Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 85.1% → 98.0%

Time: 11.4s

Alternatives: 17

Speedup: 2.0×

• 64.5% 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: 85.1% 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: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 8.6 \cdot 10^{+150}:\\ \;\;\;\;\frac{y\_m \cdot \left(\left(e^{x} + e^{-x}\right) \cdot \frac{0.5}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \left(\frac{y\_m \cdot \left(x \cdot x\right)}{z} \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right)\right), \mathsf{fma}\left(x, x \cdot 0.5, 1\right) \cdot \frac{y\_m}{z}\right)}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 8.6e+150)
    (/ (* y_m (* (+ (exp x) (exp (- x))) (/ 0.5 x))) z)
    (/
     (fma
      x
      (*
       x
       (*
        (/ (* y_m (* x x)) z)
        (fma (* x x) 0.001388888888888889 0.041666666666666664)))
      (* (fma x (* x 0.5) 1.0) (/ y_m z)))
     x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 8.6e+150) {
		tmp = (y_m * ((exp(x) + exp(-x)) * (0.5 / x))) / z;
	} else {
		tmp = fma(x, (x * (((y_m * (x * x)) / z) * fma((x * x), 0.001388888888888889, 0.041666666666666664))), (fma(x, (x * 0.5), 1.0) * (y_m / z))) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 8.6e+150)
		tmp = Float64(Float64(y_m * Float64(Float64(exp(x) + exp(Float64(-x))) * Float64(0.5 / x))) / z);
	else
		tmp = Float64(fma(x, Float64(x * Float64(Float64(Float64(y_m * Float64(x * x)) / z) * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664))), Float64(fma(x, Float64(x * 0.5), 1.0) * Float64(y_m / z))) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 8.6e+150], N[(N[(y$95$m * N[(N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] * N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(x * N[(x * N[(N[(N[(y$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * N[(x * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 8.59999999999999994e150

   1. Initial program 79.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 97.1

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

   5. Simplified 97.1%

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

   if 8.59999999999999994e150 < y

   1. Initial program 90.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Simplified 90.5%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 99.9%

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

Alternative 2: 95.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{y\_m}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x, x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right), 1\right)}{x}}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ y_m x))))
   (*
    y_s
    (if (<= t_0 INFINITY)
      (/ t_0 z)
      (/
       (/
        (* y_m (fma x (* x (* (* x x) (* (* x x) 0.001388888888888889))) 1.0))
        x)
       z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = cosh(x) * (y_m / x);
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 / z;
	} else {
		tmp = ((y_m * fma(x, (x * ((x * x) * ((x * x) * 0.001388888888888889))), 1.0)) / x) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(cosh(x) * Float64(y_m / x))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(Float64(Float64(y_m * fma(x, Float64(x * Float64(Float64(x * x) * Float64(Float64(x * x) * 0.001388888888888889))), 1.0)) / x) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{y \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{720} \cdot {x}^{4}\right)}, 1\right)}{x}}{z} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 100.0

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

   8. Simplified 100.0%

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

Alternative 3: 87.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((cosh(x) * (y_m / x)) <= ((double) INFINITY)) {
		tmp = ((y_m / x) * fma(x, (x * fma((x * x), 0.041666666666666664, 0.5)), 1.0)) / z;
	} else {
		tmp = (y_m * (0.041666666666666664 * (x * (x * x)))) / z;
	}
	return y_s * tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 86.9

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

   5. Simplified 86.9%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 84.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 0.0

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

   5. Simplified 0.0%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot y}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 97.7

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

   8. Simplified 97.7%

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

4. Final simplification 86.3%

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

Alternative 4: 86.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 84.1

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

   5. Simplified 84.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 83.8

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

   8. Simplified 83.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 0.0

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

   5. Simplified 0.0%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot y}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 97.7

