Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.5% → 99.4%

Time: 7.7s

Alternatives: 9

Speedup: 1.0×

• 63.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 10^{+85}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y\_m}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1e+85)
    (/ (/ (* y_m (cosh x)) x) z)
    (* (/ (cosh x) x) (/ y_m z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1e+85) {
		tmp = ((y_m * cosh(x)) / x) / z;
	} else {
		tmp = (cosh(x) / x) * (y_m / z);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 1d+85) then
        tmp = ((y_m * cosh(x)) / x) / z
    else
        tmp = (cosh(x) / x) * (y_m / z)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1e+85) {
		tmp = ((y_m * Math.cosh(x)) / x) / z;
	} else {
		tmp = (Math.cosh(x) / x) * (y_m / z);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 1e+85:
		tmp = ((y_m * math.cosh(x)) / x) / z
	else:
		tmp = (math.cosh(x) / x) * (y_m / z)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 1e+85)
		tmp = Float64(Float64(Float64(y_m * cosh(x)) / x) / z);
	else
		tmp = Float64(Float64(cosh(x) / x) * Float64(y_m / z));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 1e+85)
		tmp = ((y_m * cosh(x)) / x) / z;
	else
		tmp = (cosh(x) / x) * (y_m / z);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1e+85], N[(N[(N[(y$95$m * N[Cosh[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1e85

   1. Initial program 86.7%

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

      1. associate-*r/ 97.7%

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

   4. Applied egg-rr 97.7%

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

   if 1e85 < y

   1. Initial program 87.5%

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

      1. associate-*r/ 87.5%

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

   4. Applied egg-rr 87.5%

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

      1. associate-/r* 86.8%

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

      2. times-frac 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+85}:\\ \;\;\;\;\frac{\frac{y \cdot \cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y\_m}{x}}{z} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{y\_m}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y\_m}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (* (cosh x) (/ y_m x)) z) 5e-16)
    (/ y_m (* x z))
    (* (/ (cosh x) x) (/ y_m z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((cosh(x) * (y_m / x)) / z) <= 5e-16) {
		tmp = y_m / (x * z);
	} else {
		tmp = (cosh(x) / x) * (y_m / z);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((cosh(x) * (y_m / x)) / z) <= 5d-16) then
        tmp = y_m / (x * z)
    else
        tmp = (cosh(x) / x) * (y_m / z)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((Math.cosh(x) * (y_m / x)) / z) <= 5e-16) {
		tmp = y_m / (x * z);
	} else {
		tmp = (Math.cosh(x) / x) * (y_m / z);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if ((math.cosh(x) * (y_m / x)) / z) <= 5e-16:
		tmp = y_m / (x * z)
	else:
		tmp = (math.cosh(x) / x) * (y_m / z)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x) * Float64(y_m / x)) / z) <= 5e-16)
		tmp = Float64(y_m / Float64(x * z));
	else
		tmp = Float64(Float64(cosh(x) / x) * Float64(y_m / z));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (((cosh(x) * (y_m / x)) / z) <= 5e-16)
		tmp = y_m / (x * z);
	else
		tmp = (cosh(x) / x) * (y_m / z);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 5e-16], N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 5.0000000000000004e-16

   1. Initial program 95.3%

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

      1. associate-/l* 87.1%

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

   3. Simplified 87.1%

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

   5. Taylor expanded in x around 0 57.6%

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

   if 5.0000000000000004e-16 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 78.9%

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

      1. associate-*r/ 97.0%

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

   4. Applied egg-rr 97.0%

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

      1. associate-/r* 81.5%

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

      2. times-frac 92.3%

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

   6. Applied egg-rr 92.3%

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

Alternative 3: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 3.1 \cdot 10^{-68}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y\_m}{z}}{x}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+82}:\\ \;\;\;\;y\_m \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x} \cdot \frac{\cosh x}{z}\\ \end{array} \end{array} \]

