Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.4% → 99.8%

Time: 13.7s

Alternatives: 21

Speedup: 2.3×

• 64.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\cosh x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.15 \cdot 10^{-24}:\\ \;\;\;\;\frac{y\_m \cdot t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{y\_m}{z}\\ \end{array} \end{array} \end{array} \]

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) x)))
   (* y_s (if (<= y_m 2.15e-24) (/ (* y_m t_0) z) (* t_0 (/ 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 t_0 = cosh(x) / x;
	double tmp;
	if (y_m <= 2.15e-24) {
		tmp = (y_m * t_0) / z;
	} else {
		tmp = t_0 * (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) :: t_0
    real(8) :: tmp
    t_0 = cosh(x) / x
    if (y_m <= 2.15d-24) then
        tmp = (y_m * t_0) / z
    else
        tmp = t_0 * (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 t_0 = Math.cosh(x) / x;
	double tmp;
	if (y_m <= 2.15e-24) {
		tmp = (y_m * t_0) / z;
	} else {
		tmp = t_0 * (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):
	t_0 = math.cosh(x) / x
	tmp = 0
	if y_m <= 2.15e-24:
		tmp = (y_m * t_0) / z
	else:
		tmp = t_0 * (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)
	t_0 = Float64(cosh(x) / x)
	tmp = 0.0
	if (y_m <= 2.15e-24)
		tmp = Float64(Float64(y_m * t_0) / z);
	else
		tmp = Float64(t_0 * 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)
	t_0 = cosh(x) / x;
	tmp = 0.0;
	if (y_m <= 2.15e-24)
		tmp = (y_m * t_0) / z;
	else
		tmp = t_0 * (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_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[y$95$m, 2.15e-24], N[(N[(y$95$m * t$95$0), $MachinePrecision] / z), $MachinePrecision], N[(t$95$0 * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1500000000000002e-24

   1. Initial program 78.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 96.8

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

   4. Applied rewrites 96.8%

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

   if 2.1500000000000002e-24 < y

   1. Initial program 92.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. div-inv N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. div-inv N/A

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

      14. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{-24}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-217}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+51}:\\ \;\;\;\;\frac{y\_m \cdot \cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}{z}\\ \end{array} \end{array} \]

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 2.4e-217)
    (/ (/ y_m z) x)
    (if (<= x 7e+51)
      (/ (* y_m (cosh x)) (* x z))
      (/
       (*
        y_m
        (/
         (fma
          (* x x)
          (fma
           x
           (* x (fma (* x x) 0.001388888888888889 0.041666666666666664))
           0.5)
          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 tmp;
	if (x <= 2.4e-217) {
		tmp = (y_m / z) / x;
	} else if (x <= 7e+51) {
		tmp = (y_m * cosh(x)) / (x * z);
	} else {
		tmp = (y_m * (fma((x * x), fma(x, (x * fma((x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 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)
	tmp = 0.0
	if (x <= 2.4e-217)
		tmp = Float64(Float64(y_m / z) / x);
	elseif (x <= 7e+51)
		tmp = Float64(Float64(y_m * cosh(x)) / Float64(x * z));
	else
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 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_] := N[(y$95$s * If[LessEqual[x, 2.4e-217], N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 7e+51], N[(N[(y$95$m * N[Cosh[x], $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 2.3999999999999999e-217

   1. Initial program 84.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 57.3

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

   5. Applied rewrites 57.3%

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

   7. Applied rewrites 59.5%

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

   if 2.3999999999999999e-217 < x < 7e51

   1. Initial program 95.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 95.2

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

   4. Applied rewrites 95.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 91.6

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

   6. Applied rewrites 91.6%

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

   if 7e51 < x

   1. Initial program 65.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 5.4%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 95.3

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

   10. Applied rewrites 95.3%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 100.0

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

   13. Applied rewrites 100.0%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-217}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+51}:\\ \;\;\;\;\frac{y \cdot \cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 4.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y\_m}{z}\\ \end{array} \end{array} \]