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

   8. Simplified 97.7%

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

4. Final simplification 86.0%

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

Alternative 5: 93.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, t\_0, 0.5\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \left(\frac{y\_m \cdot \left(x \cdot x\right)}{z} \cdot t\_0\right), \mathsf{fma}\left(x, x \cdot 0.5, 1\right) \cdot \frac{y\_m}{z}\right)}{x}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (fma (* x x) 0.001388888888888889 0.041666666666666664)))
   (*
    y_s
    (if (<= y_m 5.5e+86)
      (/ (/ (* y_m (fma x (* x (fma (* x x) t_0 0.5)) 1.0)) x) z)
      (/
       (fma
        x
        (* x (* (/ (* y_m (* x x)) z) t_0))
        (* (fma x (* x 0.5) 1.0) (/ y_m z)))
       x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = fma((x * x), 0.001388888888888889, 0.041666666666666664);
	double tmp;
	if (y_m <= 5.5e+86) {
		tmp = ((y_m * fma(x, (x * fma((x * x), t_0, 0.5)), 1.0)) / x) / z;
	} else {
		tmp = fma(x, (x * (((y_m * (x * x)) / z) * t_0)), (fma(x, (x * 0.5), 1.0) * (y_m / z))) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)
	tmp = 0.0
	if (y_m <= 5.5e+86)
		tmp = Float64(Float64(Float64(y_m * fma(x, Float64(x * fma(Float64(x * x), t_0, 0.5)), 1.0)) / x) / z);
	else
		tmp = Float64(fma(x, Float64(x * Float64(Float64(Float64(y_m * Float64(x * x)) / z) * t_0)), Float64(fma(x, Float64(x * 0.5), 1.0) * Float64(y_m / z))) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]}, N[(y$95$s * If[LessEqual[y$95$m, 5.5e+86], N[(N[(N[(y$95$m * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * t$95$0 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(x * N[(x * N[(N[(N[(y$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(x * N[(x * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.5000000000000002e86

   1. Initial program 78.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Simplified 91.2%

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

   if 5.5000000000000002e86 < y

   1. Initial program 89.7%

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

      1. lower-/.f64 N/A

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

   5. Simplified 84.5%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 94.7%

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

Alternative 6: 69.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+97}:\\ \;\;\;\;\frac{y\_m \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 4.2e-97)
    (/ (/ y_m x) z)
    (if (<= x 8e+97)
      (/
       (*
        y_m
        (fma
         x
         (*
          x
          (fma
           (* x x)
           (fma (* x x) 0.001388888888888889 0.041666666666666664)
           0.5))
         1.0))
       (* x z))
      (/ (* y_m (* 0.041666666666666664 (* x (* x x)))) z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 4.2e-97) {
		tmp = (y_m / x) / z;
	} else if (x <= 8e+97) {
		tmp = (y_m * fma(x, (x * fma((x * x), fma((x * x), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0)) / (x * z);
	} else {
		tmp = (y_m * (0.041666666666666664 * (x * (x * x)))) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 4.2e-97)
		tmp = Float64(Float64(y_m / x) / z);
	elseif (x <= 8e+97)
		tmp = Float64(Float64(y_m * fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0)) / Float64(x * z));
	else
		tmp = Float64(Float64(y_m * Float64(0.041666666666666664 * Float64(x * Float64(x * x)))) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 4.2e-97], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[x, 8e+97], N[(N[(y$95$m * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(0.041666666666666664 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 4.2000000000000002e-97

   1. Initial program 86.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 52.5

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

   5. Simplified 52.5%

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

   if 4.2000000000000002e-97 < x < 8.0000000000000006e97

   1. Initial program 91.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 78.2%

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

   if 8.0000000000000006e97 < x

   1. Initial program 50.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 50.0

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

   5. Simplified 50.0%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot y}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 98.0

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

   8. Simplified 98.0%

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

Alternative 7: 91.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 7 \cdot 10^{+186}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 7e+186)
    (/
     (/
      (*
       y_m
       (fma
        x
        (*
         x
         (fma
          (* x x)
          (fma (* x x) 0.001388888888888889 0.041666666666666664)
          0.5))
        1.0))
      x)
     z)
    (/ (* (/ y_m z) (fma 0.5 (* x x) 1.0)) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 7e+186) {
		tmp = ((y_m * fma(x, (x * fma((x * x), fma((x * x), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0)) / x) / z;
	} else {
		tmp = ((y_m / z) * fma(0.5, (x * x), 1.0)) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 7e+186)
		tmp = Float64(Float64(Float64(y_m * fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0)) / x) / z);
	else
		tmp = Float64(Float64(Float64(y_m / z) * fma(0.5, Float64(x * x), 1.0)) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 7e+186], N[(N[(N[(y$95$m * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y$95$m / z), $MachinePrecision] * N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 6.99999999999999974e186

   1. Initial program 79.4%

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

      1. lower-/.f64 N/A

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

   5. Simplified 90.0%

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

   if 6.99999999999999974e186 < y

   1. Initial program 93.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 93.6

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

   5. Simplified 93.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{x}} \]
   7. 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. lower-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 100.0

