Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.5% → 99.4%

Time: 12.0s

Alternatives: 20

Speedup: 2.3×

• Could not converge on a ground truth

  1. x = 1.1855507636500148e+70
  2. y = 6.0099056334628074e+212
  3. z = 6.19776303863707e+243

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    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: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 5000000000000:\\ \;\;\;\;\left(-y\_m\right) \cdot \frac{-1}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{--1}{\frac{z\_m}{\cosh x\_m \cdot y\_m} \cdot x\_m}\\ \end{array}\right)\right) \end{array} \]

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) 5000000000000.0)
      (* (- y_m) (/ -1.0 (* z_m x_m)))
      (/ (- -1.0) (* (/ z_m (* (cosh x_m) y_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) <= 5000000000000.0) {
		tmp = -y_m * (-1.0 / (z_m * x_m));
	} else {
		tmp = -(-1.0) / ((z_m / (cosh(x_m) * y_m)) * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
    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) <= 5000000000000.0d0) then
        tmp = -y_m * ((-1.0d0) / (z_m * x_m))
    else
        tmp = -(-1.0d0) / ((z_m / (cosh(x_m) * y_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) <= 5000000000000.0) {
		tmp = -y_m * (-1.0 / (z_m * x_m));
	} else {
		tmp = -(-1.0) / ((z_m / (Math.cosh(x_m) * y_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) <= 5000000000000.0:
		tmp = -y_m * (-1.0 / (z_m * x_m))
	else:
		tmp = -(-1.0) / ((z_m / (math.cosh(x_m) * y_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) <= 5000000000000.0)
		tmp = Float64(Float64(-y_m) * Float64(-1.0 / Float64(z_m * x_m)));
	else
		tmp = Float64(Float64(-(-1.0)) / Float64(Float64(z_m / Float64(cosh(x_m) * y_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) <= 5000000000000.0)
		tmp = -y_m * (-1.0 / (z_m * x_m));
	else
		tmp = -(-1.0) / ((z_m / (cosh(x_m) * y_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], 5000000000000.0], N[((-y$95$m) * N[(-1.0 / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((--1.0) / N[(N[(z$95$m / N[(N[Cosh[x$95$m], $MachinePrecision] * y$95$m), $MachinePrecision]), $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) < 5e12

   1. Initial program 92.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 66.6%

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

   if 5e12 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   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. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-2neg N/A

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

      6. metadata-eval N/A

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

      7. frac-times N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 100.0

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

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{-1}{\frac{z}{\cosh x \cdot y} \cdot \left(-x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 5000000000000:\\ \;\;\;\;\left(-y\right) \cdot \frac{-1}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{--1}{\frac{z}{\cosh x \cdot y} \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m}\\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5000000000000:\\ \;\;\;\;\left(-y\_m\right) \cdot \frac{-1}{z\_m \cdot x\_m}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\frac{y\_m}{z\_m} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x\_m \cdot x\_m\right), x\_m \cdot x\_m, 1\right)}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \end{array} \]

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 (/ (* (cosh x_m) (/ y_m x_m)) z_m)))
   (*
    x_s
    (*
     y_s
     (*
      z_s
      (if (<= t_0 5000000000000.0)
        (* (- y_m) (/ -1.0 (* z_m x_m)))
        (if (<= t_0 INFINITY)
          (/
           (*
            (/ y_m z_m)
            (fma (* 0.041666666666666664 (* x_m x_m)) (* x_m x_m) 1.0))
           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 t_0 = (cosh(x_m) * (y_m / x_m)) / z_m;
	double tmp;
	if (t_0 <= 5000000000000.0) {
		tmp = -y_m * (-1.0 / (z_m * x_m));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((y_m / z_m) * fma((0.041666666666666664 * (x_m * x_m)), (x_m * x_m), 1.0)) / 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)
	t_0 = Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m)
	tmp = 0.0
	if (t_0 <= 5000000000000.0)
		tmp = Float64(Float64(-y_m) * Float64(-1.0 / Float64(z_m * x_m)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(y_m / z_m) * fma(Float64(0.041666666666666664 * Float64(x_m * x_m)), Float64(x_m * x_m), 1.0)) / x_m);
	else
		tmp = Float64(Float64(Float64(y_m * 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_] := 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, 5000000000000.0], N[((-y$95$m) * N[(-1.0 / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(y$95$m / z$95$m), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 5e12