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 (<= z 3.1e-68)
    (* (cosh x) (/ (/ y_m z) x))
    (if (<= z 5.8e+82)
      (* y_m (/ (cosh x) (* x z)))
      (* (/ y_m x) (/ (cosh 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 (z <= 3.1e-68) {
		tmp = cosh(x) * ((y_m / z) / x);
	} else if (z <= 5.8e+82) {
		tmp = y_m * (cosh(x) / (x * z));
	} else {
		tmp = (y_m / x) * (cosh(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 (z <= 3.1d-68) then
        tmp = cosh(x) * ((y_m / z) / x)
    else if (z <= 5.8d+82) then
        tmp = y_m * (cosh(x) / (x * z))
    else
        tmp = (y_m / x) * (cosh(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 (z <= 3.1e-68) {
		tmp = Math.cosh(x) * ((y_m / z) / x);
	} else if (z <= 5.8e+82) {
		tmp = y_m * (Math.cosh(x) / (x * z));
	} else {
		tmp = (y_m / x) * (Math.cosh(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 z <= 3.1e-68:
		tmp = math.cosh(x) * ((y_m / z) / x)
	elif z <= 5.8e+82:
		tmp = y_m * (math.cosh(x) / (x * z))
	else:
		tmp = (y_m / x) * (math.cosh(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 (z <= 3.1e-68)
		tmp = Float64(cosh(x) * Float64(Float64(y_m / z) / x));
	elseif (z <= 5.8e+82)
		tmp = Float64(y_m * Float64(cosh(x) / Float64(x * z)));
	else
		tmp = Float64(Float64(y_m / x) * Float64(cosh(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 (z <= 3.1e-68)
		tmp = cosh(x) * ((y_m / z) / x);
	elseif (z <= 5.8e+82)
		tmp = y_m * (cosh(x) / (x * z));
	else
		tmp = (y_m / x) * (cosh(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[z, 3.1e-68], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.8e+82], N[(y$95$m * N[(N[Cosh[x], $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / x), $MachinePrecision] * N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 3.0999999999999999e-68

   1. Initial program 90.9%

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

      1. associate-/l* 87.6%

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

   3. Simplified 87.6%

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

      1. associate-/l/ 81.9%

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

      2. associate-/r* 90.4%

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

   6. Applied egg-rr 90.4%

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

   if 3.0999999999999999e-68 < z < 5.8000000000000003e82

   1. Initial program 79.8%

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

      1. associate-/l* 76.5%

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

   3. Simplified 76.5%

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

      1. add-log-exp 41.7%

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

      2. *-un-lft-identity 41.7%

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

      3. log-prod 41.7%

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

      4. metadata-eval 41.7%

         \[\leadsto \color{blue}{0} + \log \left(e^{\cosh x \cdot \frac{\frac{y}{x}}{z}}\right) \]

      5. add-log-exp 76.5%

         \[\leadsto 0 + \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      6. div-inv 76.5%

         \[\leadsto 0 + \cosh x \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{1}{z}\right)} \]

      7. frac-times 76.5%

         \[\leadsto 0 + \cosh x \cdot \color{blue}{\frac{y \cdot 1}{x \cdot z}} \]

      8. *-commutative 76.5%

         \[\leadsto 0 + \cosh x \cdot \frac{\color{blue}{1 \cdot y}}{x \cdot z} \]

      9. *-un-lft-identity 76.5%

         \[\leadsto 0 + \cosh x \cdot \frac{\color{blue}{y}}{x \cdot z} \]

   6. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{0 + \cosh x \cdot \frac{y}{x \cdot z}} \]
   7. Step-by-step derivation