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

   8. Simplified 100.0%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+186}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+97}:\\ \;\;\;\;\frac{y\_m \cdot \mathsf{fma}\left(x, x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right), 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 4.2e-97)
    (/ (/ y_m x) z)
    (if (<= x 8e+97)
      (/
       (* y_m (fma x (* x (* (* x x) (* (* x x) 0.001388888888888889))) 1.0))
       (* x z))
      (/ (* y_m (* 0.041666666666666664 (* x (* x x)))) z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 4.2e-97) {
		tmp = (y_m / x) / z;
	} else if (x <= 8e+97) {
		tmp = (y_m * fma(x, (x * ((x * x) * ((x * x) * 0.001388888888888889))), 1.0)) / (x * z);
	} else {
		tmp = (y_m * (0.041666666666666664 * (x * (x * x)))) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 4.2e-97)
		tmp = Float64(Float64(y_m / x) / z);
	elseif (x <= 8e+97)
		tmp = Float64(Float64(y_m * fma(x, Float64(x * Float64(Float64(x * x) * Float64(Float64(x * x) * 0.001388888888888889))), 1.0)) / Float64(x * z));
	else
		tmp = Float64(Float64(y_m * Float64(0.041666666666666664 * Float64(x * Float64(x * x)))) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 4.2e-97], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[x, 8e+97], N[(N[(y$95$m * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(0.041666666666666664 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 4.2000000000000002e-97

   1. Initial program 86.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 52.5

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

   5. Simplified 52.5%

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

   if 4.2000000000000002e-97 < x < 8.0000000000000006e97

   1. Initial program 91.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 78.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{y \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{720} \cdot {x}^{4}\right)}, 1\right)}{x \cdot z} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 77.9

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

   8. Simplified 77.9%

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

   if 8.0000000000000006e97 < x

   1. Initial program 50.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 50.0

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

   5. Simplified 50.0%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot y}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 98.0

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

   8. Simplified 98.0%

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

Alternative 9: 91.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 7.4 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(x, x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 7.4e+152)
    (/
     (/
      (* y_m (fma x (* x (* (* x x) (* (* x x) 0.001388888888888889))) 1.0))
      x)
     z)
    (/ (* (/ y_m z) (fma 0.5 (* x x) 1.0)) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 7.4e+152) {
		tmp = ((y_m * fma(x, (x * ((x * x) * ((x * x) * 0.001388888888888889))), 1.0)) / x) / z;
	} else {
		tmp = ((y_m / z) * fma(0.5, (x * x), 1.0)) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 7.4e+152)
		tmp = Float64(Float64(Float64(y_m * fma(x, Float64(x * Float64(Float64(x * x) * Float64(Float64(x * x) * 0.001388888888888889))), 1.0)) / x) / z);
	else
		tmp = Float64(Float64(Float64(y_m / z) * fma(0.5, Float64(x * x), 1.0)) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 7.4e+152], N[(N[(N[(y$95$m * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y$95$m / z), $MachinePrecision] * N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 7.39999999999999992e152

   1. Initial program 79.3%

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

      1. lower-/.f64 N/A

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

   5. Simplified 90.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{y \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{720} \cdot {x}^{4}\right)}, 1\right)}{x}}{z} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 90.1

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

   8. Simplified 90.1%

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

   if 7.39999999999999992e152 < y

   1. Initial program 90.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 90.5

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

   5. Simplified 90.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{x}} \]
   7. 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. lower-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 97.9

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

   8. Simplified 97.9%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.4 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x, x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 0.029:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m, x \cdot 0.5, \frac{y\_m}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 0.029)
    (/ (fma y_m (* x 0.5) (/ y_m x)) z)
    (/ (* y_m (* 0.041666666666666664 (* x (* x x)))) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 0.029) {
		tmp = fma(y_m, (x * 0.5), (y_m / x)) / z;
	} else {
		tmp = (y_m * (0.041666666666666664 * (x * (x * x)))) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 0.029)
		tmp = Float64(fma(y_m, Float64(x * 0.5), Float64(y_m / x)) / z);
	else
		tmp = Float64(Float64(y_m * Float64(0.041666666666666664 * Float64(x * Float64(x * x)))) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 0.029], N[(N[(y$95$m * N[(x * 0.5), $MachinePrecision] + N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m * N[(0.041666666666666664 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.0290000000000000015

   1. Initial program 86.8%

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

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      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. associate-/l* N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 68.3

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

   5. Simplified 68.3%

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

   if 0.0290000000000000015 < x

   1. Initial program 62.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 51.1

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

   5. Simplified 51.1%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot y}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.5