   1. Initial program 92.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 66.6%

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

   if 5e12 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < +inf.0

   1. Initial program 98.8%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   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. distribute-lft-in N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. distribute-lft1-in N/A

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

   7. Applied rewrites 87.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 87.8%

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

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

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 81.4

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

   7. Applied rewrites 81.4%

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

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{z\_m}}{x\_m} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \cosh x\_m}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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-40)
      (*
       (/
        (/
         (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0)
         z_m)
        x_m)
       y_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-40) {
		tmp = ((fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) / z_m) / x_m) * y_m;
	} else {
		tmp = ((y_m * 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-40)
		tmp = Float64(Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / z_m) / x_m) * y_m);
	else
		tmp = Float64(Float64(Float64(y_m * cosh(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[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 5e-40], N[(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] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(y$95$m * N[Cosh[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 (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 4.99999999999999965e-40

   1. Initial program 92.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.2%

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

   if 4.99999999999999965e-40 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 79.4%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 4: 94.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 10^{+14}:\\ \;\;\;\;\left(-y\_m\right) \cdot \frac{-1}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right) \cdot x\_m, x\_m, 1\right)}{z\_m} \cdot y\_m}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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) 1e+14)
      (* (- y_m) (/ -1.0 (* z_m x_m)))
      (/
       (*
        (/
         (fma
          (*
           (fma
            (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664)
            (* x_m x_m)
            0.5)
           x_m)
          x_m
          1.0)
         z_m)
        y_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) <= 1e+14) {
		tmp = -y_m * (-1.0 / (z_m * x_m));
	} else {
		tmp = ((fma((fma(fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664), (x_m * x_m), 0.5) * x_m), x_m, 1.0) / z_m) * y_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) <= 1e+14)
		tmp = Float64(Float64(-y_m) * Float64(-1.0 / Float64(z_m * x_m)));
	else
		tmp = Float64(Float64(Float64(fma(Float64(fma(fma(Float64(x_m * x_m), 0.001388888888888889, 0.041666666666666664), Float64(x_m * x_m), 0.5) * x_m), x_m, 1.0) / z_m) * y_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], 1e+14], N[((-y$95$m) * N[(-1.0 / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision] * y$95$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) < 1e14

   1. Initial program 92.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.2

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

   5. Applied rewrites 67.2%

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

   7. Applied rewrites 66.9%

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

   if 1e14 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 78.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 72.5

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

   7. Applied rewrites 72.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   9. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right)}{z} \cdot y}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 91.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{z\_m}}{x\_m} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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-40)
      (*
       (/
        (/
         (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0)
         z_m)
        x_m)
       y_m)
      (/
       (/
        (*
         (fma (fma (* x_m x_m) 0.041666666666666664 0.5) (* x_m x_m) 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) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5e-40) {
		tmp = ((fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) / z_m) / x_m) * y_m;
	} else {
		tmp = ((fma(fma((x_m * x_m), 0.041666666666666664, 0.5), (x_m * x_m), 1.0) * 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-40)
		tmp = Float64(Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / z_m) / x_m) * y_m);
	else
		tmp = Float64(Float64(Float64(fma(fma(Float64(x_m * x_m), 0.041666666666666664, 0.5), Float64(x_m * x_m), 1.0) * y_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[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 5e-40], N[(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] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / 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) < 4.99999999999999965e-40

   1. Initial program 92.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.2%

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

   if 4.99999999999999965e-40 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 79.4%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   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. distribute-lft-in N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. distribute-lft1-in N/A

