Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.3% → 99.7%

Time: 13.0s

Alternatives: 19

Speedup: 2.6×

• 64.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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: 84.3% 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: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 8 \cdot 10^{+42}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{\frac{x}{\cosh x}}\\ \end{array} \end{array} \]

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 8e+42)
    (/ (* y_m (/ (cosh x) x)) z)
    (/ (/ y_m z) (/ x (cosh 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 <= 8e+42) {
		tmp = (y_m * (cosh(x) / x)) / z;
	} else {
		tmp = (y_m / z) / (x / cosh(x));
	}
	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 (y_m <= 8d+42) then
        tmp = (y_m * (cosh(x) / x)) / z
    else
        tmp = (y_m / z) / (x / cosh(x))
    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 (y_m <= 8e+42) {
		tmp = (y_m * (Math.cosh(x) / x)) / z;
	} else {
		tmp = (y_m / z) / (x / Math.cosh(x));
	}
	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 y_m <= 8e+42:
		tmp = (y_m * (math.cosh(x) / x)) / z
	else:
		tmp = (y_m / z) / (x / math.cosh(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 <= 8e+42)
		tmp = Float64(Float64(y_m * Float64(cosh(x) / x)) / z);
	else
		tmp = Float64(Float64(y_m / z) / Float64(x / cosh(x)));
	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 (y_m <= 8e+42)
		tmp = (y_m * (cosh(x) / x)) / z;
	else
		tmp = (y_m / z) / (x / cosh(x));
	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[y$95$m, 8e+42], N[(N[(y$95$m * N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] / N[(x / N[Cosh[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 8.00000000000000036e42

   1. Initial program 81.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 99.0

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

   4. Applied rewrites 99.0%

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

   if 8.00000000000000036e42 < y

   1. Initial program 90.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-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. clear-num N/A

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

      9. div-inv N/A

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

      10. lift-cosh.f64 N/A

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

      11. cosh-def N/A

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. clear-num N/A

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

      17. cosh-def N/A

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

      18. lift-cosh.f64 N/A

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

      19. div-inv N/A

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

      20. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+42}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{x}{\cosh x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.5% accurate, 0.5× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ y_m x))))
   (*
    y_s
    (if (<= t_0 INFINITY)
      (/ t_0 z)
      (/
       (*
        y_m
        (/
         (fma
          (* x x)
          (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 t_0 = cosh(x) * (y_m / x);
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 / z;
	} else {
		tmp = (y_m * (fma((x * x), 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)
	t_0 = Float64(cosh(x) * Float64(y_m / x))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x * x), fma(Float64(x * 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_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, Infinity], N[(t$95$0 / z), $MachinePrecision], N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 92.1%

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

   7. Applied rewrites 92.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 97.3%

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

Alternative 3: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{z}{y\_m}}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{y\_m \cdot \cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)}{x}}{z}\\ \end{array} \end{array} \]

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 1.35e-84)
    (/ (/ 1.0 x) (/ z y_m))
    (if (<= x 2.6e+77)
      (/ (* y_m (cosh x)) (* x z))
      (/ (/ (* y_m (* (* x x) (* x (* x 0.041666666666666664)))) 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 <= 1.35e-84) {
		tmp = (1.0 / x) / (z / y_m);
	} else if (x <= 2.6e+77) {
		tmp = (y_m * cosh(x)) / (x * z);
	} else {
		tmp = ((y_m * ((x * x) * (x * (x * 0.041666666666666664)))) / 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 <= 1.35d-84) then
        tmp = (1.0d0 / x) / (z / y_m)
    else if (x <= 2.6d+77) then
        tmp = (y_m * cosh(x)) / (x * z)
    else
        tmp = ((y_m * ((x * x) * (x * (x * 0.041666666666666664d0)))) / 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 <= 1.35e-84) {
		tmp = (1.0 / x) / (z / y_m);
	} else if (x <= 2.6e+77) {
		tmp = (y_m * Math.cosh(x)) / (x * z);
	} else {
		tmp = ((y_m * ((x * x) * (x * (x * 0.041666666666666664)))) / 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 <= 1.35e-84:
		tmp = (1.0 / x) / (z / y_m)
	elif x <= 2.6e+77:
		tmp = (y_m * math.cosh(x)) / (x * z)
	else:
		tmp = ((y_m * ((x * x) * (x * (x * 0.041666666666666664)))) / 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 <= 1.35e-84)
		tmp = Float64(Float64(1.0 / x) / Float64(z / y_m));
	elseif (x <= 2.6e+77)
		tmp = Float64(Float64(y_m * cosh(x)) / Float64(x * z));
	else
		tmp = Float64(Float64(Float64(y_m * Float64(Float64(x * x) * Float64(x * Float64(x * 0.041666666666666664)))) / 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 <= 1.35e-84)
		tmp = (1.0 / x) / (z / y_m);
	elseif (x <= 2.6e+77)
		tmp = (y_m * cosh(x)) / (x * z);
	else
		tmp = ((y_m * ((x * x) * (x * (x * 0.041666666666666664)))) / 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, 1.35e-84], N[(N[(1.0 / x), $MachinePrecision] / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6e+77], N[(N[(y$95$m * N[Cosh[x], $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.35e-84