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

   8. Simplified 84.5%

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

Alternative 11: 67.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 0.029:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 0.029)
    (/ (/ y_m x) z)
    (/ (* y_m (* 0.041666666666666664 (* x (* x x)))) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 0.029) {
		tmp = (y_m / x) / z;
	} else {
		tmp = (y_m * (0.041666666666666664 * (x * (x * x)))) / z;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 0.029d0) then
        tmp = (y_m / x) / z
    else
        tmp = (y_m * (0.041666666666666664d0 * (x * (x * x)))) / z
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 0.029) {
		tmp = (y_m / x) / z;
	} else {
		tmp = (y_m * (0.041666666666666664 * (x * (x * x)))) / z;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 0.029:
		tmp = (y_m / x) / z
	else:
		tmp = (y_m * (0.041666666666666664 * (x * (x * x)))) / z
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 0.029)
		tmp = Float64(Float64(y_m / x) / z);
	else
		tmp = Float64(Float64(y_m * Float64(0.041666666666666664 * Float64(x * Float64(x * x)))) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 0.029)
		tmp = (y_m / x) / z;
	else
		tmp = (y_m * (0.041666666666666664 * (x * (x * x)))) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 0.029], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m * N[(0.041666666666666664 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.0290000000000000015

   1. Initial program 86.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 55.8

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

   5. Simplified 55.8%

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

   if 0.0290000000000000015 < x

   1. Initial program 62.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 51.1

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

   5. Simplified 51.1%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot y}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.5

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

   8. Simplified 84.5%

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

Alternative 12: 66.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 0.029:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(x \cdot \left(y\_m \cdot \left(x \cdot x\right)\right)\right)}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 0.029)
    (/ (/ y_m x) z)
    (/ (* 0.041666666666666664 (* x (* y_m (* x x)))) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 0.029) {
		tmp = (y_m / x) / z;
	} else {
		tmp = (0.041666666666666664 * (x * (y_m * (x * x)))) / z;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 0.029d0) then
        tmp = (y_m / x) / z
    else
        tmp = (0.041666666666666664d0 * (x * (y_m * (x * x)))) / z
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 0.029) {
		tmp = (y_m / x) / z;
	} else {
		tmp = (0.041666666666666664 * (x * (y_m * (x * x)))) / z;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 0.029:
		tmp = (y_m / x) / z
	else:
		tmp = (0.041666666666666664 * (x * (y_m * (x * x)))) / z
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 0.029)
		tmp = Float64(Float64(y_m / x) / z);
	else
		tmp = Float64(Float64(0.041666666666666664 * Float64(x * Float64(y_m * Float64(x * x)))) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 0.029)
		tmp = (y_m / x) / z;
	else
		tmp = (0.041666666666666664 * (x * (y_m * (x * x)))) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 0.029], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(0.041666666666666664 * N[(x * N[(y$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.0290000000000000015

   1. Initial program 86.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 55.8

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

   5. Simplified 55.8%

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

   if 0.0290000000000000015 < x

   1. Initial program 62.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 51.1

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

   5. Simplified 51.1%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot y}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.5

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

   8. Simplified 84.5%

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

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}}{z} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 80.3

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

   11. Simplified 80.3%

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

Alternative 13: 65.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 0.029:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(y\_m \cdot \left(x \cdot x\right)\right) \cdot 0.041666666666666664}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 0.029)
    (/ (/ y_m x) z)
    (* x (/ (* (* y_m (* x x)) 0.041666666666666664) z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 0.029) {
		tmp = (y_m / x) / z;
	} else {
		tmp = x * (((y_m * (x * x)) * 0.041666666666666664) / z);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 0.029d0) then
        tmp = (y_m / x) / z
    else
        tmp = x * (((y_m * (x * x)) * 0.041666666666666664d0) / z)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 0.029) {
		tmp = (y_m / x) / z;
	} else {
		tmp = x * (((y_m * (x * x)) * 0.041666666666666664) / z);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 0.029:
		tmp = (y_m / x) / z
	else:
		tmp = x * (((y_m * (x * x)) * 0.041666666666666664) / z)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 0.029)
		tmp = Float64(Float64(y_m / x) / z);
	else
		tmp = Float64(x * Float64(Float64(Float64(y_m * Float64(x * x)) * 0.041666666666666664) / z));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 0.029)
		tmp = (y_m / x) / z;
	else
		tmp = x * (((y_m * (x * x)) * 0.041666666666666664) / z);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 0.029], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], N[(x * N[(N[(N[(y$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.0290000000000000015

