Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 85.0% → 99.9%

Time: 5.0s

Alternatives: 21

Speedup: 2.6×

• 64.1% 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: 85.0% 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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m}\\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\cosh x\_m}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (/ (* (cosh x_m) (/ y_m x_m)) z_m)))
   (*
    x_s
    (*
     y_s
     (* z_s (if (<= t_0 5e+286) t_0 (/ (* y_m (/ (cosh x_m) z_m)) x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double t_0 = (cosh(x_m) * (y_m / x_m)) / z_m;
	double tmp;
	if (t_0 <= 5e+286) {
		tmp = t_0;
	} else {
		tmp = (y_m * (cosh(x_m) / z_m)) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

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

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

real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (cosh(x_m) * (y_m / x_m)) / z_m
    if (t_0 <= 5d+286) then
        tmp = t_0
    else
        tmp = (y_m * (cosh(x_m) / z_m)) / x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double t_0 = (Math.cosh(x_m) * (y_m / x_m)) / z_m;
	double tmp;
	if (t_0 <= 5e+286) {
		tmp = t_0;
	} else {
		tmp = (y_m * (Math.cosh(x_m) / z_m)) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, z_s, x_m, y_m, z_m):
	t_0 = (math.cosh(x_m) * (y_m / x_m)) / z_m
	tmp = 0
	if t_0 <= 5e+286:
		tmp = t_0
	else:
		tmp = (y_m * (math.cosh(x_m) / z_m)) / x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	t_0 = Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m)
	tmp = 0.0
	if (t_0 <= 5e+286)
		tmp = t_0;
	else
		tmp = Float64(Float64(y_m * Float64(cosh(x_m) / z_m)) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m)
	t_0 = (cosh(x_m) * (y_m / x_m)) / z_m;
	tmp = 0.0;
	if (t_0 <= 5e+286)
		tmp = t_0;
	else
		tmp = (y_m * (cosh(x_m) / z_m)) / x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[t$95$0, 5e+286], t$95$0, N[(N[(y$95$m * N[(N[Cosh[x$95$m], $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 5.0000000000000004e286

   1. Initial program 95.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   if 5.0000000000000004e286 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 70.0%

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

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

Alternative 2: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 5 \cdot 10^{+110}:\\ \;\;\;\;\frac{y\_m \cdot \cosh x\_m}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\cosh x\_m}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 5e+110)
      (/ (* y_m (cosh x_m)) (* z_m x_m))
      (/ (* y_m (/ (cosh x_m) z_m)) x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5e+110) {
		tmp = (y_m * cosh(x_m)) / (z_m * x_m);
	} else {
		tmp = (y_m * (cosh(x_m) / z_m)) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

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

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

real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5d+110) then
        tmp = (y_m * cosh(x_m)) / (z_m * x_m)
    else
        tmp = (y_m * (cosh(x_m) / z_m)) / x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((Math.cosh(x_m) * (y_m / x_m)) / z_m) <= 5e+110) {
		tmp = (y_m * Math.cosh(x_m)) / (z_m * x_m);
	} else {
		tmp = (y_m * (Math.cosh(x_m) / z_m)) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, z_s, x_m, y_m, z_m):
	tmp = 0
	if ((math.cosh(x_m) * (y_m / x_m)) / z_m) <= 5e+110:
		tmp = (y_m * math.cosh(x_m)) / (z_m * x_m)
	else:
		tmp = (y_m * (math.cosh(x_m) / z_m)) / x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 5e+110)
		tmp = Float64(Float64(y_m * cosh(x_m)) / Float64(z_m * x_m));
	else
		tmp = Float64(Float64(y_m * Float64(cosh(x_m) / z_m)) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5e+110)
		tmp = (y_m * cosh(x_m)) / (z_m * x_m);
	else
		tmp = (y_m * (cosh(x_m) / z_m)) / x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 5e+110], N[(N[(y$95$m * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[Cosh[x$95$m], $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 4.99999999999999978e110

   1. Initial program 95.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. 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-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 90.7

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

   4. Applied rewrites 90.7%

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

   if 4.99999999999999978e110 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 72.0%