   1. Initial program 85.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 55.7

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

   5. Applied rewrites 55.7%

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

   7. Applied rewrites 59.4%

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

   if 1.35e-84 < x < 2.6000000000000002e77

   1. Initial program 93.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 99.8

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

   4. Applied rewrites 99.8%

      \[\leadsto \frac{\color{blue}{\frac{\cosh x}{x} \cdot y}}{z} \]
   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 99.9

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

   6. Applied rewrites 99.9%

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

   if 2.6000000000000002e77 < x

   1. Initial program 70.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)}{x}}}{z} \]
   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 96.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{z}{y}}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{y \cdot \cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 10^{-84}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{z}{y\_m}}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;y\_m \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)}{x}}{z}\\ \end{array} \end{array} \]

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 1e-84)
    (/ (/ 1.0 x) (/ z y_m))
    (if (<= x 2.6e+77)
      (* y_m (/ (cosh x) (* x z)))
      (/ (/ (* y_m (* (* x x) (* x (* x 0.041666666666666664)))) 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 <= 1e-84) {
		tmp = (1.0 / x) / (z / y_m);
	} else if (x <= 2.6e+77) {
		tmp = y_m * (cosh(x) / (x * z));
	} else {
		tmp = ((y_m * ((x * x) * (x * (x * 0.041666666666666664)))) / 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 <= 1d-84) then
        tmp = (1.0d0 / x) / (z / y_m)
    else if (x <= 2.6d+77) then
        tmp = y_m * (cosh(x) / (x * z))
    else
        tmp = ((y_m * ((x * x) * (x * (x * 0.041666666666666664d0)))) / 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 <= 1e-84) {
		tmp = (1.0 / x) / (z / y_m);
	} else if (x <= 2.6e+77) {
		tmp = y_m * (Math.cosh(x) / (x * z));
	} else {
		tmp = ((y_m * ((x * x) * (x * (x * 0.041666666666666664)))) / 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 <= 1e-84:
		tmp = (1.0 / x) / (z / y_m)
	elif x <= 2.6e+77:
		tmp = y_m * (math.cosh(x) / (x * z))
	else:
		tmp = ((y_m * ((x * x) * (x * (x * 0.041666666666666664)))) / 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 <= 1e-84)
		tmp = Float64(Float64(1.0 / x) / Float64(z / y_m));
	elseif (x <= 2.6e+77)
		tmp = Float64(y_m * Float64(cosh(x) / Float64(x * z)));
	else
		tmp = Float64(Float64(Float64(y_m * Float64(Float64(x * x) * Float64(x * Float64(x * 0.041666666666666664)))) / 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 <= 1e-84)
		tmp = (1.0 / x) / (z / y_m);
	elseif (x <= 2.6e+77)
		tmp = y_m * (cosh(x) / (x * z));
	else
		tmp = ((y_m * ((x * x) * (x * (x * 0.041666666666666664)))) / 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, 1e-84], N[(N[(1.0 / x), $MachinePrecision] / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6e+77], N[(y$95$m * N[(N[Cosh[x], $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1e-84