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 (<= y_m 4.5e-25)
    (/
     (*
      y_m
      (/
       (fma
        (* x x)
        (fma
         x
         (* x (fma (* x x) 0.001388888888888889 0.041666666666666664))
         0.5)
        1.0)
       x))
     z)
    (* (/ (cosh x) x) (/ 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 (y_m <= 4.5e-25) {
		tmp = (y_m * (fma((x * x), fma(x, (x * fma((x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0) / x)) / z;
	} else {
		tmp = (cosh(x) / x) * (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 (y_m <= 4.5e-25)
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0) / x)) / z);
	else
		tmp = Float64(Float64(cosh(x) / x) * Float64(y_m / 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[y$95$m, 4.5e-25], N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.5000000000000001e-25

   1. Initial program 78.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 96.8

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 53.9%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 88.5

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

   10. Applied rewrites 88.5%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 91.8

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

   13. Applied rewrites 91.8%

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

   if 4.5000000000000001e-25 < y

   1. Initial program 92.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. div-inv N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. div-inv N/A

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

      14. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y\_m \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), y\_m\right)}{z}}{x}\\ \end{array} \end{array} \]

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 3.4e-13)
    (/
     (*
      y_m
      (/
       (fma
        (* x x)
        (fma
         x
         (* x (fma (* x x) 0.001388888888888889 0.041666666666666664))
         0.5)
        1.0)
       x))
     z)
    (/
     (/ (fma (* y_m x) (* x (fma x (* x 0.041666666666666664) 0.5)) 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 <= 3.4e-13) {
		tmp = (y_m * (fma((x * x), fma(x, (x * fma((x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0) / x)) / z;
	} else {
		tmp = (fma((y_m * x), (x * fma(x, (x * 0.041666666666666664), 0.5)), 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 <= 3.4e-13)
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0) / x)) / z);
	else
		tmp = Float64(Float64(fma(Float64(y_m * x), Float64(x * fma(x, Float64(x * 0.041666666666666664), 0.5)), 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, 3.4e-13], N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(y$95$m * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.40000000000000015e-13

   1. Initial program 79.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 96.8

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 53.4%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 88.1

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

   10. Applied rewrites 88.1%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 91.4

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

   13. Applied rewrites 91.4%

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

   if 3.40000000000000015e-13 < y

   1. Initial program 92.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 84.4%

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

   7. Applied rewrites 95.5%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), y\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y\_m \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), y\_m\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y\_m \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), y\_m\right)}{z}}{x}\\ \end{array} \end{array} \]

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 5e-9)
    (/
     (/
      (fma
       (* y_m x)
       (*
        x
        (fma
         x
         (* x (fma x (* x 0.001388888888888889) 0.041666666666666664))
         0.5))
       y_m)
      x)
     z)
    (/
     (/ (fma (* y_m x) (* x (fma x (* x 0.041666666666666664) 0.5)) 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 <= 5e-9) {
		tmp = (fma((y_m * x), (x * fma(x, (x * fma(x, (x * 0.001388888888888889), 0.041666666666666664)), 0.5)), y_m) / x) / z;
	} else {
		tmp = (fma((y_m * x), (x * fma(x, (x * 0.041666666666666664), 0.5)), 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 <= 5e-9)
		tmp = Float64(Float64(fma(Float64(y_m * x), Float64(x * fma(x, Float64(x * fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664)), 0.5)), y_m) / x) / z);
	else
		tmp = Float64(Float64(fma(Float64(y_m * x), Float64(x * fma(x, Float64(x * 0.041666666666666664), 0.5)), 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, 5e-9], N[(N[(N[(N[(y$95$m * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(y$95$m * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000001e-9

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 96.8

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 89.4%

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

   9. Applied rewrites 90.4%

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

   if 5.0000000000000001e-9 < y

   1. Initial program 91.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 84.2%

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

   7. Applied rewrites 95.5%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), y\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), y\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, y\_m \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), y\_m\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y\_m \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), y\_m\right)}{z}}{x}\\ \end{array} \end{array} \]