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

   7. Applied rewrites 83.4%

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

   9. Applied rewrites 90.8%

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

Alternative 6: 91.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{z\_m}}{x\_m} \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

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)) 2e+297)
      (/ (/ y_m x_m) z_m)
      (*
       (/
        (/
         (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0)
         z_m)
        x_m)
       y_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)) <= 2e+297) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = ((fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) / z_m) / x_m) * y_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(cosh(x_m) * Float64(y_m / x_m)) <= 2e+297)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / z_m) / x_m) * y_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[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 2e+297], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(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] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2e297

   1. Initial program 96.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 71.1

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

   5. Applied rewrites 71.1%

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

   if 2e297 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 67.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 88.5%

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

Alternative 7: 84.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 5000000000000:\\ \;\;\;\;\left(-y\_m\right) \cdot \frac{-1}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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) 5000000000000.0)
      (* (- y_m) (/ -1.0 (* 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) <= 5000000000000.0) {
		tmp = -y_m * (-1.0 / (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) <= 5000000000000.0)
		tmp = Float64(Float64(-y_m) * Float64(-1.0 / Float64(z_m * x_m)));
	else
		tmp = Float64(Float64(Float64(y_m * 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], 5000000000000.0], N[((-y$95$m) * N[(-1.0 / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 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) < 5e12

   1. Initial program 92.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 66.6%

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

   if 5e12 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   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. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 84.5

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

   7. Applied rewrites 84.5%

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

Alternative 8: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 5000000000000:\\ \;\;\;\;\left(-y\_m\right) \cdot \frac{-1}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right) \cdot \frac{y\_m}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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) 5000000000000.0)
      (* (- y_m) (/ -1.0 (* z_m x_m)))
      (/ (* (fma (* x_m x_m) 0.5 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) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 5000000000000.0) {
		tmp = -y_m * (-1.0 / (z_m * x_m));
	} else {
		tmp = (fma((x_m * x_m), 0.5, 1.0) * (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) <= 5000000000000.0)
		tmp = Float64(Float64(-y_m) * Float64(-1.0 / Float64(z_m * x_m)));
	else
		tmp = Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) * Float64(y_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[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 5000000000000.0], N[((-y$95$m) * N[(-1.0 / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(y$95$m / 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) < 5e12

   1. Initial program 92.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 66.6%

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

   if 5e12 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   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. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 73.8

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

   7. Applied rewrites 73.8%

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

Alternative 9: 71.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m} \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right) \cdot y\_m}{z\_m \cdot x\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= (/ (* (cosh x_m) (/ y_m x_m)) z_m) 2e+256)
      (/ (/ y_m x_m) z_m)
      (/ (* (fma (* x_m x_m) 0.5 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) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z_m) <= 2e+256) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = (fma((x_m * x_m), 0.5, 1.0) * 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) <= 2e+256)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) * y_m) / 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], 2e+256], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $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 (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 2.0000000000000001e256

   1. Initial program 93.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 65.5

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

   5. Applied rewrites 65.5%

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

   if 2.0000000000000001e256 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 73.9%

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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 76.4

