Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.5% → 99.8%

Time: 4.6s

Alternatives: 16

Speedup: 2.3×

• 63.9% 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: 84.5% 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

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \cosh x\_m \cdot \frac{y\_m}{x\_m}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+275}:\\ \;\;\;\;\frac{t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2 \cdot \cosh x\_m}{x\_m}}{z} \cdot y\_m}{2}\\ \end{array}\right) \end{array} \end{array} \]

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 5e+275)
       (/ 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 <= 5e+275) {
		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 <= 5d+275) 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 <= 5e+275) {
		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 <= 5e+275:
		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 <= 5e+275)
		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 <= 5e+275)
		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, 5e+275], 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)) < 5.0000000000000003e275

   1. Initial program 99.7%

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

   if 5.0000000000000003e275 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 76.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cosh.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/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: 79.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq 2 \cdot 10^{+276}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot \frac{\frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot y\_m, 0.5, y\_m\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

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+276)
     (* (fma 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 (((cosh(x_m) * (y_m / x_m)) / z) <= 2e+276) {
		tmp = fma(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 (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 2e+276)
		tmp = Float64(fma(0.5, Float64(x_m * x_m), 1.0) * 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+276], N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $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) < 2.0000000000000001e276

   1. Initial program 96.1%

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

   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 88.9

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

   4. Applied rewrites 88.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \mathsf{Rewrite=>}\left(associate-/r*, \left(\frac{\frac{y}{x}}{z}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \mathsf{Rewrite=>}\left(lower-/.f64, \left(\frac{\frac{y}{x}}{z}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{\mathsf{Rewrite<=}\left(lift-/.f64, \left(\frac{y}{x}\right)\right)}{z} \]

   6. Applied rewrites 84.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 79.0%

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

   if 2.0000000000000001e276 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 68.1%

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

   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 78.9

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

   4. Applied rewrites 78.9%

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

Alternative 3: 82.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 5 \cdot 10^{+275}:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664 \cdot \left(x\_m \cdot x\_m\right), x\_m \cdot x\_m, 1\right) \cdot \frac{\frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(x\_m, x\_m, 2\right)}{z}}{x\_m} \cdot y\_m}{2}\\ \end{array}\right) \end{array} \]

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)) 5e+275)
     (*
      (fma (* 0.041666666666666664 (* x_m x_m)) (* 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)) <= 5e+275) {
		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, 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(cosh(x_m) * Float64(y_m / x_m)) <= 5e+275)
		tmp = Float64(fma(Float64(0.041666666666666664 * Float64(x_m * x_m)), Float64(x_m * x_m), 1.0) * Float64(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[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 5e+275], 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] * N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $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 (cosh.f64 x) (/.f64 y x)) < 5.0000000000000003e275

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   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 98.7

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

   4. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \mathsf{Rewrite=>}\left(associate-/r*, \left(\frac{\frac{y}{x}}{z}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \mathsf{Rewrite=>}\left(lower-/.f64, \left(\frac{\frac{y}{x}}{z}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{\mathsf{Rewrite<=}\left(lift-/.f64, \left(\frac{y}{x}\right)\right)}{z} \]

   6. Applied rewrites 98.7%

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

   7. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 98.0

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

   9. Applied rewrites 98.0%

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

   if 5.0000000000000003e275 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 76.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cosh.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/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

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot \frac{1}{z} + \frac{{x}^{2}}{z}}{x}} \cdot y}{2} \]
   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 74.5

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

   6. Applied rewrites 74.5%

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

Alternative 4: 83.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 5 \cdot 10^{+275}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot \frac{\frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(x\_m, x\_m, 2\right)}{z}}{x\_m} \cdot y\_m}{2}\\ \end{array}\right) \end{array} \]

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)) 5e+275)
     (* (fma 0.5 (* 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)) <= 5e+275) {
		tmp = fma(0.5, (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(cosh(x_m) * Float64(y_m / x_m)) <= 5e+275)
		tmp = Float64(fma(0.5, Float64(x_m * x_m), 1.0) * Float64(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[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 5e+275], N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $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 (cosh.f64 x) (/.f64 y x)) < 5.0000000000000003e275

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   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 98.7