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 5e-9)
    (/
     (/
      (fma
       (* x x)
       (*
        y_m
        (fma
         x
         (* x (fma x (* x 0.001388888888888889) 0.041666666666666664))
         0.5))
       y_m)
      x)
     z)
    (/
     (/ (fma (* y_m x) (* x (fma x (* x 0.041666666666666664) 0.5)) 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 <= 5e-9) {
		tmp = (fma((x * x), (y_m * fma(x, (x * fma(x, (x * 0.001388888888888889), 0.041666666666666664)), 0.5)), y_m) / x) / z;
	} else {
		tmp = (fma((y_m * x), (x * fma(x, (x * 0.041666666666666664), 0.5)), 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 <= 5e-9)
		tmp = Float64(Float64(fma(Float64(x * x), Float64(y_m * fma(x, Float64(x * fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664)), 0.5)), y_m) / x) / z);
	else
		tmp = Float64(Float64(fma(Float64(y_m * x), Float64(x * fma(x, Float64(x * 0.041666666666666664), 0.5)), 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, 5e-9], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(y$95$m * N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(y$95$m * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000001e-9

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 96.8

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 89.4%

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

   if 5.0000000000000001e-9 < y

   1. Initial program 91.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 84.2%

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

   7. Applied rewrites 95.5%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, y \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), y\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), y\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, y\_m \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right), y\_m\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y\_m \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), y\_m\right)}{z}}{x}\\ \end{array} \end{array} \]

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 5e-9)
    (/
     (/
      (fma (* x x) (* y_m (* x (* x (* (* x x) 0.001388888888888889)))) y_m)
      x)
     z)
    (/
     (/ (fma (* y_m x) (* x (fma x (* x 0.041666666666666664) 0.5)) 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 <= 5e-9) {
		tmp = (fma((x * x), (y_m * (x * (x * ((x * x) * 0.001388888888888889)))), y_m) / x) / z;
	} else {
		tmp = (fma((y_m * x), (x * fma(x, (x * 0.041666666666666664), 0.5)), 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 <= 5e-9)
		tmp = Float64(Float64(fma(Float64(x * x), Float64(y_m * Float64(x * Float64(x * Float64(Float64(x * x) * 0.001388888888888889)))), y_m) / x) / z);
	else
		tmp = Float64(Float64(fma(Float64(y_m * x), Float64(x * fma(x, Float64(x * 0.041666666666666664), 0.5)), 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, 5e-9], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(y$95$m * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(y$95$m * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000001e-9

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 96.8

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 89.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 89.0%

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

   if 5.0000000000000001e-9 < y

   1. Initial program 91.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 84.2%

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

   7. Applied rewrites 95.5%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, y \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right), y\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), y\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-217}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-121}:\\ \;\;\;\;\frac{y\_m}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(y\_m \cdot \left(x \cdot 0.041666666666666664\right)\right)\right), y\_m\right)}{x}}{z}\\ \end{array} \end{array} \]

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 2.4e-217)
    (/ (/ y_m z) x)
    (if (<= x 2.25e-121)
      (/ y_m (* x z))
      (/
       (/ (fma x (* x (* x (* y_m (* x 0.041666666666666664)))) 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 <= 2.4e-217) {
		tmp = (y_m / z) / x;
	} else if (x <= 2.25e-121) {
		tmp = y_m / (x * z);
	} else {
		tmp = (fma(x, (x * (x * (y_m * (x * 0.041666666666666664)))), 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 <= 2.4e-217)
		tmp = Float64(Float64(y_m / z) / x);
	elseif (x <= 2.25e-121)
		tmp = Float64(y_m / Float64(x * z));
	else
		tmp = Float64(Float64(fma(x, Float64(x * Float64(x * Float64(y_m * Float64(x * 0.041666666666666664)))), y_m) / 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, 2.4e-217], N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.25e-121], N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(x * N[(x * N[(y$95$m * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 2.3999999999999999e-217

   1. Initial program 84.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 57.3

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

   5. Applied rewrites 57.3%

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

   7. Applied rewrites 59.5%

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

   if 2.3999999999999999e-217 < x < 2.2500000000000002e-121

   1. Initial program 89.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 96.4

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

   5. Applied rewrites 96.4%

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

   if 2.2500000000000002e-121 < x

   1. Initial program 78.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 89.4%

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

Alternative 9: 69.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-217}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{y\_m \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)}{z}\\ \end{array} \end{array} \]