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

   7. Applied rewrites 76.4%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 66.1

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

   10. Applied rewrites 66.1%

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

Alternative 10: 95.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\left(-y\_m\right) \cdot \frac{-1}{z\_m \cdot x\_m}\\ \mathbf{elif}\;x\_m \leq 5 \cdot 10^{+51}:\\ \;\;\;\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x\_m \cdot x\_m\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y\_m}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 1.55e-162)
      (* (- y_m) (/ -1.0 (* z_m x_m)))
      (if (<= x_m 5e+51)
        (/ (* (cosh x_m) (/ y_m x_m)) z_m)
        (/
         (*
          (/
           (fma
            (fma (* 0.001388888888888889 (* x_m x_m)) (* x_m x_m) 0.5)
            (* x_m x_m)
            1.0)
           x_m)
          y_m)
         z_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 <= 1.55e-162) {
		tmp = -y_m * (-1.0 / (z_m * x_m));
	} else if (x_m <= 5e+51) {
		tmp = (cosh(x_m) * (y_m / x_m)) / z_m;
	} else {
		tmp = ((fma(fma((0.001388888888888889 * (x_m * x_m)), (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m) * y_m) / z_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 <= 1.55e-162)
		tmp = Float64(Float64(-y_m) * Float64(-1.0 / Float64(z_m * x_m)));
	elseif (x_m <= 5e+51)
		tmp = Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z_m);
	else
		tmp = Float64(Float64(Float64(fma(fma(Float64(0.001388888888888889 * Float64(x_m * x_m)), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / x_m) * y_m) / z_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, 1.55e-162], N[((-y$95$m) * N[(-1.0 / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 5e+51], N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.5499999999999999e-162

   1. Initial program 84.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 58.4

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

   5. Applied rewrites 58.4%

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

   7. Applied rewrites 58.5%

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

   if 1.5499999999999999e-162 < x < 5e51

   1. Initial program 93.2%

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

   if 5e51 < x

   1. Initial program 88.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

Alternative 11: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.9 \cdot 10^{+51}:\\ \;\;\;\;\frac{y\_m \cdot \cosh x\_m}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x\_m \cdot x\_m\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y\_m}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 3.9e+51)
      (/ (* y_m (cosh x_m)) (* z_m x_m))
      (/
       (*
        (/
         (fma
          (fma (* 0.001388888888888889 (* x_m x_m)) (* x_m x_m) 0.5)
          (* x_m x_m)
          1.0)
         x_m)
        y_m)
       z_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 <= 3.9e+51) {
		tmp = (y_m * cosh(x_m)) / (z_m * x_m);
	} else {
		tmp = ((fma(fma((0.001388888888888889 * (x_m * x_m)), (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m) * y_m) / z_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 <= 3.9e+51)
		tmp = Float64(Float64(y_m * cosh(x_m)) / Float64(z_m * x_m));
	else
		tmp = Float64(Float64(Float64(fma(fma(Float64(0.001388888888888889 * Float64(x_m * x_m)), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / x_m) * y_m) / z_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, 3.9e+51], N[(N[(y$95$m * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.89999999999999984e51

   1. Initial program 86.6%

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

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 89.1

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

   4. Applied rewrites 89.1%

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

   if 3.89999999999999984e51 < x

   1. Initial program 88.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

Alternative 12: 93.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.3 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= y_m 2.3e+27)
      (/
       (*
        (/
         (fma
          (fma
           (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664)
           (* x_m x_m)
           0.5)
          (* x_m x_m)
          1.0)
         x_m)
        y_m)
       z_m)
      (/
       (/
        (*
         (fma (fma (* x_m x_m) 0.041666666666666664 0.5) (* x_m x_m) 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) {
	double tmp;
	if (y_m <= 2.3e+27) {
		tmp = ((fma(fma(fma(0.001388888888888889, (x_m * x_m), 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m) * y_m) / z_m;
	} else {
		tmp = ((fma(fma((x_m * x_m), 0.041666666666666664, 0.5), (x_m * x_m), 1.0) * 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 (y_m <= 2.3e+27)
		tmp = Float64(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) / x_m) * y_m) / z_m);
	else
		tmp = Float64(Float64(Float64(fma(fma(Float64(x_m * x_m), 0.041666666666666664, 0.5), Float64(x_m * x_m), 1.0) * y_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[y$95$m, 2.3e+27], N[(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] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.3000000000000001e27

   1. Initial program 86.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.4%

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

   if 2.3000000000000001e27 < y

   1. Initial program 89.5%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   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. distribute-lft-in N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. distribute-lft1-in N/A