      1. +-lft-identity 76.5%

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

      2. *-commutative 76.5%

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

      3. associate-*l/ 96.6%

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

      4. associate-/l* 96.5%

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

   8. Simplified 96.5%

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

   if 5.8000000000000003e82 < z

   1. Initial program 75.9%

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

      1. associate-/l* 54.9%

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

   3. Simplified 54.9%

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

      1. associate-*r/ 75.9%

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

      2. *-commutative 75.9%

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

      3. associate-/l* 77.8%

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

   6. Applied egg-rr 77.8%

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

Alternative 4: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.6 \cdot 10^{-215}:\\ \;\;\;\;y\_m \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y\_m}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 2.6e-215)
    (* y_m (/ (cosh x) (* x z)))
    (* (cosh x) (/ (/ y_m z) x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2.6e-215) {
		tmp = y_m * (cosh(x) / (x * z));
	} else {
		tmp = cosh(x) * ((y_m / z) / 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 <= 2.6d-215) then
        tmp = y_m * (cosh(x) / (x * z))
    else
        tmp = cosh(x) * ((y_m / z) / 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 <= 2.6e-215) {
		tmp = y_m * (Math.cosh(x) / (x * z));
	} else {
		tmp = Math.cosh(x) * ((y_m / z) / 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 <= 2.6e-215:
		tmp = y_m * (math.cosh(x) / (x * z))
	else:
		tmp = math.cosh(x) * ((y_m / z) / x)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 2.6e-215)
		tmp = Float64(y_m * Float64(cosh(x) / Float64(x * z)));
	else
		tmp = Float64(cosh(x) * Float64(Float64(y_m / z) / 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 <= 2.6e-215)
		tmp = y_m * (cosh(x) / (x * z));
	else
		tmp = cosh(x) * ((y_m / z) / 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, 2.6e-215], N[(y$95$m * N[(N[Cosh[x], $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.6e-215

   1. Initial program 83.5%

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

      1. associate-/l* 75.5%

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

   3. Simplified 75.5%

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

      1. add-log-exp 51.2%

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

      2. *-un-lft-identity 51.2%

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

      3. log-prod 51.2%

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

      4. metadata-eval 51.2%

         \[\leadsto \color{blue}{0} + \log \left(e^{\cosh x \cdot \frac{\frac{y}{x}}{z}}\right) \]

      5. add-log-exp 75.5%

         \[\leadsto 0 + \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      6. div-inv 75.5%

         \[\leadsto 0 + \cosh x \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{1}{z}\right)} \]

      7. frac-times 73.2%

         \[\leadsto 0 + \cosh x \cdot \color{blue}{\frac{y \cdot 1}{x \cdot z}} \]

      8. *-commutative 73.2%

         \[\leadsto 0 + \cosh x \cdot \frac{\color{blue}{1 \cdot y}}{x \cdot z} \]

      9. *-un-lft-identity 73.2%

         \[\leadsto 0 + \cosh x \cdot \frac{\color{blue}{y}}{x \cdot z} \]

   6. Applied egg-rr 73.2%

      \[\leadsto \color{blue}{0 + \cosh x \cdot \frac{y}{x \cdot z}} \]
   7. Step-by-step derivation

      1. +-lft-identity 73.2%

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

      2. *-commutative 73.2%

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

      3. associate-*l/ 80.5%

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

      4. associate-/l* 80.6%

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

   8. Simplified 80.6%

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

   if 2.6e-215 < y

   1. Initial program 90.7%

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

      1. associate-/l* 85.6%

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

   3. Simplified 85.6%

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

      1. associate-/l/ 81.9%

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

      2. associate-/r* 88.9%

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

   6. Applied egg-rr 88.9%

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

Alternative 5: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 5.6 \cdot 10^{-188}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\cosh x}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= z 5.6e-188)
    (* (cosh x) (/ (/ y_m x) z))
    (* y_m (/ (cosh 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 (z <= 5.6e-188) {
		tmp = cosh(x) * ((y_m / x) / z);
	} else {
		tmp = y_m * (cosh(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 (z <= 5.6d-188) then
        tmp = cosh(x) * ((y_m / x) / z)
    else
        tmp = y_m * (cosh(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 (z <= 5.6e-188) {
		tmp = Math.cosh(x) * ((y_m / x) / z);
	} else {
		tmp = y_m * (Math.cosh(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 z <= 5.6e-188:
		tmp = math.cosh(x) * ((y_m / x) / z)
	else:
		tmp = y_m * (math.cosh(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 (z <= 5.6e-188)
		tmp = Float64(cosh(x) * Float64(Float64(y_m / x) / z));
	else
		tmp = Float64(y_m * Float64(cosh(x) / Float64(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 (z <= 5.6e-188)
		tmp = cosh(x) * ((y_m / x) / z);
	else
		tmp = y_m * (cosh(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[z, 5.6e-188], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[Cosh[x], $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.6000000000000002e-188