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 2.4e-217)
    (/ (/ y_m z) x)
    (if (<= x 2.8e+94)
      (/
       (* y_m (fma x (* x (fma x (* x 0.041666666666666664) 0.5)) 1.0))
       (* x z))
      (/ (* y_m (* x (* (* 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 <= 2.4e-217) {
		tmp = (y_m / z) / x;
	} else if (x <= 2.8e+94) {
		tmp = (y_m * fma(x, (x * fma(x, (x * 0.041666666666666664), 0.5)), 1.0)) / (x * z);
	} else {
		tmp = (y_m * (x * ((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 <= 2.4e-217)
		tmp = Float64(Float64(y_m / z) / x);
	elseif (x <= 2.8e+94)
		tmp = Float64(Float64(y_m * fma(x, Float64(x * fma(x, Float64(x * 0.041666666666666664), 0.5)), 1.0)) / Float64(x * z));
	else
		tmp = Float64(Float64(y_m * Float64(x * Float64(Float64(x * x) * 0.041666666666666664))) / 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, 2.4e-217], N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.8e+94], N[(N[(y$95$m * N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 2.3999999999999999e-217

   1. Initial program 84.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 57.3

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

   5. Applied rewrites 57.3%

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

   7. Applied rewrites 59.5%

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

   if 2.3999999999999999e-217 < x < 2.79999999999999998e94

   1. Initial program 95.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 95.6

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

   4. Applied rewrites 95.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 74.9%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 80.4

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

   10. Applied rewrites 80.4%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-*l/ N/A

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

       5. associate-/l/ N/A

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

       6. lower-/.f64 N/A

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

   12. Applied rewrites 81.4%

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

   if 2.79999999999999998e94 < x

   1. Initial program 61.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.2%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-217}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-217}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), y\_m, y\_m\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)}{z}\\ \end{array} \end{array} \]

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 2.4e-217)
    (/ (/ y_m z) x)
    (if (<= x 2.8e+94)
      (/
       (fma (* (* x x) (fma x (* x 0.041666666666666664) 0.5)) y_m y_m)
       (* x z))
      (/ (* y_m (* x (* (* 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 <= 2.4e-217) {
		tmp = (y_m / z) / x;
	} else if (x <= 2.8e+94) {
		tmp = fma(((x * x) * fma(x, (x * 0.041666666666666664), 0.5)), y_m, y_m) / (x * z);
	} else {
		tmp = (y_m * (x * ((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 <= 2.4e-217)
		tmp = Float64(Float64(y_m / z) / x);
	elseif (x <= 2.8e+94)
		tmp = Float64(fma(Float64(Float64(x * x) * fma(x, Float64(x * 0.041666666666666664), 0.5)), y_m, y_m) / Float64(x * z));
	else
		tmp = Float64(Float64(y_m * Float64(x * Float64(Float64(x * x) * 0.041666666666666664))) / 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, 2.4e-217], N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.8e+94], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * y$95$m + y$95$m), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 2.3999999999999999e-217

   1. Initial program 84.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 57.3

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

   5. Applied rewrites 57.3%

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

   7. Applied rewrites 59.5%

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

   if 2.3999999999999999e-217 < x < 2.79999999999999998e94

   1. Initial program 95.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 81.4%

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

   7. Applied rewrites 81.4%

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

   if 2.79999999999999998e94 < x

   1. Initial program 61.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.2%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-217}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), y, y\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-217}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m \cdot \left(x \cdot x\right), \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), y\_m\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)}{z}\\ \end{array} \end{array} \]

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 2.4e-217)
    (/ (/ y_m z) x)
    (if (<= x 8.2e+92)
      (/
       (fma (* y_m (* x x)) (fma (* x x) 0.041666666666666664 0.5) y_m)
       (* x z))
      (/ (* y_m (* x (* (* 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 <= 2.4e-217) {
		tmp = (y_m / z) / x;
	} else if (x <= 8.2e+92) {
		tmp = fma((y_m * (x * x)), fma((x * x), 0.041666666666666664, 0.5), y_m) / (x * z);
	} else {
		tmp = (y_m * (x * ((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 <= 2.4e-217)
		tmp = Float64(Float64(y_m / z) / x);
	elseif (x <= 8.2e+92)
		tmp = Float64(fma(Float64(y_m * Float64(x * x)), fma(Float64(x * x), 0.041666666666666664, 0.5), y_m) / Float64(x * z));
	else
		tmp = Float64(Float64(y_m * Float64(x * Float64(Float64(x * x) * 0.041666666666666664))) / 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, 2.4e-217], N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 8.2e+92], N[(N[(N[(y$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + y$95$m), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 2.3999999999999999e-217