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

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 3: 91.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot \frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 5e+286)
      (/
       (*
        (fma
         (fma
          (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664)
          (* x_m x_m)
          0.5)
         (* x_m x_m)
         1.0)
        (/ y_m x_m))
       z_m)
      (/
       (/
        (*
         y_m
         (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0))
        z_m)
       x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5e+286) {
		tmp = (fma(fma(fma(0.001388888888888889, (x_m * x_m), 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0) * (y_m / x_m)) / z_m;
	} else {
		tmp = ((y_m * fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0)) / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 5e+286)
		tmp = Float64(Float64(fma(fma(fma(0.001388888888888889, Float64(x_m * x_m), 0.041666666666666664), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) * Float64(y_m / x_m)) / z_m);
	else
		tmp = Float64(Float64(Float64(y_m * fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0)) / z_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 5e+286], N[(N[(N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 5.0000000000000004e286

   1. Initial program 95.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 91.0

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

   5. Applied rewrites 91.0%

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

   if 5.0000000000000004e286 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 70.0%

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

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 93.1%

      \[\leadsto \frac{\color{blue}{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{z}}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 91.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 5 \cdot 10^{-58}:\\ \;\;\;\;y\_m \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 5e-58)
      (*
       y_m
       (/
        (fma
         (fma
          (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664)
          (* x_m x_m)
          0.5)
         (* x_m x_m)
         1.0)
        (* z_m x_m)))
      (/
       (/
        (*
         y_m
         (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0))
        z_m)
       x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5e-58) {
		tmp = y_m * (fma(fma(fma(0.001388888888888889, (x_m * x_m), 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0) / (z_m * x_m));
	} else {
		tmp = ((y_m * fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0)) / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 5e-58)
		tmp = Float64(y_m * Float64(fma(fma(fma(0.001388888888888889, Float64(x_m * x_m), 0.041666666666666664), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / Float64(z_m * x_m)));
	else
		tmp = Float64(Float64(Float64(y_m * fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0)) / z_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 5e-58], N[(y$95$m * N[(N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 4.99999999999999977e-58

   1. Initial program 95.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. 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-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 89.9

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

   4. Applied rewrites 89.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. +-commutative N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 85.4

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

   7. Applied rewrites 85.4%

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

   if 4.99999999999999977e-58 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 74.6%

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

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 94.1%

      \[\leadsto \frac{\color{blue}{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{z}}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 91.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)\\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 5 \cdot 10^{-58}:\\ \;\;\;\;\frac{t\_0 \cdot y\_m}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot t\_0}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0)))
   (*
    x_s
    (*
     y_s
     (*
      z_s
      (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 5e-58)
        (/ (* t_0 y_m) (* z_m x_m))
        (/ (/ (* y_m t_0) z_m) x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double t_0 = fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0);
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5e-58) {
		tmp = (t_0 * y_m) / (z_m * x_m);
	} else {
		tmp = ((y_m * t_0) / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	t_0 = fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 5e-58)
		tmp = Float64(Float64(t_0 * y_m) / Float64(z_m * x_m));
	else
		tmp = Float64(Float64(Float64(y_m * t_0) / z_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 5e-58], N[(N[(t$95$0 * y$95$m), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * t$95$0), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 4.99999999999999977e-58

   1. Initial program 95.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 85.0

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

   5. Applied rewrites 85.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. 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-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 80.6

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

   7. Applied rewrites 80.6%

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

   if 4.99999999999999977e-58 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 74.6%

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

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 94.1%

      \[\leadsto \frac{\color{blue}{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{z}}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 91.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), x\_m \cdot x\_m, 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 5e+286)
      (/ (/ (* (fma 0.5 (* x_m x_m) 1.0) y_m) x_m) z_m)
      (*
       y_m
       (/
        (/
         (fma (fma (* x_m x_m) 0.041666666666666664 0.5) (* x_m x_m) 1.0)
         z_m)
        x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5e+286) {
		tmp = ((fma(0.5, (x_m * x_m), 1.0) * y_m) / x_m) / z_m;
	} else {
		tmp = y_m * ((fma(fma((x_m * x_m), 0.041666666666666664, 0.5), (x_m * x_m), 1.0) / z_m) / x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 5e+286)
		tmp = Float64(Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y_m) / x_m) / z_m);
	else
		tmp = Float64(y_m * Float64(Float64(fma(fma(Float64(x_m * x_m), 0.041666666666666664, 0.5), Float64(x_m * x_m), 1.0) / z_m) / x_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 5e+286], N[(N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m * 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] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 5.0000000000000004e286