   1. Initial program 85.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 55.7

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

   5. Applied rewrites 55.7%

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

   7. Applied rewrites 59.4%

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

   if 1e-84 < x < 2.6000000000000002e77

   1. Initial program 93.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if 2.6000000000000002e77 < x

   1. Initial program 70.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)}{x}}}{z} \]
   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 96.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

Alternative 5: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\cosh x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3 \cdot 10^{+43}:\\ \;\;\;\;\frac{y\_m \cdot t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{y\_m}{z}\\ \end{array} \end{array} \end{array} \]

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 3e+43) (/ (* 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 <= 3e+43) {
		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 <= 3d+43) 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 <= 3e+43) {
		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 <= 3e+43:
		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 <= 3e+43)
		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 <= 3e+43)
		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, 3e+43], 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 < 3.00000000000000017e43

   1. Initial program 81.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 99.0

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

   4. Applied rewrites 99.0%

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

   if 3.00000000000000017e43 < y

   1. Initial program 90.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-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 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+43}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := x \cdot \mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.55 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{\frac{y\_m \cdot \mathsf{fma}\left(x, t\_0 \cdot \left(x \cdot t\_0\right), -1\right)}{\mathsf{fma}\left(x, t\_0, -1\right)}}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)}{x}}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* x (fma 0.041666666666666664 (* x x) 0.5))))
   (*
    y_s
    (if (<= x 2.55e+77)
      (/ (/ (/ (* y_m (fma x (* t_0 (* x t_0)) -1.0)) (fma x t_0 -1.0)) z) x)
      (/ (/ (* y_m (* (* x x) (* x (* x 0.041666666666666664)))) x) z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = x * fma(0.041666666666666664, (x * x), 0.5);
	double tmp;
	if (x <= 2.55e+77) {
		tmp = (((y_m * fma(x, (t_0 * (x * t_0)), -1.0)) / fma(x, t_0, -1.0)) / z) / x;
	} else {
		tmp = ((y_m * ((x * x) * (x * (x * 0.041666666666666664)))) / x) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(x * fma(0.041666666666666664, Float64(x * x), 0.5))
	tmp = 0.0
	if (x <= 2.55e+77)
		tmp = Float64(Float64(Float64(Float64(y_m * fma(x, Float64(t_0 * Float64(x * t_0)), -1.0)) / fma(x, t_0, -1.0)) / z) / x);
	else
		tmp = Float64(Float64(Float64(y_m * Float64(Float64(x * x) * Float64(x * Float64(x * 0.041666666666666664)))) / x) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(x * N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[x, 2.55e+77], N[(N[(N[(N[(y$95$m * N[(x * N[(t$95$0 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(x * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.54999999999999985e77

   1. Initial program 86.9%

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

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

   7. Applied rewrites 81.6%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 85.6%

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

   12. Applied rewrites 67.0%

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

   if 2.54999999999999985e77 < x

   1. Initial program 70.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)}{x}}}{z} \]
   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 96.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

Alternative 7: 92.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 4.3 \cdot 10^{-125}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, t\_0, 0.5\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, t\_0 \cdot \frac{y\_m \cdot \left(x \cdot x\right)}{z}, \frac{y\_m}{z} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\right)}{x}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (fma (* x x) 0.001388888888888889 0.041666666666666664)))
   (*
    y_s
    (if (<= y_m 4.3e-125)
      (/ (* y_m (/ (fma (* x x) (fma (* x x) t_0 0.5) 1.0) x)) z)
      (/
       (fma
        (* x x)
        (* t_0 (/ (* y_m (* x x)) z))
        (* (/ y_m z) (fma 0.5 (* x x) 1.0)))
       x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = fma((x * x), 0.001388888888888889, 0.041666666666666664);
	double tmp;
	if (y_m <= 4.3e-125) {
		tmp = (y_m * (fma((x * x), fma((x * x), t_0, 0.5), 1.0) / x)) / z;
	} else {
		tmp = fma((x * x), (t_0 * ((y_m * (x * x)) / z)), ((y_m / z) * fma(0.5, (x * x), 1.0))) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)
	tmp = 0.0
	if (y_m <= 4.3e-125)
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x * x), fma(Float64(x * x), t_0, 0.5), 1.0) / x)) / z);
	else
		tmp = Float64(fma(Float64(x * x), Float64(t_0 * Float64(Float64(y_m * Float64(x * x)) / z)), Float64(Float64(y_m / z) * fma(0.5, Float64(x * x), 1.0))) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]}, N[(y$95$s * If[LessEqual[y$95$m, 4.3e-125], N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * t$95$0 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * N[(N[(y$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + N[(N[(y$95$m / z), $MachinePrecision] * N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.3000000000000002e-125