   1. Initial program 84.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 57.3

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

   5. Applied rewrites 57.3%

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

   7. Applied rewrites 59.5%

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

   if 2.3999999999999999e-217 < x < 8.20000000000000047e92

   1. Initial program 95.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. Applied rewrites 84.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.4%

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

   if 8.20000000000000047e92 < x

   1. Initial program 61.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.2%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-217}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot \left(x \cdot x\right), \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), y\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.9 \cdot 10^{-98}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x \cdot x, t\_0, 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y\_m \cdot x, x \cdot t\_0, y\_m\right)}{z}}{x}\\ \end{array} \end{array} \end{array} \]

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 0.041666666666666664) 0.5)))
   (*
    y_s
    (if (<= y_m 2.9e-98)
      (/ (* y_m (/ (fma (* x x) t_0 1.0) x)) z)
      (/ (/ (fma (* y_m x) (* x t_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 * 0.041666666666666664), 0.5);
	double tmp;
	if (y_m <= 2.9e-98) {
		tmp = (y_m * (fma((x * x), t_0, 1.0) / x)) / z;
	} else {
		tmp = (fma((y_m * x), (x * t_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(x, Float64(x * 0.041666666666666664), 0.5)
	tmp = 0.0
	if (y_m <= 2.9e-98)
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x * x), t_0, 1.0) / x)) / z);
	else
		tmp = Float64(Float64(fma(Float64(y_m * x), Float64(x * t_0), 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[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(y$95$s * If[LessEqual[y$95$m, 2.9e-98], N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(y$95$m * x), $MachinePrecision] * N[(x * t$95$0), $MachinePrecision] + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.9e-98

   1. Initial program 78.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 96.5

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 88.0

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

   7. Applied rewrites 88.0%

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

   if 2.9e-98 < y

   1. Initial program 90.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 84.3%

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

   7. Applied rewrites 94.2%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{-98}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), y\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.7 \cdot 10^{-98}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot 0.041666666666666664\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y\_m \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), y\_m\right)}{z}}{x}\\ \end{array} \end{array} \]

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 2.7e-98)
    (/ (* y_m (/ (fma (* x x) (* x (* x 0.041666666666666664)) 1.0) x)) z)
    (/
     (/ (fma (* y_m x) (* x (fma x (* x 0.041666666666666664) 0.5)) 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 <= 2.7e-98) {
		tmp = (y_m * (fma((x * x), (x * (x * 0.041666666666666664)), 1.0) / x)) / z;
	} else {
		tmp = (fma((y_m * x), (x * fma(x, (x * 0.041666666666666664), 0.5)), 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 <= 2.7e-98)
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x * x), Float64(x * Float64(x * 0.041666666666666664)), 1.0) / x)) / z);
	else
		tmp = Float64(Float64(fma(Float64(y_m * x), Float64(x * fma(x, Float64(x * 0.041666666666666664), 0.5)), 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, 2.7e-98], N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(y$95$m * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.6999999999999999e-98

   1. Initial program 78.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 96.5

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 53.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 88.0

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

   10. Applied rewrites 88.0%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 87.8%

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

   if 2.6999999999999999e-98 < y

   1. Initial program 90.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 84.4%

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

   7. Applied rewrites 94.3%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-98}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot 0.041666666666666664\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), y\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot 0.041666666666666664\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-217}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \left(y\_m \cdot t\_0\right), y\_m\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x \cdot t\_0\right)}{z}\\ \end{array} \end{array} \end{array} \]