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

   4. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \mathsf{Rewrite=>}\left(associate-/r*, \left(\frac{\frac{y}{x}}{z}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \mathsf{Rewrite=>}\left(lower-/.f64, \left(\frac{\frac{y}{x}}{z}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{\mathsf{Rewrite<=}\left(lift-/.f64, \left(\frac{y}{x}\right)\right)}{z} \]

   6. Applied rewrites 98.7%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 98.5%

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

   if 5.0000000000000003e275 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 76.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cosh.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/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

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot \frac{1}{z} + \frac{{x}^{2}}{z}}{x}} \cdot y}{2} \]
   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 74.5

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

   6. Applied rewrites 74.5%

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

Alternative 5: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot y\_m, 0.5, y\_m\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

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)) 5e+292)
     (/ (/ 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)) <= 5e+292) {
		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(cosh(x_m) * Float64(y_m / x_m)) <= 5e+292)
		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[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 5e+292], 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 (cosh.f64 x) (/.f64 y x)) < 4.9999999999999996e292

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0

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

      1. lift-/.f64 98.0

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

   4. Applied rewrites 98.0%

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

   if 4.9999999999999996e292 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 75.8%

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

   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 76.5

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

   4. Applied rewrites 76.5%

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

Alternative 6: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 6 \cdot 10^{+51}:\\ \;\;\;\;\frac{\cosh x\_m \cdot y\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{x\_m}}{z}\\ \end{array}\right) \end{array} \]

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 6e+51)
     (/ (* (cosh x_m) y_m) (* z x_m))
     (/
      (/
       (*
        (fma
         (fma (* (* x_m x_m) 0.001388888888888889) (* x_m x_m) 0.5)
         (* x_m x_m)
         1.0)
        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 <= 6e+51) {
		tmp = (cosh(x_m) * y_m) / (z * x_m);
	} else {
		tmp = ((fma(fma(((x_m * x_m) * 0.001388888888888889), (x_m * x_m), 0.5), (x_m * x_m), 1.0) * 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 <= 6e+51)
		tmp = Float64(Float64(cosh(x_m) * y_m) / Float64(z * x_m));
	else
		tmp = Float64(Float64(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) * y_m) / x_m) / z);
	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[x$95$m, 6e+51], N[(N[(N[Cosh[x$95$m], $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(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] * y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 6e51

   1. Initial program 92.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cosh.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 93.2

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

   3. Applied rewrites 93.2%

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

   if 6e51 < x

   1. Initial program 73.7%

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

   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 95.6%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 100.0

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

   10. Applied rewrites 100.0%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.001388888888888889, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{x}}{z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 92.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3.55 \cdot 10^{+64}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x\_m \cdot x\_m, 1\right), x\_m \cdot x\_m, 2\right)}{x\_m}}{z} \cdot y\_m}{2}\\ \end{array}\right) \end{array} \]

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 (<= y_m 3.55e+64)
     (/
      (/
       (*
        (fma
         (fma
          (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664)
          (* x_m x_m)
          0.5)
         (* x_m x_m)
         1.0)
        y_m)
       x_m)
      z)
     (/
      (*
       (/
        (/ (fma (fma 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 (y_m <= 3.55e+64) {
		tmp = ((fma(fma(fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0) * y_m) / x_m) / z;
	} else {
		tmp = (((fma(fma(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 (y_m <= 3.55e+64)
		tmp = Float64(Float64(Float64(fma(fma(fma(Float64(x_m * x_m), 0.001388888888888889, 0.041666666666666664), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) * y_m) / x_m) / z);
	else
		tmp = Float64(Float64(Float64(Float64(fma(fma(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[y$95$m, 3.55e+64], N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + 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] * y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.08333333333333333 * 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 y < 3.55000000000000029e64

   1. Initial program 80.7%

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

   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 88.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 92.0%

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

   if 3.55000000000000029e64 < y

   1. Initial program 90.2%

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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 93.6

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

   6. Applied rewrites 93.6%

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

Alternative 8: 92.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.35 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot y\_m, 0.5, y\_m\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

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 2.35e+65)
     (/
      (/
       (*
        (fma
         (fma
          (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664)
          (* x_m x_m)
          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 <= 2.35e+65) {
		tmp = ((fma(fma(fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664), (x_m * x_m), 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 <= 2.35e+65)
		tmp = Float64(Float64(Float64(fma(fma(fma(Float64(x_m * x_m), 0.001388888888888889, 0.041666666666666664), Float64(x_m * x_m), 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, 2.35e+65], N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + 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] * 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 < 2.3500000000000001e65