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

   7. Applied rewrites 98.0%

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

   9. Applied rewrites 98.0%

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

Alternative 13: 93.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.3 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x\_m \cdot x\_m\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= y_m 2.3e+27)
      (/
       (*
        (/
         (fma
          (fma (* 0.001388888888888889 (* x_m x_m)) (* x_m x_m) 0.5)
          (* x_m x_m)
          1.0)
         x_m)
        y_m)
       z_m)
      (/
       (/
        (*
         (fma (fma (* x_m x_m) 0.041666666666666664 0.5) (* x_m x_m) 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) {
	double tmp;
	if (y_m <= 2.3e+27) {
		tmp = ((fma(fma((0.001388888888888889 * (x_m * x_m)), (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m) * y_m) / z_m;
	} else {
		tmp = ((fma(fma((x_m * x_m), 0.041666666666666664, 0.5), (x_m * x_m), 1.0) * 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 (y_m <= 2.3e+27)
		tmp = Float64(Float64(Float64(fma(fma(Float64(0.001388888888888889 * Float64(x_m * x_m)), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / x_m) * y_m) / z_m);
	else
		tmp = Float64(Float64(Float64(fma(fma(Float64(x_m * x_m), 0.041666666666666664, 0.5), Float64(x_m * x_m), 1.0) * y_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[y$95$m, 2.3e+27], N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.3000000000000001e27

   1. Initial program 86.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 90.4%

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

   if 2.3000000000000001e27 < y

   1. Initial program 89.5%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   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. distribute-lft-in N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. distribute-lft1-in N/A

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

   7. Applied rewrites 98.0%

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

   9. Applied rewrites 98.0%

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

Alternative 14: 90.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.25 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= y_m 2.25e+27)
      (/
       (/
        (*
         (fma (fma (* x_m x_m) 0.041666666666666664 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 (y_m <= 2.25e+27) {
		tmp = ((fma(fma((x_m * x_m), 0.041666666666666664, 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 (y_m <= 2.25e+27)
		tmp = Float64(Float64(Float64(fma(fma(Float64(x_m * x_m), 0.041666666666666664, 0.5), Float64(x_m * x_m), 1.0) * y_m) / x_m) / z_m);
	else
		tmp = Float64(Float64(Float64(y_m * 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[y$95$m, 2.25e+27], N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.25e27

   1. Initial program 86.2%

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 87.9%

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

   7. Applied rewrites 87.9%

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

   if 2.25e27 < y

   1. Initial program 89.5%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 96.2

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

   7. Applied rewrites 96.2%

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

Alternative 15: 90.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.25 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)}{z\_m}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= y_m 2.25e+27)
      (/
       (*
        (/
         (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0)
         x_m)
        y_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 (y_m <= 2.25e+27) {
		tmp = ((fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m) * y_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 (y_m <= 2.25e+27)
		tmp = Float64(Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / x_m) * y_m) / z_m);
	else
		tmp = Float64(Float64(Float64(y_m * 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[y$95$m, 2.25e+27], N[(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] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.25e27