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 (* (* x x) 0.041666666666666664)))
   (*
    y_s
    (if (<= x 2.4e-217)
      (/ (/ y_m z) x)
      (if (<= x 8.2e+92)
        (/ (fma x (* x (* y_m t_0)) y_m) (* x z))
        (/ (* y_m (* x t_0)) 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 = (x * x) * 0.041666666666666664;
	double tmp;
	if (x <= 2.4e-217) {
		tmp = (y_m / z) / x;
	} else if (x <= 8.2e+92) {
		tmp = fma(x, (x * (y_m * t_0)), y_m) / (x * z);
	} else {
		tmp = (y_m * (x * t_0)) / 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(Float64(x * x) * 0.041666666666666664)
	tmp = 0.0
	if (x <= 2.4e-217)
		tmp = Float64(Float64(y_m / z) / x);
	elseif (x <= 8.2e+92)
		tmp = Float64(fma(x, Float64(x * Float64(y_m * t_0)), y_m) / Float64(x * z));
	else
		tmp = Float64(Float64(y_m * Float64(x * t_0)) / 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[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]}, N[(y$95$s * If[LessEqual[x, 2.4e-217], N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 8.2e+92], N[(N[(x * N[(x * N[(y$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x < 2.3999999999999999e-217

   1. Initial program 84.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 57.3

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

   5. Applied rewrites 57.3%

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

   7. Applied rewrites 59.5%

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

   if 2.3999999999999999e-217 < x < 8.20000000000000047e92

   1. Initial program 95.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 80.4%

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

   if 8.20000000000000047e92 < x

   1. Initial program 61.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.2%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-217}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \left(y \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right), y\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 87.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 7.2 \cdot 10^{+255}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot 0.041666666666666664\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x \cdot z}\\ \end{array} \end{array} \]

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 (<= y_m 7.2e+255)
    (/ (* y_m (/ (fma (* x x) (* x (* x 0.041666666666666664)) 1.0) x)) z)
    (/
     (* y_m (fma x (* x (fma x (* x 0.041666666666666664) 0.5)) 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 tmp;
	if (y_m <= 7.2e+255) {
		tmp = (y_m * (fma((x * x), (x * (x * 0.041666666666666664)), 1.0) / x)) / z;
	} else {
		tmp = (y_m * fma(x, (x * fma(x, (x * 0.041666666666666664), 0.5)), 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)
	tmp = 0.0
	if (y_m <= 7.2e+255)
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x * x), Float64(x * Float64(x * 0.041666666666666664)), 1.0) / x)) / z);
	else
		tmp = Float64(Float64(y_m * fma(x, Float64(x * fma(x, Float64(x * 0.041666666666666664), 0.5)), 1.0)) / Float64(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[y$95$m, 7.2e+255], N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m * N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 7.1999999999999998e255

   1. Initial program 82.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 51.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 87.5

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

   10. Applied rewrites 87.5%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 87.2%

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

   if 7.1999999999999998e255 < y

   1. Initial program 85.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 85.4

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

   4. Applied rewrites 85.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 76.6

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

   10. Applied rewrites 76.6%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-*l/ N/A

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

       5. associate-/l/ N/A

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

       6. lower-/.f64 N/A

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

   12. Applied rewrites 90.8%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+255}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot 0.041666666666666664\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 69.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(y\_m \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{z}\\ \end{array} \end{array} \]

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 2.2)
    (* (/ y_m z) (/ 1.0 x))
    (/ (* 0.041666666666666664 (* y_m (* 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 <= 2.2) {
		tmp = (y_m / z) * (1.0 / x);
	} else {
		tmp = (0.041666666666666664 * (y_m * (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 <= 2.2d0) then
        tmp = (y_m / z) * (1.0d0 / x)
    else
        tmp = (0.041666666666666664d0 * (y_m * (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 <= 2.2) {
		tmp = (y_m / z) * (1.0 / x);
	} else {
		tmp = (0.041666666666666664 * (y_m * (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 <= 2.2:
		tmp = (y_m / z) * (1.0 / x)
	else:
		tmp = (0.041666666666666664 * (y_m * (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 <= 2.2)
		tmp = Float64(Float64(y_m / z) * Float64(1.0 / x));
	else
		tmp = Float64(Float64(0.041666666666666664 * Float64(y_m * 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 <= 2.2)
		tmp = (y_m / z) * (1.0 / x);
	else
		tmp = (0.041666666666666664 * (y_m * (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, 2.2], N[(N[(y$95$m / z), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(0.041666666666666664 * N[(y$95$m * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 87.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 68.9%

