Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.8% → 99.5%

Time: 9.1s

Alternatives: 12

Speedup: 1.0×

• 64.6% 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: 84.8% 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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 4.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{y_m \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x}{\frac{z}{y_m}}}{x}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 4.4e-45)
    (/ (* y_m (/ (cosh x) x)) z)
    (/ (/ (cosh x) (/ z y_m)) 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 <= 4.4e-45) {
		tmp = (y_m * (cosh(x) / x)) / z;
	} else {
		tmp = (cosh(x) / (z / y_m)) / 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 <= 4.4d-45) then
        tmp = (y_m * (cosh(x) / x)) / z
    else
        tmp = (cosh(x) / (z / y_m)) / 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 <= 4.4e-45) {
		tmp = (y_m * (Math.cosh(x) / x)) / z;
	} else {
		tmp = (Math.cosh(x) / (z / y_m)) / 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 <= 4.4e-45:
		tmp = (y_m * (math.cosh(x) / x)) / z
	else:
		tmp = (math.cosh(x) / (z / y_m)) / 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 <= 4.4e-45)
		tmp = Float64(Float64(y_m * Float64(cosh(x) / x)) / z);
	else
		tmp = Float64(Float64(cosh(x) / Float64(z / y_m)) / 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 <= 4.4e-45)
		tmp = (y_m * (cosh(x) / x)) / z;
	else
		tmp = (cosh(x) / (z / y_m)) / 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, 4.4e-45], N[(N[(y$95$m * N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[Cosh[x], $MachinePrecision] / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.39999999999999987e-45

   1. Initial program 78.8%

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

      1. expm1-log1p-u 41.1%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x \cdot \frac{y}{x}\right)\right)}}{z} \]

      2. expm1-udef 32.3%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\cosh x \cdot \frac{y}{x}\right)} - 1}}{z} \]

   4. Applied egg-rr 32.3%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\cosh x \cdot \frac{y}{x}\right)} - 1}}{z} \]
   5. Step-by-step derivation

      1. expm1-def 41.1%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x \cdot \frac{y}{x}\right)\right)}}{z} \]

      2. expm1-log1p 78.8%

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

      3. associate-*r/ 96.3%

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

      4. associate-*l/ 96.3%

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

      5. *-commutative 96.3%

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

   6. Simplified 96.3%

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

   if 4.39999999999999987e-45 < y

   1. Initial program 94.4%

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

      1. associate-*l/ 94.4%

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

   3. Simplified 94.4%

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

      1. associate-/r/ 89.9%

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

      2. associate-/r/ 95.5%

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

      3. associate-/r* 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x}{\frac{z}{y}}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 69.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.45 \cdot 10^{-213}:\\ \;\;\;\;\frac{\frac{y_m}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y_m}{x}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 2.45e-213) (/ (/ y_m z) x) (* (/ (cosh x) z) (/ y_m 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 (x <= 2.45e-213) {
		tmp = (y_m / z) / x;
	} else {
		tmp = (cosh(x) / z) * (y_m / 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 (x <= 2.45d-213) then
        tmp = (y_m / z) / x
    else
        tmp = (cosh(x) / z) * (y_m / 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 (x <= 2.45e-213) {
		tmp = (y_m / z) / x;
	} else {
		tmp = (Math.cosh(x) / z) * (y_m / 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 x <= 2.45e-213:
		tmp = (y_m / z) / x
	else:
		tmp = (math.cosh(x) / z) * (y_m / 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 (x <= 2.45e-213)
		tmp = Float64(Float64(y_m / z) / x);
	else
		tmp = Float64(Float64(cosh(x) / z) * Float64(y_m / 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 (x <= 2.45e-213)
		tmp = (y_m / z) / x;
	else
		tmp = (cosh(x) / z) * (y_m / 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[x, 2.45e-213], N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.4499999999999999e-213

   1. Initial program 79.5%

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

      1. associate-*l/ 79.5%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in x around 0 49.0%

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

      1. associate-*r/ 58.7%

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

      2. associate-*l/ 58.7%

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

      3. *-un-lft-identity 58.7%

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

   7. Applied egg-rr 58.7%

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

   if 2.4499999999999999e-213 < x

   1. Initial program 88.2%

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

      1. associate-*l/ 88.1%

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

   3. Simplified 88.1%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.45 \cdot 10^{-213}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 8.2 \cdot 10^{+34}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y_m}{x}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= z 8.2e+34)
    (/ (cosh x) (* x (/ z y_m)))
    (* (/ (cosh x) z) (/ y_m 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 (z <= 8.2e+34) {
		tmp = cosh(x) / (x * (z / y_m));
	} else {
		tmp = (cosh(x) / z) * (y_m / 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 (z <= 8.2d+34) then
        tmp = cosh(x) / (x * (z / y_m))
    else
        tmp = (cosh(x) / z) * (y_m / 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 (z <= 8.2e+34) {
		tmp = Math.cosh(x) / (x * (z / y_m));
	} else {
		tmp = (Math.cosh(x) / z) * (y_m / 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 z <= 8.2e+34:
		tmp = math.cosh(x) / (x * (z / y_m))
	else:
		tmp = (math.cosh(x) / z) * (y_m / 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 (z <= 8.2e+34)
		tmp = Float64(cosh(x) / Float64(x * Float64(z / y_m)));
	else
		tmp = Float64(Float64(cosh(x) / z) * Float64(y_m / 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 (z <= 8.2e+34)
		tmp = cosh(x) / (x * (z / y_m));
	else
		tmp = (cosh(x) / z) * (y_m / 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[z, 8.2e+34], N[(N[Cosh[x], $MachinePrecision] / N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 8.1999999999999997e34

