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}{x}}{z} \]

   7. Applied rewrites 91.1%

      \[\leadsto \frac{y \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x}}}{z} \]

   8. Taylor expanded in x around 0

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

      1. *-commutative 81.1

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

      2. *-commutative 81.1

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

      3. *-commutative 81.1

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

   10. Applied rewrites 81.1%

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

   if 1.00000000000000005e32 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 72.6%

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

   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 80.6

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

   4. Applied rewrites 80.6%

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

Alternative 8: 70.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq 2 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot y\_m, 0.5, y\_m\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) 2e-36)
     (/ (/ y_m x_m) z)
     (/ (/ (fma (* (* x_m x_m) y_m) 0.5 y_m) z) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 2e-36) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = (fma(((x_m * x_m) * y_m), 0.5, y_m) / z) / x_m;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 2e-36)
		tmp = Float64(Float64(y_m / x_m) / z);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x_m * x_m) * y_m), 0.5, y_m) / z) / x_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 2e-36], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5 + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.9999999999999999e-36

   1. Initial program 96.0%

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

   2. Taylor expanded in x around 0

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

      1. lift-/.f64 60.0

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

   4. Applied rewrites 60.0%

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

   if 1.9999999999999999e-36 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 73.6%

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

   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 81.0

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

   4. Applied rewrites 81.0%

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

Alternative 9: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3 \cdot 10^{+48}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot x\_m, x\_m, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x\_m} \cdot \cosh x\_m\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= y_m 3e+48)
     (/
      (*
       y_m
       (/
        (fma
         (fma
          (fma (* 0.001388888888888889 x_m) x_m 0.041666666666666664)
          (* x_m x_m)
          0.5)
         (* x_m x_m)
         1.0)
        x_m))
      z)
     (* (/ (/ y_m z) x_m) (cosh x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 3e+48) {
		tmp = (y_m * (fma(fma(fma((0.001388888888888889 * x_m), x_m, 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m)) / z;
	} else {
		tmp = ((y_m / z) / x_m) * cosh(x_m);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 3e+48)
		tmp = Float64(Float64(y_m * Float64(fma(fma(fma(Float64(0.001388888888888889 * x_m), x_m, 0.041666666666666664), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / x_m)) / z);
	else
		tmp = Float64(Float64(Float64(y_m / z) / x_m) * cosh(x_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 3e+48], N[(N[(y$95$m * N[(N[(N[(N[(N[(0.001388888888888889 * x$95$m), $MachinePrecision] * x$95$m + 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y$95$m / z), $MachinePrecision] / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3e48

   1. Initial program 80.5%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{x}}}{z} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{\color{blue}{x}}}{z} \]

   4. Applied rewrites 89.8%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5 \cdot y\right), x \cdot x, y\right)}{x}}}{z} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}{\color{blue}{x}}}{z} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{y \cdot \frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}{\color{blue}{x}}}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}{\color{blue}{x}}}{z} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot \frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}{x}}{z} \]

   7. Applied rewrites 92.0%

      \[\leadsto \frac{y \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x}}}{z} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{x}}{z} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{x}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot \left(x \cdot x\right) + \frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{x}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{1}{720} \cdot x\right) \cdot x + \frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot x, x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{x}}{z} \]

      6. lower-*.f64 92.0

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot x, x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x}}{z} \]

   9. Applied rewrites 92.0%

      \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot x, x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x}}{z} \]