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

   if 2.2000000000000002 < x

   1. Initial program 69.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 83.2%

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

Alternative 17: 69.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)}{z}\\ \end{array} \end{array} \]

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 2.2)
    (* (/ y_m z) (/ 1.0 x))
    (/ (* y_m (* x (* (* 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 <= 2.2) {
		tmp = (y_m / z) * (1.0 / x);
	} else {
		tmp = (y_m * (x * ((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 <= 2.2d0) then
        tmp = (y_m / z) * (1.0d0 / x)
    else
        tmp = (y_m * (x * ((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 <= 2.2) {
		tmp = (y_m / z) * (1.0 / x);
	} else {
		tmp = (y_m * (x * ((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 <= 2.2:
		tmp = (y_m / z) * (1.0 / x)
	else:
		tmp = (y_m * (x * ((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 <= 2.2)
		tmp = Float64(Float64(y_m / z) * Float64(1.0 / x));
	else
		tmp = Float64(Float64(y_m * Float64(x * Float64(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 <= 2.2)
		tmp = (y_m / z) * (1.0 / x);
	else
		tmp = (y_m * (x * ((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, 2.2], N[(N[(y$95$m / z), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 87.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 68.9%

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

   if 2.2000000000000002 < x

   1. Initial program 69.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 83.2%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\frac{y}{z} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 59.3% accurate, 3.8× speedup

Math

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

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 1.4) (* (/ y_m z) (/ 1.0 x)) (/ (* 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 <= 1.4) {
		tmp = (y_m / z) * (1.0 / x);
	} 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 <= 1.4d0) then
        tmp = (y_m / z) * (1.0d0 / x)
    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 <= 1.4) {
		tmp = (y_m / z) * (1.0 / x);
	} 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 <= 1.4:
		tmp = (y_m / z) * (1.0 / x)
	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 <= 1.4)
		tmp = Float64(Float64(y_m / z) * Float64(1.0 / x));
	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 <= 1.4)
		tmp = (y_m / z) * (1.0 / x);
	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, 1.4], N[(N[(y$95$m / z), $MachinePrecision] * N[(1.0 / x), $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 < 1.3999999999999999

   1. Initial program 87.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 68.9%

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

   if 1.3999999999999999 < x

   1. Initial program 69.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}}{z} \]
   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 37.0

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

   5. Applied rewrites 37.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 37.0%

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

Alternative 19: 59.3% accurate, 4.4× speedup

Math

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

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 1.4) (/ (/ y_m z) x) (/ (* 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 <= 1.4) {
		tmp = (y_m / z) / x;
	} 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 <= 1.4d0) then
        tmp = (y_m / z) / x
    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 <= 1.4) {
		tmp = (y_m / z) / x;
	} 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 <= 1.4:
		tmp = (y_m / z) / x
	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 <= 1.4)
		tmp = Float64(Float64(y_m / z) / x);
	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 <= 1.4)
		tmp = (y_m / z) / x;
	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, 1.4], N[(N[(y$95$m / z), $MachinePrecision] / x), $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 < 1.3999999999999999

   1. Initial program 87.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 68.9%

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

   if 1.3999999999999999 < x

   1. Initial program 69.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}}{z} \]
   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 37.0

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

   5. Applied rewrites 37.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 37.0%

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

Alternative 20: 57.6% accurate, 4.6× speedup

Math

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

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 1.4) (/ 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 <= 1.4) {
		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 <= 1.4d0) 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 <= 1.4) {
		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 <= 1.4:
		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 <= 1.4)
		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 <= 1.4)
		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, 1.4], 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 < 1.3999999999999999

   1. Initial program 87.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   if 1.3999999999999999 < x

   1. Initial program 69.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}}{z} \]
   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 37.0

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

   5. Applied rewrites 37.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 37.0%

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

Alternative 21: 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 82.6%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 50.1

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

5. Applied rewrites 50.1%

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

Developer Target 1: 97.2% 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 2024219 
(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))