   1. Initial program 90.8%

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

      1. associate-/l* 86.9%

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

   3. Simplified 86.9%

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

   if 5.6000000000000002e-188 < z

   1. Initial program 80.7%

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

      1. associate-/l* 69.8%

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

   3. Simplified 69.8%

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

      1. add-log-exp 42.2%

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

      2. *-un-lft-identity 42.2%

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

      3. log-prod 42.2%

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

      4. metadata-eval 42.2%

         \[\leadsto \color{blue}{0} + \log \left(e^{\cosh x \cdot \frac{\frac{y}{x}}{z}}\right) \]

      5. add-log-exp 69.8%

         \[\leadsto 0 + \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      6. div-inv 69.8%

         \[\leadsto 0 + \cosh x \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{1}{z}\right)} \]

      7. frac-times 74.4%

         \[\leadsto 0 + \cosh x \cdot \color{blue}{\frac{y \cdot 1}{x \cdot z}} \]

      8. *-commutative 74.4%

         \[\leadsto 0 + \cosh x \cdot \frac{\color{blue}{1 \cdot y}}{x \cdot z} \]

      9. *-un-lft-identity 74.4%

         \[\leadsto 0 + \cosh x \cdot \frac{\color{blue}{y}}{x \cdot z} \]

   6. Applied egg-rr 74.4%

      \[\leadsto \color{blue}{0 + \cosh x \cdot \frac{y}{x \cdot z}} \]
   7. Step-by-step derivation

      1. +-lft-identity 74.4%

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

      2. *-commutative 74.4%

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

      3. associate-*l/ 83.3%

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

      4. associate-/l* 84.0%

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

   8. Simplified 84.0%

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

Alternative 6: 68.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\cosh x}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 2.2e-37) (/ (/ y_m z) x) (* y_m (/ (cosh 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 <= 2.2e-37) {
		tmp = (y_m / z) / x;
	} else {
		tmp = y_m * (cosh(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 <= 2.2d-37) then
        tmp = (y_m / z) / x
    else
        tmp = y_m * (cosh(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 <= 2.2e-37) {
		tmp = (y_m / z) / x;
	} else {
		tmp = y_m * (Math.cosh(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 <= 2.2e-37:
		tmp = (y_m / z) / x
	else:
		tmp = y_m * (math.cosh(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 <= 2.2e-37)
		tmp = Float64(Float64(y_m / z) / x);
	else
		tmp = Float64(y_m * Float64(cosh(x) / Float64(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 <= 2.2e-37)
		tmp = (y_m / z) / x;
	else
		tmp = y_m * (cosh(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, 2.2e-37], N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision], N[(y$95$m * N[(N[Cosh[x], $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.20000000000000002e-37

   1. Initial program 88.1%

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

      1. associate-/l* 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in x around 0 59.6%

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

      1. *-commutative 59.6%

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

      2. associate-/r* 65.0%

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

   7. Applied egg-rr 65.0%

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

   if 2.20000000000000002e-37 < x

   1. Initial program 83.4%

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

      1. associate-/l* 73.5%

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

   3. Simplified 73.5%

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

      1. add-log-exp 64.4%

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

      2. *-un-lft-identity 64.4%

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

      3. log-prod 64.4%

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

      4. metadata-eval 64.4%

         \[\leadsto \color{blue}{0} + \log \left(e^{\cosh x \cdot \frac{\frac{y}{x}}{z}}\right) \]