   1. Initial program 80.7%

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

   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 88.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 92.0%

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

   if 2.3500000000000001e65 < y

   1. Initial program 90.1%

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

   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 94.0

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

   4. Applied rewrites 94.0%

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

Alternative 9: 92.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.35 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot y\_m, 0.5, y\_m\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

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 2.35e+65)
     (/
      (/
       (*
        (fma
         (fma (* (* x_m x_m) 0.001388888888888889) (* x_m x_m) 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 <= 2.35e+65) {
		tmp = ((fma(fma(((x_m * x_m) * 0.001388888888888889), (x_m * x_m), 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 <= 2.35e+65)
		tmp = Float64(Float64(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) * 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, 2.35e+65], N[(N[(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] * 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 < 2.3500000000000001e65

   1. Initial program 80.7%

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

   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 88.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 92.0%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 91.9

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

   10. Applied rewrites 91.9%

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

   if 2.3500000000000001e65 < y

   1. Initial program 90.1%

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

   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 94.0

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

   4. Applied rewrites 94.0%

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

Alternative 10: 84.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4 \cdot 10^{-107}:\\ \;\;\;\;\frac{y\_m}{z \cdot x\_m}\\ \mathbf{elif}\;x\_m \leq 6.2 \cdot 10^{+139}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.041666666666666664 \cdot x\_m\right) \cdot x\_m, x\_m \cdot x\_m, 1\right) \cdot \frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot y\_m, 0.5, y\_m\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

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 (<= x_m 4e-107)
     (/ y_m (* z x_m))
     (if (<= x_m 6.2e+139)
       (/
        (*
         (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 (x_m <= 4e-107) {
		tmp = y_m / (z * x_m);
	} else if (x_m <= 6.2e+139) {
		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 (x_m <= 4e-107)
		tmp = Float64(y_m / Float64(z * x_m));
	elseif (x_m <= 6.2e+139)
		tmp = Float64(Float64(fma(Float64(Float64(0.041666666666666664 * x_m) * x_m), Float64(x_m * x_m), 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[x$95$m, 4e-107], N[(y$95$m / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 6.2e+139], N[(N[(N[(N[(N[(0.041666666666666664 * x$95$m), $MachinePrecision] * x$95$m), $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), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5 + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 4e-107

   1. Initial program 88.5%

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

   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 90.1

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

   4. Applied rewrites 90.1%

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]
   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 90.3

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

   6. Applied rewrites 90.3%

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

   if 4e-107 < x < 6.2e139

   1. Initial program 93.2%

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

   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 73.1

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

   4. Applied rewrites 73.1%

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

   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 72.4

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

   7. Applied rewrites 72.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 72.4

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

   9. Applied rewrites 72.4%

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

   if 6.2e139 < x

   1. Initial program 67.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 96.4

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

   4. Applied rewrites 96.4%

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

Alternative 11: 90.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.35 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot y\_m, 0.5, y\_m\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

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 2.35e+65)
     (/
      (/
       (* (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 <= 2.35e+65) {
		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 <= 2.35e+65)
		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, 2.35e+65], 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 < 2.3500000000000001e65

   1. Initial program 80.7%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   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 69.9

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

   4. Applied rewrites 69.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   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 87.7

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

      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 87.7

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

   6. Applied rewrites 87.7%

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

   if 2.3500000000000001e65 < y

   1. Initial program 90.1%

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

   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 94.0

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

   4. Applied rewrites 94.0%

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

Alternative 12: 90.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.35 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x\_m \cdot x\_m\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot y\_m, 0.5, y\_m\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

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 2.35e+65)
     (/
      (/
       (* (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 <= 2.35e+65) {
		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 <= 2.35e+65)
		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, 2.35e+65], 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 < 2.3500000000000001e65

   1. Initial program 80.7%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   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 69.9

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

   4. Applied rewrites 69.9%

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

   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 69.6

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

   7. Applied rewrites 69.6%

      \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, \color{blue}{x} \cdot x, 1\right) \cdot \frac{y}{x}}{z} \]
   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 87.4