   1. Initial program 80.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 87.5%

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

   7. Applied rewrites 87.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 89.8%

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

   if 4.3000000000000002e-125 < y

   1. Initial program 88.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 94.4

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

   4. Applied rewrites 94.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 97.7%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-125}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right) \cdot \frac{y \cdot \left(x \cdot x\right)}{z}, \frac{y}{z} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 4 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), y\_m, y\_m\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y\_m, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right), y\_m\right)}{z}}{x}\\ \end{array} \end{array} \]

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 4e+58)
    (/
     (/
      (fma
       (*
        (* x x)
        (fma
         x
         (* x (fma x (* x 0.001388888888888889) 0.041666666666666664))
         0.5))
       y_m
       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 <= 4e+58) {
		tmp = (fma(((x * x) * fma(x, (x * fma(x, (x * 0.001388888888888889), 0.041666666666666664)), 0.5)), y_m, 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 <= 4e+58)
		tmp = Float64(Float64(fma(Float64(Float64(x * x) * fma(x, Float64(x * fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664)), 0.5)), y_m, y_m) / x) / z);
	else
		tmp = Float64(Float64(fma(y_m, Float64(x * Float64(x * fma(Float64(x * 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, 4e+58], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * y$95$m + y$95$m), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y$95$m * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.99999999999999978e58

   1. Initial program 80.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 88.9%

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

   7. Applied rewrites 91.2%

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

   if 3.99999999999999978e58 < y

   1. Initial program 92.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 92.0

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

   4. Applied rewrites 92.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 95.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 97.9%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), y, y\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right), y\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3 \cdot 10^{+43}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y\_m, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right), y\_m\right)}{z}}{x}\\ \end{array} \end{array} \]

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 3e+43)
    (/
     (*
      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 <= 3e+43) {
		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 <= 3e+43)
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0) / x)) / z);
	else
		tmp = Float64(Float64(fma(y_m, Float64(x * Float64(x * fma(Float64(x * 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, 3e+43], N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y$95$m * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.00000000000000017e43

   1. Initial program 81.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 89.2%

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

   7. Applied rewrites 89.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 91.1%

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

   if 3.00000000000000017e43 < y

   1. Initial program 90.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 90.6

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

   4. Applied rewrites 90.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 96.1%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 98.0%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+43}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right), y\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.2 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y\_m \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right), x \cdot x, y\_m\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y\_m, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right), y\_m\right)}{z}}{x}\\ \end{array} \end{array} \]

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 1.2e-144)
    (/
     (/
      (fma (* y_m (* x (* x (* (* x x) 0.001388888888888889)))) (* x x) 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 <= 1.2e-144) {
		tmp = (fma((y_m * (x * (x * ((x * x) * 0.001388888888888889)))), (x * x), 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 <= 1.2e-144)
		tmp = Float64(Float64(fma(Float64(y_m * Float64(x * Float64(x * Float64(Float64(x * x) * 0.001388888888888889)))), Float64(x * x), y_m) / x) / z);
	else
		tmp = Float64(Float64(fma(y_m, Float64(x * Float64(x * fma(Float64(x * 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, 1.2e-144], N[(N[(N[(N[(y$95$m * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + y$95$m), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y$95$m * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.19999999999999997e-144

   1. Initial program 80.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 87.7%

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

   7. Applied rewrites 87.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 87.1%

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

   if 1.19999999999999997e-144 < y

   1. Initial program 88.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 94.7

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

   4. Applied rewrites 94.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 95.7%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 96.8%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right), x \cdot x, y\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right), y\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{z}{y\_m}}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+98}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right), y\_m\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)}{z}\\ \end{array} \end{array} \]