   1. Initial program 95.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 81.1

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

   5. Applied rewrites 81.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 81.7

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

   7. Applied rewrites 81.7%

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

   if 5.0000000000000004e286 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 70.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 62.2

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

   5. Applied rewrites 62.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. 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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 62.2

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

   7. Applied rewrites 62.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 91.2

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

   9. Applied rewrites 91.2%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), x \cdot x, 1\right)}{z}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 5e+286)
      (/ (/ (* (fma 0.5 (* x_m x_m) 1.0) y_m) x_m) z_m)
      (/ (* y_m (/ (fma (* x_m x_m) 0.5 1.0) z_m)) x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5e+286) {
		tmp = ((fma(0.5, (x_m * x_m), 1.0) * y_m) / x_m) / z_m;
	} else {
		tmp = (y_m * (fma((x_m * x_m), 0.5, 1.0) / z_m)) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 5e+286)
		tmp = Float64(Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y_m) / x_m) / z_m);
	else
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x_m * x_m), 0.5, 1.0) / z_m)) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 5e+286], N[(N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(y$95$m * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 5.0000000000000004e286

   1. Initial program 95.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 81.1

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

   5. Applied rewrites 81.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 81.7

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

   7. Applied rewrites 81.7%

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

   if 5.0000000000000004e286 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 70.0%

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

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 84.2

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

   7. Applied rewrites 84.2%

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

Alternative 8: 83.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 5 \cdot 10^{+110}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y\_m}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 5e+110)
      (/ (* (fma 0.5 (* x_m x_m) 1.0) y_m) (* z_m x_m))
      (/ (* y_m (/ (fma (* x_m x_m) 0.5 1.0) z_m)) x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5e+110) {
		tmp = (fma(0.5, (x_m * x_m), 1.0) * y_m) / (z_m * x_m);
	} else {
		tmp = (y_m * (fma((x_m * x_m), 0.5, 1.0) / z_m)) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 5e+110)
		tmp = Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y_m) / Float64(z_m * x_m));
	else
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x_m * x_m), 0.5, 1.0) / z_m)) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 5e+110], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 4.99999999999999978e110

   1. Initial program 95.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 80.2

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

   5. Applied rewrites 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 76.8

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

   7. Applied rewrites 76.8%

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

   if 4.99999999999999978e110 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 72.0%

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

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 85.2

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

   7. Applied rewrites 85.2%

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

Alternative 9: 71.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)}{z\_m \cdot x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 5e+286)
      (/ (/ y_m x_m) z_m)
      (* y_m (/ (fma (* x_m x_m) 0.5 1.0) (* z_m x_m))))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5e+286) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = y_m * (fma((x_m * x_m), 0.5, 1.0) / (z_m * x_m));
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 5e+286)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(y_m * Float64(fma(Float64(x_m * x_m), 0.5, 1.0) / Float64(z_m * x_m)));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 5e+286], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 5.0000000000000004e286

   1. Initial program 95.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 65.3

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

   5. Applied rewrites 65.3%

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

   if 5.0000000000000004e286 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 70.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. 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-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 81.8

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

   4. Applied rewrites 81.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 66.1

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

   7. Applied rewrites 66.1%

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

Alternative 10: 57.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 5 \cdot 10^{-62}:\\ \;\;\;\;\frac{1 \cdot y\_m}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 5e-62)
      (/ (* 1.0 y_m) (* z_m x_m))
      (/ (/ y_m z_m) x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5e-62) {
		tmp = (1.0 * y_m) / (z_m * x_m);
	} else {
		tmp = (y_m / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

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

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

real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5d-62) then
        tmp = (1.0d0 * y_m) / (z_m * x_m)
    else
        tmp = (y_m / z_m) / x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((Math.cosh(x_m) * (y_m / x_m)) / z_m) <= 5e-62) {
		tmp = (1.0 * y_m) / (z_m * x_m);
	} else {
		tmp = (y_m / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, z_s, x_m, y_m, z_m):
	tmp = 0
	if ((math.cosh(x_m) * (y_m / x_m)) / z_m) <= 5e-62:
		tmp = (1.0 * y_m) / (z_m * x_m)
	else:
		tmp = (y_m / z_m) / x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m) <= 5e-62)
		tmp = Float64(Float64(1.0 * y_m) / Float64(z_m * x_m));
	else
		tmp = Float64(Float64(y_m / z_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5e-62)
		tmp = (1.0 * y_m) / (z_m * x_m);
	else
		tmp = (y_m / z_m) / x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 5e-62], N[(N[(1.0 * y$95$m), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 5.0000000000000002e-62