   1. Initial program 86.2%

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 87.9%

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

   if 2.25e27 < y

   1. Initial program 89.5%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 96.2

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

   7. Applied rewrites 96.2%

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

Alternative 16: 64.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\left(-y\_m\right) \cdot \frac{-1}{z\_m \cdot x\_m}\\ \mathbf{elif}\;x\_m \leq 1.42:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\_m\right) \cdot y\_m}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 1.55e-162)
      (* (- y_m) (/ -1.0 (* z_m x_m)))
      (if (<= x_m 1.42) (/ (/ y_m x_m) z_m) (/ (* (* 0.5 x_m) y_m) z_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 <= 1.55e-162) {
		tmp = -y_m * (-1.0 / (z_m * x_m));
	} else if (x_m <= 1.42) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = ((0.5 * x_m) * y_m) / z_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
    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 <= 1.55d-162) then
        tmp = -y_m * ((-1.0d0) / (z_m * x_m))
    else if (x_m <= 1.42d0) then
        tmp = (y_m / x_m) / z_m
    else
        tmp = ((0.5d0 * x_m) * y_m) / z_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 <= 1.55e-162) {
		tmp = -y_m * (-1.0 / (z_m * x_m));
	} else if (x_m <= 1.42) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = ((0.5 * x_m) * y_m) / z_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 <= 1.55e-162:
		tmp = -y_m * (-1.0 / (z_m * x_m))
	elif x_m <= 1.42:
		tmp = (y_m / x_m) / z_m
	else:
		tmp = ((0.5 * x_m) * y_m) / z_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 <= 1.55e-162)
		tmp = Float64(Float64(-y_m) * Float64(-1.0 / Float64(z_m * x_m)));
	elseif (x_m <= 1.42)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(Float64(Float64(0.5 * x_m) * y_m) / z_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 <= 1.55e-162)
		tmp = -y_m * (-1.0 / (z_m * x_m));
	elseif (x_m <= 1.42)
		tmp = (y_m / x_m) / z_m;
	else
		tmp = ((0.5 * x_m) * y_m) / z_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, 1.55e-162], N[((-y$95$m) * N[(-1.0 / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 1.42], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(0.5 * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.5499999999999999e-162

   1. Initial program 84.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 58.4

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

   5. Applied rewrites 58.4%

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

   7. Applied rewrites 58.5%

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

   if 1.5499999999999999e-162 < x < 1.4199999999999999

   1. Initial program 92.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 92.1

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

   5. Applied rewrites 92.1%

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

   if 1.4199999999999999 < x

   1. Initial program 89.8%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. distribute-lft-out N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. *-rgt-identity N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-/.f64 43.1

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

   5. Applied rewrites 43.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 43.1%

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

Alternative 17: 65.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.15 \cdot 10^{-162}:\\ \;\;\;\;\frac{y\_m}{z\_m \cdot x\_m}\\ \mathbf{elif}\;x\_m \leq 1.42:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\_m\right) \cdot y\_m}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 2.15e-162)
      (/ y_m (* z_m x_m))
      (if (<= x_m 1.42) (/ (/ y_m x_m) z_m) (/ (* (* 0.5 x_m) y_m) z_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 <= 2.15e-162) {
		tmp = y_m / (z_m * x_m);
	} else if (x_m <= 1.42) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = ((0.5 * x_m) * y_m) / z_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
    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 <= 2.15d-162) then
        tmp = y_m / (z_m * x_m)
    else if (x_m <= 1.42d0) then
        tmp = (y_m / x_m) / z_m
    else
        tmp = ((0.5d0 * x_m) * y_m) / z_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 <= 2.15e-162) {
		tmp = y_m / (z_m * x_m);
	} else if (x_m <= 1.42) {
		tmp = (y_m / x_m) / z_m;
	} else {
		tmp = ((0.5 * x_m) * y_m) / z_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 <= 2.15e-162:
		tmp = y_m / (z_m * x_m)
	elif x_m <= 1.42:
		tmp = (y_m / x_m) / z_m
	else:
		tmp = ((0.5 * x_m) * y_m) / z_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 <= 2.15e-162)
		tmp = Float64(y_m / Float64(z_m * x_m));
	elseif (x_m <= 1.42)
		tmp = Float64(Float64(y_m / x_m) / z_m);
	else
		tmp = Float64(Float64(Float64(0.5 * x_m) * y_m) / z_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 <= 2.15e-162)
		tmp = y_m / (z_m * x_m);
	elseif (x_m <= 1.42)
		tmp = (y_m / x_m) / z_m;
	else
		tmp = ((0.5 * x_m) * y_m) / z_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, 2.15e-162], N[(y$95$m / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 1.42], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(0.5 * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 2.14999999999999998e-162

   1. Initial program 84.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 58.4

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

   5. Applied rewrites 58.4%

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

   if 2.14999999999999998e-162 < x < 1.4199999999999999

   1. Initial program 92.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 92.1

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

   5. Applied rewrites 92.1%

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

   if 1.4199999999999999 < x

   1. Initial program 89.8%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. distribute-lft-out N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. *-rgt-identity N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-/.f64 43.1