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 1.35e-84)
    (/ (/ 1.0 x) (/ z y_m))
    (if (<= x 1.15e+98)
      (/
       (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 <= 1.35e-84) {
		tmp = (1.0 / x) / (z / y_m);
	} else if (x <= 1.15e+98) {
		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 <= 1.35e-84)
		tmp = Float64(Float64(1.0 / x) / Float64(z / y_m));
	elseif (x <= 1.15e+98)
		tmp = Float64(fma(y_m, Float64(x * Float64(x * fma(Float64(x * x), 0.041666666666666664, 0.5))), y_m) / Float64(x * z));
	else
		tmp = Float64(Float64(y_m * Float64(x * Float64(x * Float64(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, 1.35e-84], N[(N[(1.0 / x), $MachinePrecision] / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.15e+98], N[(N[(y$95$m * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.35e-84

   1. Initial program 85.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 55.7

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

   5. Applied rewrites 55.7%

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

   7. Applied rewrites 59.4%

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

   if 1.35e-84 < x < 1.15000000000000007e98

   1. Initial program 93.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 67.7%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 70.6%

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

   if 1.15000000000000007e98 < x

   1. Initial program 69.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 98.4%

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

   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 95.4%

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

   10. Applied rewrites 100.0%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{z}{y}}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+98}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\right), y\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.6% accurate, 2.4× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 86.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 58.8

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

   5. Applied rewrites 58.8%

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

   7. Applied rewrites 61.7%

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

   if 2.2000000000000002 < x

   1. Initial program 74.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 86.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 88.8%

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

Alternative 13: 89.0% accurate, 2.6× speedup

Math

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

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 (/ (fma x (* x (fma 0.041666666666666664 (* x x) 0.5)) 1.0) 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) {
	return y_s * ((y_m * (fma(x, (x * fma(0.041666666666666664, (x * x), 0.5)), 1.0) / z)) / x);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(Float64(y_m * Float64(fma(x, Float64(x * fma(0.041666666666666664, Float64(x * x), 0.5)), 1.0) / z)) / x))
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[(N[(y$95$m * N[(N[(x * N[(x * N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
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 97.3

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

4. Applied rewrites 97.3%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 84.3%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 89.0%

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

12. Applied rewrites 89.3%

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

Alternative 14: 69.6% accurate, 3.2× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 86.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 58.8

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

   5. Applied rewrites 58.8%

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

   7. Applied rewrites 61.7%

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

   if 2.2000000000000002 < x

   1. Initial program 74.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 86.3%

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

   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 84.0%

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

   10. Applied rewrites 87.5%

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

4. Final simplification 69.5%

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

Alternative 15: 70.3% accurate, 3.4× speedup

Math

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

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) 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) / 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) / 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) / 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) / 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) / x);
	else
		tmp = Float64(Float64(y_m * Float64(x * Float64(x * Float64(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) / 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] / x), $MachinePrecision], N[(N[(y$95$m * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 86.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 58.8

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

   5. Applied rewrites 58.8%

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

   7. Applied rewrites 61.7%

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

   if 2.2000000000000002 < x

   1. Initial program 74.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 86.3%

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

   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 84.0%

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

   10. Applied rewrites 87.5%

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

4. Final simplification 69.5%

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

Alternative 16: 69.3% accurate, 3.4× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 2.2)
    (/ (/ y_m z) x)
    (/ (* x (* 0.041666666666666664 (* y_m (* x x)))) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.2) {
		tmp = (y_m / z) / x;
	} else {
		tmp = (x * (0.041666666666666664 * (y_m * (x * x)))) / z;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 2.2d0) then
        tmp = (y_m / z) / x
    else
        tmp = (x * (0.041666666666666664d0 * (y_m * (x * x)))) / z
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 2.2) {
		tmp = (y_m / z) / x;
	} else {
		tmp = (x * (0.041666666666666664 * (y_m * (x * x)))) / z;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 2.2:
		tmp = (y_m / z) / x
	else:
		tmp = (x * (0.041666666666666664 * (y_m * (x * x)))) / z
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 2.2)
		tmp = Float64(Float64(y_m / z) / x);
	else
		tmp = Float64(Float64(x * Float64(0.041666666666666664 * Float64(y_m * Float64(x * x)))) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = (y_m / z) / x;
	else
		tmp = (x * (0.041666666666666664 * (y_m * (x * x)))) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 2.2], N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision], N[(N[(x * N[(0.041666666666666664 * N[(y$95$m * 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 86.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 58.8