   1. Initial program 95.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. 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-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 89.9

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

   4. Applied rewrites 89.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 62.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 62.3

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

   9. Applied rewrites 62.3%

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

   if 5.0000000000000002e-62 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 74.6%

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

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 45.7

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

   7. Applied rewrites 45.7%

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

Alternative 11: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{y\_m \cdot \cosh x\_m}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 1.2e+72)
      (/ (* y_m (cosh x_m)) (* z_m x_m))
      (/
       (/
        (*
         y_m
         (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0))
        z_m)
       x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 1.2e+72) {
		tmp = (y_m * cosh(x_m)) / (z_m * x_m);
	} else {
		tmp = ((y_m * fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0)) / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 1.2e+72)
		tmp = Float64(Float64(y_m * cosh(x_m)) / Float64(z_m * x_m));
	else
		tmp = Float64(Float64(Float64(y_m * fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0)) / z_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 1.2e+72], N[(N[(y$95$m * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.20000000000000005e72

   1. Initial program 87.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. 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-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 89.6

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

   4. Applied rewrites 89.6%

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

   if 1.20000000000000005e72 < x

   1. Initial program 78.2%

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

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 98.2%

      \[\leadsto \frac{\color{blue}{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{z}}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.2 \cdot 10^{+72}:\\ \;\;\;\;y\_m \cdot \frac{\cosh x\_m}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 1.2e+72)
      (* y_m (/ (cosh x_m) (* z_m x_m)))
      (/
       (/
        (*
         y_m
         (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0))
        z_m)
       x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 1.2e+72) {
		tmp = y_m * (cosh(x_m) / (z_m * x_m));
	} else {
		tmp = ((y_m * fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0)) / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 1.2e+72)
		tmp = Float64(y_m * Float64(cosh(x_m) / Float64(z_m * x_m)));
	else
		tmp = Float64(Float64(Float64(y_m * fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0)) / z_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 1.2e+72], N[(y$95$m * N[(N[Cosh[x$95$m], $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.20000000000000005e72

   1. Initial program 87.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. 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-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 89.1

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

   4. Applied rewrites 89.1%

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

   if 1.20000000000000005e72 < x

   1. Initial program 78.2%

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

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 98.2%

      \[\leadsto \frac{\color{blue}{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{z}}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 91.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := y\_m \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)\\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{t\_0}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0
         (*
          y_m
          (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0))))
   (*
    x_s
    (*
     y_s
     (* z_s (if (<= y_m 3.5e-43) (/ (/ t_0 x_m) z_m) (/ (/ t_0 z_m) x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double t_0 = y_m * fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0);
	double tmp;
	if (y_m <= 3.5e-43) {
		tmp = (t_0 / x_m) / z_m;
	} else {
		tmp = (t_0 / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	t_0 = Float64(y_m * fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0))
	tmp = 0.0
	if (y_m <= 3.5e-43)
		tmp = Float64(Float64(t_0 / x_m) / z_m);
	else
		tmp = Float64(Float64(t_0 / z_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(y$95$m * N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[y$95$m, 3.5e-43], N[(N[(t$95$0 / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(t$95$0 / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.49999999999999997e-43