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

      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 87.4

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

   9. Applied rewrites 87.4%

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

   if 2.3500000000000001e65 < y

   1. Initial program 90.1%

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

   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 94.0

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

   4. Applied rewrites 94.0%

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

Alternative 13: 79.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 33000:\\ \;\;\;\;\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot \frac{y\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(x\_m \cdot x\_m\right) \cdot 0.5\right) \cdot y\_m}{x\_m}}{z}\\ \end{array}\right) \end{array} \]

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 33000.0)
     (* (fma 0.5 (* x_m x_m) 1.0) (/ y_m (* z x_m)))
     (/ (/ (* (* (* 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 <= 33000.0) {
		tmp = fma(0.5, (x_m * x_m), 1.0) * (y_m / (z * x_m));
	} 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 <= 33000.0)
		tmp = Float64(fma(0.5, Float64(x_m * x_m), 1.0) * Float64(y_m / Float64(z * x_m)));
	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

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, 33000.0], N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y$95$m / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $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 < 33000

   1. Initial program 91.3%

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

   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 90.2

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

   4. Applied rewrites 90.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \mathsf{Rewrite=>}\left(associate-/r*, \left(\frac{\frac{y}{x}}{z}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \mathsf{Rewrite=>}\left(lower-/.f64, \left(\frac{\frac{y}{x}}{z}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{\mathsf{Rewrite<=}\left(lift-/.f64, \left(\frac{y}{x}\right)\right)}{z} \]

   6. Applied rewrites 90.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 90.0%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 91.3

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

   11. Applied rewrites 91.3%

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

   if 33000 < x

   1. Initial program 77.6%

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

   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 68.3

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

   4. Applied rewrites 68.3%

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

   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 68.3

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

   7. Applied rewrites 68.3%

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

Alternative 14: 66.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.28 \cdot 10^{+238}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot \frac{y\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y\_m \cdot x\_m\right) \cdot 0.5}{z}\\ \end{array}\right) \end{array} \]

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 1.28e+238)
     (* (fma 0.5 (* x_m x_m) 1.0) (/ 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 <= 1.28e+238) {
		tmp = fma(0.5, (x_m * x_m), 1.0) * (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 <= 1.28e+238)
		tmp = Float64(fma(0.5, Float64(x_m * x_m), 1.0) * 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

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, 1.28e+238], N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y$95$m / N[(z * x$95$m), $MachinePrecision]), $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 < 1.28000000000000007e238

   1. Initial program 87.6%

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

   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 78.5

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

   4. Applied rewrites 78.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \mathsf{Rewrite=>}\left(associate-/r*, \left(\frac{\frac{y}{x}}{z}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \mathsf{Rewrite=>}\left(lower-/.f64, \left(\frac{\frac{y}{x}}{z}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{\mathsf{Rewrite<=}\left(lift-/.f64, \left(\frac{y}{x}\right)\right)}{z} \]

   6. Applied rewrites 73.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 66.7%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 67.0

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

   11. Applied rewrites 67.0%

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

   if 1.28000000000000007e238 < x

   1. Initial program 60.2%

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

   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 100.0

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

   4. Applied rewrites 100.0%

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

   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 59.9

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

   7. Applied rewrites 59.9%

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

Alternative 15: 65.7% accurate, 4.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0054:\\ \;\;\;\;\frac{y\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y\_m \cdot x\_m\right) \cdot 0.5}{z}\\ \end{array}\right) \end{array} \]

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 0.0054) (/ 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 <= 0.0054) {
		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 <= 0.0054d0) 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 <= 0.0054) {
		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 <= 0.0054:
		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 <= 0.0054)
		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 <= 0.0054)
		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, 0.0054], 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 < 0.0054000000000000003

   1. Initial program 91.2%

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

   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 92.1

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

   4. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]
   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 92.2

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

   6. Applied rewrites 92.2%

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

   if 0.0054000000000000003 < x

   1. Initial program 78.0%

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

   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 67.7

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

   4. Applied rewrites 67.7%

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

   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 40.0

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

   7. Applied rewrites 40.0%

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

Alternative 16: 49.2% 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 84.5%

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

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 53.1

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

4. Applied rewrites 53.1%

   \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]
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 49.2

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

6. Applied rewrites 49.2%

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

Developer Target 1: 97.5% 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 2025093 
(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))