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

   5. Applied rewrites 58.8%

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

   7. Applied rewrites 61.7%

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

   if 2.2000000000000002 < x

   1. Initial program 74.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 86.3%

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

   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 84.0%

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

4. Final simplification 68.4%

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

Alternative 17: 60.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x \cdot 0.5\right)}{z}\\ \end{array} \end{array} \]

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 1.45) (/ (/ y_m z) x) (/ (* y_m (* x 0.5)) 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 <= 1.45) {
		tmp = (y_m / z) / x;
	} else {
		tmp = (y_m * (x * 0.5)) / 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 <= 1.45d0) then
        tmp = (y_m / z) / x
    else
        tmp = (y_m * (x * 0.5d0)) / 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 <= 1.45) {
		tmp = (y_m / z) / x;
	} else {
		tmp = (y_m * (x * 0.5)) / 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 <= 1.45:
		tmp = (y_m / z) / x
	else:
		tmp = (y_m * (x * 0.5)) / 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 <= 1.45)
		tmp = Float64(Float64(y_m / z) / x);
	else
		tmp = Float64(Float64(y_m * Float64(x * 0.5)) / 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 <= 1.45)
		tmp = (y_m / z) / x;
	else
		tmp = (y_m * (x * 0.5)) / 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, 1.45], N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision], N[(N[(y$95$m * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 86.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 58.8

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

   5. Applied rewrites 58.8%

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

   7. Applied rewrites 61.7%

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

   if 1.44999999999999996 < x

   1. Initial program 74.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}}{z} \]
   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 42.8

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

   5. Applied rewrites 42.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 42.8%

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

Alternative 18: 58.0% accurate, 4.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\frac{y\_m}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(x \cdot 0.5\right)}{z}\\ \end{array} \end{array} \]

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 1.45) (/ y_m (* x z)) (/ (* y_m (* x 0.5)) 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 <= 1.45) {
		tmp = y_m / (x * z);
	} else {
		tmp = (y_m * (x * 0.5)) / 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 <= 1.45d0) then
        tmp = y_m / (x * z)
    else
        tmp = (y_m * (x * 0.5d0)) / 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 <= 1.45) {
		tmp = y_m / (x * z);
	} else {
		tmp = (y_m * (x * 0.5)) / 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 <= 1.45:
		tmp = y_m / (x * z)
	else:
		tmp = (y_m * (x * 0.5)) / 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 <= 1.45)
		tmp = Float64(y_m / Float64(x * z));
	else
		tmp = Float64(Float64(y_m * Float64(x * 0.5)) / 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 <= 1.45)
		tmp = y_m / (x * z);
	else
		tmp = (y_m * (x * 0.5)) / 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, 1.45], N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 86.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 58.8

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

   5. Applied rewrites 58.8%

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

   if 1.44999999999999996 < x

   1. Initial program 74.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}{x}}}{z} \]
   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 42.8

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

   5. Applied rewrites 42.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 42.8%

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

Alternative 19: 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 82.9%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 43.5

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

5. Applied rewrites 43.5%

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

Developer Target 1: 96.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y / z) / x) * cosh(x)
    if (y < (-4.618902267687042d-52)) then
        tmp = t_0
    else if (y < 1.038530535935153d-39) then
        tmp = ((cosh(x) * y) / x) / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z):
	t_0 = ((y / z) / x) * math.cosh(x)
	tmp = 0
	if y < -4.618902267687042e-52:
		tmp = t_0
	elif y < 1.038530535935153e-39:
		tmp = ((math.cosh(x) * y) / x) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y / z) / x) * cosh(x))
	tmp = 0.0
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = Float64(Float64(Float64(cosh(x) * y) / x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y / z) / x) * cosh(x);
	tmp = 0.0;
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = ((cosh(x) * y) / x) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -4.618902267687042e-52], t$95$0, If[Less[y, 1.038530535935153e-39], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024233 
(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))