   if 3e48 < y

   1. Initial program 91.1%

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

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-cosh.f64 97.0

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

   3. Applied rewrites 97.0%

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

Alternative 10: 92.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 10^{+153}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot x\_m, x\_m, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot y\_m, 0.5, y\_m\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= y_m 1e+153)
     (/
      (*
       y_m
       (/
        (fma
         (fma
          (fma (* 0.001388888888888889 x_m) x_m 0.041666666666666664)
          (* x_m x_m)
          0.5)
         (* x_m x_m)
         1.0)
        x_m))
      z)
     (/ (/ (fma (* (* x_m x_m) y_m) 0.5 y_m) z) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 1e+153) {
		tmp = (y_m * (fma(fma(fma((0.001388888888888889 * x_m), x_m, 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m)) / z;
	} else {
		tmp = (fma(((x_m * x_m) * y_m), 0.5, y_m) / z) / x_m;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 1e+153)
		tmp = Float64(Float64(y_m * Float64(fma(fma(fma(Float64(0.001388888888888889 * x_m), x_m, 0.041666666666666664), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / x_m)) / z);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x_m * x_m) * y_m), 0.5, y_m) / z) / x_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 1e+153], N[(N[(y$95$m * N[(N[(N[(N[(N[(0.001388888888888889 * x$95$m), $MachinePrecision] * x$95$m + 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5 + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1e153

   1. Initial program 83.4%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{x}}}{z} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{\color{blue}{x}}}{z} \]

   4. Applied rewrites 89.8%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5 \cdot y\right), x \cdot x, y\right)}{x}}}{z} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}{\color{blue}{x}}}{z} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{y \cdot \frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}{\color{blue}{x}}}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}{\color{blue}{x}}}{z} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot \frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}{x}}{z} \]

   7. Applied rewrites 91.4%

      \[\leadsto \frac{y \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x}}}{z} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{x}}{z} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{x}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot \left(x \cdot x\right) + \frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{x}}{z} \]

      4. associate-*r* N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{1}{720} \cdot x\right) \cdot x + \frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot x, x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{x}}{z} \]

      6. lower-*.f64 91.4

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot x, x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x}}{z} \]

   9. Applied rewrites 91.4%

      \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot x, x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x}}{z} \]

   if 1e153 < y

   1. Initial program 89.5%

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

   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 96.2

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

   4. Applied rewrites 96.2%

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

Alternative 11: 92.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 10^{+153}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot y\_m, 0.5, y\_m\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= y_m 1e+153)
     (/
      (*
       y_m
       (/
        (fma
         (fma (* (* x_m x_m) 0.001388888888888889) (* x_m x_m) 0.5)
         (* x_m x_m)
         1.0)
        x_m))
      z)
     (/ (/ (fma (* (* x_m x_m) y_m) 0.5 y_m) z) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 1e+153) {
		tmp = (y_m * (fma(fma(((x_m * x_m) * 0.001388888888888889), (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m)) / z;
	} else {
		tmp = (fma(((x_m * x_m) * y_m), 0.5, y_m) / z) / x_m;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 1e+153)
		tmp = Float64(Float64(y_m * Float64(fma(fma(Float64(Float64(x_m * x_m) * 0.001388888888888889), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / x_m)) / z);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x_m * x_m) * y_m), 0.5, y_m) / z) / x_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 1e+153], N[(N[(y$95$m * N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5 + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1e153

   1. Initial program 83.4%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{x}}}{z} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y + {x}^{2} \cdot \left(\frac{1}{2} \cdot y + {x}^{2} \cdot \left(\frac{1}{720} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{24} \cdot y\right)\right)}{\color{blue}{x}}}{z} \]

   4. Applied rewrites 89.8%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5 \cdot y\right), x \cdot x, y\right)}{x}}}{z} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}{\color{blue}{x}}}{z} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{y \cdot \frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}{\color{blue}{x}}}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}{\color{blue}{x}}}{z} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot \frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}{x}}{z} \]

   7. Applied rewrites 91.4%

      \[\leadsto \frac{y \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x}}}{z} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{x}}{z} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot \left(x \cdot x\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{720}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{x}}{z} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{720}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)}{x}}{z} \]

      4. lift-*.f64 91.3

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.001388888888888889, x \cdot x, 0.5\right), x \cdot x, 1\right)}{x}}{z} \]