   1. Initial program 83.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 85.2%

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

   if 3.49999999999999997e-43 < y

   1. Initial program 92.3%

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

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 97.3%

      \[\leadsto \frac{\color{blue}{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{z}}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 88.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, x\_m \cdot x\_m, 1\right) \cdot y\_m}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(\left(t\_0 \cdot x\_m\right) \cdot x\_m\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (fma 0.041666666666666664 (* x_m x_m) 0.5)))
   (*
    x_s
    (*
     y_s
     (*
      z_s
      (if (<= x_m 5000000.0)
        (/ (* (fma t_0 (* x_m x_m) 1.0) y_m) (* z_m x_m))
        (/ (/ (* y_m (* (* t_0 x_m) x_m)) z_m) x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double t_0 = fma(0.041666666666666664, (x_m * x_m), 0.5);
	double tmp;
	if (x_m <= 5000000.0) {
		tmp = (fma(t_0, (x_m * x_m), 1.0) * y_m) / (z_m * x_m);
	} else {
		tmp = ((y_m * ((t_0 * x_m) * x_m)) / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	t_0 = fma(0.041666666666666664, Float64(x_m * x_m), 0.5)
	tmp = 0.0
	if (x_m <= 5000000.0)
		tmp = Float64(Float64(fma(t_0, Float64(x_m * x_m), 1.0) * y_m) / Float64(z_m * x_m));
	else
		tmp = Float64(Float64(Float64(y_m * Float64(Float64(t_0 * x_m) * x_m)) / z_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 5000000.0], N[(N[(N[(t$95$0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(t$95$0 * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e6

   1. Initial program 87.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 80.6

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

   5. Applied rewrites 80.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. 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-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 83.1

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

   7. Applied rewrites 83.1%

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

   if 5e6 < x

   1. Initial program 82.1%

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

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 87.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{y \cdot \left({x}^{4} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}{z}}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 87.0%

       \[\leadsto \frac{\frac{y \cdot \left(\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x\right) \cdot x\right)}{z}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x\right) \cdot x\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 88.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.6 \cdot 10^{+36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right) \cdot x\_m\right) \cdot x\_m}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 3.6e+36)
      (/
       (* (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0) y_m)
       (* z_m x_m))
      (*
       y_m
       (/
        (/ (* (* (fma (* x_m x_m) 0.041666666666666664 0.5) x_m) x_m) z_m)
        x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 3.6e+36) {
		tmp = (fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) * y_m) / (z_m * x_m);
	} else {
		tmp = y_m * ((((fma((x_m * x_m), 0.041666666666666664, 0.5) * x_m) * x_m) / z_m) / x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 3.6e+36)
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) * y_m) / Float64(z_m * x_m));
	else
		tmp = Float64(y_m * Float64(Float64(Float64(Float64(fma(Float64(x_m * x_m), 0.041666666666666664, 0.5) * x_m) * x_m) / z_m) / x_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 3.6e+36], N[(N[(N[(N[(0.041666666666666664 * 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] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.5999999999999997e36

   1. Initial program 87.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 79.2

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

   5. Applied rewrites 79.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. 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-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 81.6

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

   7. Applied rewrites 81.6%

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

   if 3.5999999999999997e36 < x

   1. Initial program 80.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 70.8

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

   5. Applied rewrites 70.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. 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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 70.8

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

   7. Applied rewrites 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 90.5

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

   9. Applied rewrites 90.5%

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

   10. Taylor expanded in x around inf

       \[\leadsto y \cdot \frac{\frac{{x}^{4} \cdot \color{blue}{\left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)}}{z}}{x} \]
   11. Step-by-step derivation

   12. Applied rewrites 90.5%

       \[\leadsto y \cdot \frac{\frac{\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right) \cdot x\right) \cdot \color{blue}{x}}{z}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 85.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4.6 \cdot 10^{+150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(\left(0.5 \cdot x\_m\right) \cdot x\_m\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 4.6e+150)
      (/
       (* (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0) y_m)
       (* z_m x_m))
      (/ (/ (* y_m (* (* 0.5 x_m) x_m)) z_m) x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 4.6e+150) {
		tmp = (fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) * y_m) / (z_m * x_m);
	} else {
		tmp = ((y_m * ((0.5 * x_m) * x_m)) / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 4.6e+150)
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) * y_m) / Float64(z_m * x_m));
	else
		tmp = Float64(Float64(Float64(y_m * Float64(Float64(0.5 * x_m) * x_m)) / z_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 4.6e+150], N[(N[(N[(N[(0.041666666666666664 * 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] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(0.5 * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.60000000000000002e150

   1. Initial program 87.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 77.6

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

   5. Applied rewrites 77.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. 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-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 80.1