   1. Initial program 86.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 55.8

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

   5. Simplified 55.8%

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

   if 0.0290000000000000015 < x

   1. Initial program 62.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 51.1

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

   5. Simplified 51.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \frac{{x}^{3} \cdot y}{z}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} \cdot y}{z} \cdot \frac{1}{24}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left({x}^{3} \cdot \frac{y}{z}\right)} \cdot \frac{1}{24} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{y}{z} \cdot \frac{1}{24}\right)} \]

      4. *-commutative N/A

         \[\leadsto {x}^{3} \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{y}{z}\right)} \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 75.1

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

   8. Simplified 75.1%

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

Alternative 14: 57.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 0.029:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(y\_m \cdot x\right)}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 0.029) (/ (/ y_m x) z) (/ (* 0.5 (* y_m x)) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 0.029) {
		tmp = (y_m / x) / z;
	} else {
		tmp = (0.5 * (y_m * x)) / z;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 0.029d0) then
        tmp = (y_m / x) / z
    else
        tmp = (0.5d0 * (y_m * x)) / z
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 0.029) {
		tmp = (y_m / x) / z;
	} else {
		tmp = (0.5 * (y_m * x)) / z;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 0.029:
		tmp = (y_m / x) / z
	else:
		tmp = (0.5 * (y_m * x)) / z
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 0.029)
		tmp = Float64(Float64(y_m / x) / z);
	else
		tmp = Float64(Float64(0.5 * Float64(y_m * x)) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 0.029)
		tmp = (y_m / x) / z;
	else
		tmp = (0.5 * (y_m * x)) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 0.029], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(0.5 * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.0290000000000000015

   1. Initial program 86.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 55.8

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

   5. Simplified 55.8%

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

   if 0.0290000000000000015 < x

   1. Initial program 62.3%

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

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      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. associate-/l* N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 31.8

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

   5. Simplified 31.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 31.8

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

   8. Simplified 31.8%

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

Alternative 15: 57.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 0.029:\\ \;\;\;\;\frac{y\_m}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(y\_m \cdot x\right)}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 0.029) (/ y_m (* x z)) (/ (* 0.5 (* y_m x)) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 0.029) {
		tmp = y_m / (x * z);
	} else {
		tmp = (0.5 * (y_m * x)) / z;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 0.029d0) then
        tmp = y_m / (x * z)
    else
        tmp = (0.5d0 * (y_m * x)) / z
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 0.029) {
		tmp = y_m / (x * z);
	} else {
		tmp = (0.5 * (y_m * x)) / z;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 0.029:
		tmp = y_m / (x * z)
	else:
		tmp = (0.5 * (y_m * x)) / z
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 0.029)
		tmp = Float64(y_m / Float64(x * z));
	else
		tmp = Float64(Float64(0.5 * Float64(y_m * x)) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 0.029)
		tmp = y_m / (x * z);
	else
		tmp = (0.5 * (y_m * x)) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 0.029], N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.0290000000000000015

   1. Initial program 86.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. lower-*.f64 54.0

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

   5. Simplified 54.0%

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

   if 0.0290000000000000015 < x

   1. Initial program 62.3%

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

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      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. associate-/l* N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 31.8

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

   5. Simplified 31.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 31.8

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

   8. Simplified 31.8%

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

Alternative 16: 55.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 0.029:\\ \;\;\;\;\frac{y\_m}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{y\_m}{z}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 0.029) (/ y_m (* x z)) (* x (* 0.5 (/ y_m z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 0.029) {
		tmp = y_m / (x * z);
	} else {
		tmp = x * (0.5 * (y_m / z));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 0.029d0) then
        tmp = y_m / (x * z)
    else
        tmp = x * (0.5d0 * (y_m / z))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 0.029) {
		tmp = y_m / (x * z);
	} else {
		tmp = x * (0.5 * (y_m / z));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 0.029:
		tmp = y_m / (x * z)
	else:
		tmp = x * (0.5 * (y_m / z))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 0.029)
		tmp = Float64(y_m / Float64(x * z));
	else
		tmp = Float64(x * Float64(0.5 * Float64(y_m / z)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 0.029)
		tmp = y_m / (x * z);
	else
		tmp = x * (0.5 * (y_m / z));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 0.029], N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.5 * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.0290000000000000015

   1. Initial program 86.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. lower-*.f64 54.0

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

   5. Simplified 54.0%

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

   if 0.0290000000000000015 < x

   1. Initial program 62.3%

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

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      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. associate-/l* N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 31.8

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

   5. Simplified 31.8%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 27.6

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

   8. Simplified 27.6%

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

Alternative 17: 49.6% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (y_m / (x * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
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. lower-*.f64 41.2

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

5. Simplified 41.2%

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

Developer Target 1: 96.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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 2024215 
(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))