   10. Applied rewrites 91.3%

       \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.001388888888888889, x \cdot x, 0.5\right), x \cdot x, 1\right)}{x}}{z} \]

   if 1e153 < y

   1. Initial program 89.5%

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

   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 96.2

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

   4. Applied rewrites 96.2%

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

Alternative 12: 90.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 10^{+153}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot y\_m, 0.5, y\_m\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= y_m 1e+153)
     (/
      (/
       (* (fma (fma (* x_m x_m) 0.041666666666666664 0.5) (* x_m x_m) 1.0) y_m)
       x_m)
      z)
     (/ (/ (fma (* (* x_m x_m) y_m) 0.5 y_m) z) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 1e+153) {
		tmp = ((fma(fma((x_m * x_m), 0.041666666666666664, 0.5), (x_m * x_m), 1.0) * y_m) / x_m) / z;
	} else {
		tmp = (fma(((x_m * x_m) * y_m), 0.5, y_m) / z) / x_m;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 1e+153)
		tmp = Float64(Float64(Float64(fma(fma(Float64(x_m * x_m), 0.041666666666666664, 0.5), Float64(x_m * x_m), 1.0) * y_m) / x_m) / z);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x_m * x_m) * y_m), 0.5, y_m) / z) / x_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 1e+153], N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5 + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1e153

   1. Initial program 83.4%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + \color{blue}{1}\right) \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, \color{blue}{{x}^{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}, {\color{blue}{x}}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right), {\color{blue}{x}}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot \color{blue}{x}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-*.f64 73.8

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot \color{blue}{x}, 1\right) \cdot \frac{y}{x}}{z} \]

   4. Applied rewrites 73.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}}{z} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

      5. lower-*.f64 88.5

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}}{x}}{z} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) + \frac{1}{2}, \color{blue}{x} \cdot x, 1\right) \cdot y}{x}}{z} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) + \frac{1}{2}, x \cdot x, 1\right) \cdot y}{x}}{z} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}, x \cdot x, 1\right) \cdot y}{x}}{z} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{2} \cdot \frac{1}{24} + \frac{1}{2}, x \cdot x, 1\right) \cdot y}{x}}{z} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right), \color{blue}{x} \cdot x, 1\right) \cdot y}{x}}{z} \]

      11. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}{z} \]

      12. lift-*.f64 88.5

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), x \cdot x, 1\right) \cdot y}{x}}{z} \]

   6. Applied rewrites 88.5%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

   if 1e153 < y

   1. Initial program 89.5%

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

   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 96.2

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

   4. Applied rewrites 96.2%

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

Alternative 13: 90.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 4.8 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x\_m \cdot x\_m\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot y\_m, 0.5, y\_m\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= y_m 4.8e+48)
     (/
      (/
       (* (fma (* 0.041666666666666664 (* x_m x_m)) (* x_m x_m) 1.0) y_m)
       x_m)
      z)
     (/ (/ (fma (* (* x_m x_m) y_m) 0.5 y_m) z) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 4.8e+48) {
		tmp = ((fma((0.041666666666666664 * (x_m * x_m)), (x_m * x_m), 1.0) * y_m) / x_m) / z;
	} else {
		tmp = (fma(((x_m * x_m) * y_m), 0.5, y_m) / z) / x_m;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 4.8e+48)
		tmp = Float64(Float64(Float64(fma(Float64(0.041666666666666664 * Float64(x_m * x_m)), Float64(x_m * x_m), 1.0) * y_m) / x_m) / z);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x_m * x_m) * y_m), 0.5, y_m) / z) / x_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 4.8e+48], N[(N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5 + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.8000000000000002e48

   1. Initial program 80.5%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + \color{blue}{1}\right) \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, \color{blue}{{x}^{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}, {\color{blue}{x}}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right), {\color{blue}{x}}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot \color{blue}{x}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-*.f64 70.0

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot \color{blue}{x}, 1\right) \cdot \frac{y}{x}}{z} \]

   4. Applied rewrites 70.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2}, \color{blue}{x} \cdot x, 1\right) \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({x}^{2} \cdot \frac{1}{24}, x \cdot x, 1\right) \cdot \frac{y}{x}}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({x}^{2} \cdot \frac{1}{24}, x \cdot x, 1\right) \cdot \frac{y}{x}}{z} \]

      3. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{24}, x \cdot x, 1\right) \cdot \frac{y}{x}}{z} \]