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

   5. Applied rewrites 43.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 43.1%

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

Alternative 18: 65.5% accurate, 4.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.42:\\ \;\;\;\;\frac{y\_m}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\_m\right) \cdot y\_m}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 1.42) (/ y_m (* z_m x_m)) (/ (* (* 0.5 x_m) y_m) z_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 <= 1.42) {
		tmp = y_m / (z_m * x_m);
	} else {
		tmp = ((0.5 * x_m) * y_m) / z_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
    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 <= 1.42d0) then
        tmp = y_m / (z_m * x_m)
    else
        tmp = ((0.5d0 * x_m) * y_m) / z_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 <= 1.42) {
		tmp = y_m / (z_m * x_m);
	} else {
		tmp = ((0.5 * x_m) * y_m) / z_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 <= 1.42:
		tmp = y_m / (z_m * x_m)
	else:
		tmp = ((0.5 * x_m) * y_m) / z_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 <= 1.42)
		tmp = Float64(y_m / Float64(z_m * x_m));
	else
		tmp = Float64(Float64(Float64(0.5 * x_m) * y_m) / z_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 <= 1.42)
		tmp = y_m / (z_m * x_m);
	else
		tmp = ((0.5 * x_m) * y_m) / z_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, 1.42], N[(y$95$m / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.4199999999999999

   1. Initial program 86.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 64.9

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

   5. Applied rewrites 64.9%

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

   if 1.4199999999999999 < x

   1. Initial program 89.8%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. distribute-lft-out N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. *-rgt-identity N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-/.f64 43.1

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

   5. Applied rewrites 43.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 43.1%

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

Alternative 19: 65.7% accurate, 4.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.42:\\ \;\;\;\;\frac{y\_m}{z\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x\_m}{z\_m} \cdot y\_m\right) \cdot 0.5\\ \end{array}\right)\right) \end{array} \]

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 1.42) (/ y_m (* z_m x_m)) (* (* (/ x_m z_m) y_m) 0.5))))))

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 <= 1.42) {
		tmp = y_m / (z_m * x_m);
	} else {
		tmp = ((x_m / z_m) * y_m) * 0.5;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
    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 <= 1.42d0) then
        tmp = y_m / (z_m * x_m)
    else
        tmp = ((x_m / z_m) * y_m) * 0.5d0
    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 <= 1.42) {
		tmp = y_m / (z_m * x_m);
	} else {
		tmp = ((x_m / z_m) * y_m) * 0.5;
	}
	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 <= 1.42:
		tmp = y_m / (z_m * x_m)
	else:
		tmp = ((x_m / z_m) * y_m) * 0.5
	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 <= 1.42)
		tmp = Float64(y_m / Float64(z_m * x_m));
	else
		tmp = Float64(Float64(Float64(x_m / z_m) * y_m) * 0.5);
	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 <= 1.42)
		tmp = y_m / (z_m * x_m);
	else
		tmp = ((x_m / z_m) * y_m) * 0.5;
	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, 1.42], N[(y$95$m / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m / z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.4199999999999999

   1. Initial program 86.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 64.9

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

   5. Applied rewrites 64.9%

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

   if 1.4199999999999999 < x

   1. Initial program 89.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. +-commutative N/A

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

      11. associate-/l/ N/A

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

      12. distribute-lft1-in N/A

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

   5. Applied rewrites 41.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 41.1%

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

   10. Applied rewrites 48.7%

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

Alternative 20: 49.3% accurate, 7.5× speedup

Math

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

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 (/ 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 * (y_m / (z_m * x_m))));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
    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 / (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 / (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 / (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(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 / (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[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.9%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 54.2

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

5. Applied rewrites 54.2%

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

Developer Target 1: 97.4% 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

real(8) function code(x, y, z)
    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 2024318 
(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))