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

   7. Applied rewrites 80.1%

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

   if 4.60000000000000002e150 < x

   1. Initial program 75.0%

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

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{y \cdot \left({x}^{4} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}{z}}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 100.0%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(\left(0.5 \cdot x\right) \cdot x\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 84.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4.6 \cdot 10^{+150}:\\ \;\;\;\;y\_m \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(\left(0.5 \cdot x\_m\right) \cdot x\_m\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 4.6e+150)
      (*
       y_m
       (/
        (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0)
        (* z_m x_m)))
      (/ (/ (* y_m (* (* 0.5 x_m) x_m)) z_m) x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 4.6e+150) {
		tmp = y_m * (fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) / (z_m * x_m));
	} else {
		tmp = ((y_m * ((0.5 * x_m) * x_m)) / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 4.6e+150)
		tmp = Float64(y_m * Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / Float64(z_m * x_m)));
	else
		tmp = Float64(Float64(Float64(y_m * Float64(Float64(0.5 * x_m) * x_m)) / z_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 4.6e+150], N[(y$95$m * N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(0.5 * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.60000000000000002e150

   1. Initial program 87.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. 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-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 88.6

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

   4. Applied rewrites 88.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 79.2

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

   7. Applied rewrites 79.2%

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

   if 4.60000000000000002e150 < x

   1. Initial program 75.0%

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

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{y \cdot \left({x}^{4} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}{z}}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 100.0%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+150}:\\ \;\;\;\;y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(\left(0.5 \cdot x\right) \cdot x\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 80.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 135000:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y\_m}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(\left(0.5 \cdot x\_m\right) \cdot x\_m\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 135000.0)
      (/ (* (fma 0.5 (* x_m x_m) 1.0) y_m) (* z_m x_m))
      (/ (/ (* y_m (* (* 0.5 x_m) x_m)) z_m) x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 135000.0) {
		tmp = (fma(0.5, (x_m * x_m), 1.0) * y_m) / (z_m * x_m);
	} else {
		tmp = ((y_m * ((0.5 * x_m) * x_m)) / z_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 135000.0)
		tmp = Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y_m) / Float64(z_m * x_m));
	else
		tmp = Float64(Float64(Float64(y_m * Float64(Float64(0.5 * x_m) * x_m)) / z_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[x$95$m, 135000.0], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(0.5 * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 135000

   1. Initial program 87.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 79.0

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

   5. Applied rewrites 79.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 79.9

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

   7. Applied rewrites 79.9%

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

   if 135000 < x

   1. Initial program 82.1%

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

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 87.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{y \cdot \left({x}^{4} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}{z}}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 87.0%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 72.4%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 135000:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(\left(0.5 \cdot x\right) \cdot x\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 69.2% accurate, 3.4× speedup

Math

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

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= x_m 1.4)
      (/ (* 1.0 y_m) (* z_m x_m))
      (/ (* (* (* x_m x_m) 0.5) y_m) (* z_m x_m)))))))

C

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

Fortran

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

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

real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 1.4d0) then
        tmp = (1.0d0 * y_m) / (z_m * x_m)
    else
        tmp = (((x_m * x_m) * 0.5d0) * y_m) / (z_m * x_m)
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 86.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. 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-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 88.2

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 66.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 66.6

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

   9. Applied rewrites 66.6%

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

   if 1.3999999999999999 < x

   1. Initial program 82.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 52.4

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

   5. Applied rewrites 52.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 50.8

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

   7. Applied rewrites 50.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 50.8%

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

Alternative 20: 49.9% accurate, 5.8× speedup

Math

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

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (* x_s (* y_s (* z_s (/ (* 1.0 y_m) (* z_m x_m))))))

C

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

Fortran

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

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

real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    code = x_s * (y_s * (z_s * ((1.0d0 * y_m) / (z_m * x_m))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.7%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
2. Add Preprocessing
3. 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-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/l/ N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 85.9

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

4. Applied rewrites 85.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 50.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 51.0

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

9. Applied rewrites 51.0%

   \[\leadsto \color{blue}{\frac{1 \cdot y}{z \cdot x}} \]
10. Add Preprocessing

Alternative 21: 49.5% accurate, 5.8× speedup

Math

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

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (* x_s (* y_s (* z_s (* y_m (/ 1.0 (* z_m x_m)))))))

C

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

Fortran

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

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

real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    code = x_s * (y_s * (z_s * (y_m * (1.0d0 / (z_m * x_m)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.7%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
2. Add Preprocessing
3. 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-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/l/ N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 85.9

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

4. Applied rewrites 85.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 50.6%

   \[\leadsto y \cdot \frac{\color{blue}{1}}{z \cdot x} \]
8. Add Preprocessing

Developer Target 1: 97.1% 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 2025017 
(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))