      4. lift-*.f64 69.7

         \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x \cdot x, 1\right) \cdot \frac{y}{x}}{z} \]

   7. Applied rewrites 69.7%

      \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, \color{blue}{x} \cdot x, 1\right) \cdot \frac{y}{x}}{z} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{24}, x \cdot x, 1\right) \cdot \frac{y}{x}}}{z} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{24}, x \cdot x, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{24}, x \cdot x, 1\right) \cdot y}{x}}}{z} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{24}, x \cdot x, 1\right) \cdot y}{x}}}{z} \]

      5. lower-*.f64 88.2

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x \cdot x, 1\right) \cdot y}}{x}}{z} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{24}, x \cdot x, 1\right) \cdot y}{x}}{z} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{24}, x \cdot x, 1\right) \cdot y}{x}}{z} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right), x \cdot x, 1\right) \cdot y}{x}}{z} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right), x \cdot x, 1\right) \cdot y}{x}}{z} \]

      10. lift-*.f64 88.2

          \[\leadsto \frac{\frac{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right), x \cdot x, 1\right) \cdot y}{x}}{z} \]

   9. Applied rewrites 88.2%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

   if 4.8000000000000002e48 < y

   1. Initial program 91.1%

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

   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 93.4

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

   4. Applied rewrites 93.4%

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

Alternative 14: 79.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 92.2%

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

   2. Taylor expanded in x around 0

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

      1. lift-/.f64 91.3

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

   4. Applied rewrites 91.3%

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

   if 1.3999999999999999 < x

   1. Initial program 77.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 66.8

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

   4. Applied rewrites 66.8%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 66.8

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

   7. Applied rewrites 66.8%

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

Alternative 15: 65.8% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 92.2%

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

   2. Taylor expanded in x around 0

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

      1. lift-/.f64 91.3

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

   4. Applied rewrites 91.3%

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

   if 1.3999999999999999 < x

   1. Initial program 77.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 66.8

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

   4. Applied rewrites 66.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 40.5

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

   7. Applied rewrites 40.5%

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

Alternative 16: 66.2% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 92.2%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 92.0

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

   4. Applied rewrites 92.0%

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

   if 1.3999999999999999 < x

   1. Initial program 77.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 66.8

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

   4. Applied rewrites 66.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 40.5

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

   7. Applied rewrites 40.5%

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

Alternative 17: 65.8% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 92.2%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 92.0

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

   4. Applied rewrites 92.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 91.2

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

   6. Applied rewrites 91.2%

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

   if 1.3999999999999999 < x

   1. Initial program 77.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 66.8

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

   4. Applied rewrites 66.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 40.5

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

   7. Applied rewrites 40.5%

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

Alternative 18: 48.6% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

2. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 53.0

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

4. Applied rewrites 53.0%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. *-commutative N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 48.6

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

6. Applied rewrites 48.6%

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

Developer Target 1: 96.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (/ y z) x) (cosh x))))
   (if (< y -4.618902267687042e-52)
     t_0
     (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y / z) / x) * cosh(x)
    if (y < (-4.618902267687042d-52)) then
        tmp = t_0
    else if (y < 1.038530535935153d-39) then
        tmp = ((cosh(x) * y) / x) / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * Math.cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((Math.cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y / z) / x) * math.cosh(x)
	tmp = 0
	if y < -4.618902267687042e-52:
		tmp = t_0
	elif y < 1.038530535935153e-39:
		tmp = ((math.cosh(x) * y) / x) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y / z) / x) * cosh(x))
	tmp = 0.0
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = Float64(Float64(Float64(cosh(x) * y) / x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y / z) / x) * cosh(x);
	tmp = 0.0;
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = ((cosh(x) * y) / x) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -4.618902267687042e-52], t$95$0, If[Less[y, 1.038530535935153e-39], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2025095 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -2309451133843521/5000000000000000000000000000000000000000000000000000000000000000000) (* (/ (/ y z) x) (cosh x)) (if (< y 1038530535935153/1000000000000000000000000000000000000000000000000000000) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x)))))

  (/ (* (cosh x) (/ y x)) z))