      5. add-log-exp 73.5%

         \[\leadsto 0 + \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      6. div-inv 73.5%

         \[\leadsto 0 + \cosh x \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{1}{z}\right)} \]

      7. frac-times 72.2%

         \[\leadsto 0 + \cosh x \cdot \color{blue}{\frac{y \cdot 1}{x \cdot z}} \]

      8. *-commutative 72.2%

         \[\leadsto 0 + \cosh x \cdot \frac{\color{blue}{1 \cdot y}}{x \cdot z} \]

      9. *-un-lft-identity 72.2%

         \[\leadsto 0 + \cosh x \cdot \frac{\color{blue}{y}}{x \cdot z} \]

   6. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{0 + \cosh x \cdot \frac{y}{x \cdot z}} \]
   7. Step-by-step derivation

      1. +-lft-identity 72.2%

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

      2. *-commutative 72.2%

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

      3. associate-*l/ 79.2%

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

      4. associate-/l* 79.2%

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

   8. Simplified 79.2%

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

Alternative 7: 54.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 10^{-29}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= z 1e-29) (/ (/ y_m z) x) (/ y_m (* x z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 1e-29) {
		tmp = (y_m / z) / x;
	} else {
		tmp = y_m / (x * z);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1d-29) then
        tmp = (y_m / z) / x
    else
        tmp = y_m / (x * z)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 1e-29) {
		tmp = (y_m / z) / x;
	} else {
		tmp = y_m / (x * z);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= 1e-29:
		tmp = (y_m / z) / x
	else:
		tmp = y_m / (x * z)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= 1e-29)
		tmp = Float64(Float64(y_m / z) / x);
	else
		tmp = Float64(y_m / Float64(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 (z <= 1e-29)
		tmp = (y_m / z) / x;
	else
		tmp = y_m / (x * z);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 1e-29], N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision], N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 9.99999999999999943e-30

   1. Initial program 90.1%

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

      1. associate-/l* 86.9%

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

   3. Simplified 86.9%

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

   5. Taylor expanded in x around 0 49.8%

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

      1. *-commutative 49.8%

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

      2. associate-/r* 59.9%

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

   7. Applied egg-rr 59.9%

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

   if 9.99999999999999943e-30 < z

   1. Initial program 78.3%

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

      1. associate-/l* 62.9%

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

   3. Simplified 62.9%

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

   5. Taylor expanded in x around 0 42.7%

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

Alternative 8: 50.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 10^{+120}:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= z 1e+120) (/ (/ y_m x) z) (/ y_m (* x z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 1e+120) {
		tmp = (y_m / x) / z;
	} else {
		tmp = y_m / (x * z);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1d+120) then
        tmp = (y_m / x) / z
    else
        tmp = y_m / (x * z)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 1e+120) {
		tmp = (y_m / x) / z;
	} else {
		tmp = y_m / (x * z);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= 1e+120:
		tmp = (y_m / x) / z
	else:
		tmp = y_m / (x * z)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= 1e+120)
		tmp = Float64(Float64(y_m / x) / z);
	else
		tmp = Float64(y_m / Float64(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 (z <= 1e+120)
		tmp = (y_m / x) / z;
	else
		tmp = y_m / (x * z);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 1e+120], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 9.9999999999999998e119

   1. Initial program 89.6%

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

   3. Taylor expanded in x around 0 53.1%

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

   if 9.9999999999999998e119 < z

   1. Initial program 73.1%

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

      1. associate-/l* 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in x around 0 45.2%

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

Alternative 9: 49.3% accurate, 21.4× 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 86.8%

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

   1. associate-/l* 80.1%

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

3. Simplified 80.1%

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

5. Taylor expanded in x around 0 47.8%

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

Developer target: 97.3% 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 2024096 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :alt
  (if (< y -4.618902267687042e-52) (* (/ (/ y z) x) (cosh x)) (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

  (/ (* (cosh x) (/ y x)) z))