   1. Initial program 83.4%

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

      1. associate-/l* 79.9%

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

   3. Simplified 79.9%

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

      1. associate-/r/ 84.9%

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

   6. Applied egg-rr 84.9%

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

   if 8.1999999999999997e34 < z

   1. Initial program 81.2%

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

      1. associate-*l/ 81.2%

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

   3. Simplified 81.2%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8.2 \cdot 10^{+34}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 9.5 \cdot 10^{+91}:\\ \;\;\;\;\frac{y_m \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y_m}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 9.5e+91)
    (/ (* y_m (/ (cosh x) x)) z)
    (/ (cosh x) (* x (/ z y_m))))))

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 <= 9.5e+91) {
		tmp = (y_m * (cosh(x) / x)) / z;
	} else {
		tmp = cosh(x) / (x * (z / y_m));
	}
	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 <= 9.5d+91) then
        tmp = (y_m * (cosh(x) / x)) / z
    else
        tmp = cosh(x) / (x * (z / y_m))
    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 <= 9.5e+91) {
		tmp = (y_m * (Math.cosh(x) / x)) / z;
	} else {
		tmp = Math.cosh(x) / (x * (z / y_m));
	}
	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 <= 9.5e+91:
		tmp = (y_m * (math.cosh(x) / x)) / z
	else:
		tmp = math.cosh(x) / (x * (z / y_m))
	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 <= 9.5e+91)
		tmp = Float64(Float64(y_m * Float64(cosh(x) / x)) / z);
	else
		tmp = Float64(cosh(x) / Float64(x * Float64(z / y_m)));
	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 <= 9.5e+91)
		tmp = (y_m * (cosh(x) / x)) / z;
	else
		tmp = cosh(x) / (x * (z / y_m));
	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, 9.5e+91], N[(N[(y$95$m * N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] / N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 9.5000000000000001e91

   1. Initial program 81.4%

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

      1. expm1-log1p-u 42.4%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x \cdot \frac{y}{x}\right)\right)}}{z} \]

      2. expm1-udef 34.5%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\cosh x \cdot \frac{y}{x}\right)} - 1}}{z} \]

   4. Applied egg-rr 34.5%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\cosh x \cdot \frac{y}{x}\right)} - 1}}{z} \]
   5. Step-by-step derivation

      1. expm1-def 42.4%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x \cdot \frac{y}{x}\right)\right)}}{z} \]

      2. expm1-log1p 81.4%

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

      3. associate-*r/ 96.8%

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

      4. associate-*l/ 96.8%

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

      5. *-commutative 96.8%

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

   6. Simplified 96.8%

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

   if 9.5000000000000001e91 < y

   1. Initial program 91.1%

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

      1. associate-/l* 91.0%

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

   3. Simplified 91.0%

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

      1. associate-/r/ 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+91}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\frac{y_m}{z}}{x}\\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-213}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{y_m}{x}}{z}\\ \mathbf{elif}\;x \leq 0.0005:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+147}:\\ \;\;\;\;y_m \cdot \left(0.5 \cdot \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y_m \cdot x\right) \cdot \frac{0.5}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (/ y_m z) x)))
   (*
    y_s
    (if (<= x 7e-213)
      t_0
      (if (<= x 1.02e-126)
        (/ (/ y_m x) z)
        (if (<= x 0.0005)
          t_0
          (if (<= x 1.45e+147)
            (* y_m (* 0.5 (/ 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 t_0 = (y_m / z) / x;
	double tmp;
	if (x <= 7e-213) {
		tmp = t_0;
	} else if (x <= 1.02e-126) {
		tmp = (y_m / x) / z;
	} else if (x <= 0.0005) {
		tmp = t_0;
	} else if (x <= 1.45e+147) {
		tmp = y_m * (0.5 * (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) :: t_0
    real(8) :: tmp
    t_0 = (y_m / z) / x
    if (x <= 7d-213) then
        tmp = t_0
    else if (x <= 1.02d-126) then
        tmp = (y_m / x) / z
    else if (x <= 0.0005d0) then
        tmp = t_0
    else if (x <= 1.45d+147) then
        tmp = y_m * (0.5d0 * (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 t_0 = (y_m / z) / x;
	double tmp;
	if (x <= 7e-213) {
		tmp = t_0;
	} else if (x <= 1.02e-126) {
		tmp = (y_m / x) / z;
	} else if (x <= 0.0005) {
		tmp = t_0;
	} else if (x <= 1.45e+147) {
		tmp = y_m * (0.5 * (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):
	t_0 = (y_m / z) / x
	tmp = 0
	if x <= 7e-213:
		tmp = t_0
	elif x <= 1.02e-126:
		tmp = (y_m / x) / z
	elif x <= 0.0005:
		tmp = t_0
	elif x <= 1.45e+147:
		tmp = y_m * (0.5 * (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)
	t_0 = Float64(Float64(y_m / z) / x)
	tmp = 0.0
	if (x <= 7e-213)
		tmp = t_0;
	elseif (x <= 1.02e-126)
		tmp = Float64(Float64(y_m / x) / z);
	elseif (x <= 0.0005)
		tmp = t_0;
	elseif (x <= 1.45e+147)
		tmp = Float64(y_m * Float64(0.5 * Float64(x / z)));
	else
		tmp = Float64(Float64(y_m * x) * Float64(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)
	t_0 = (y_m / z) / x;
	tmp = 0.0;
	if (x <= 7e-213)
		tmp = t_0;
	elseif (x <= 1.02e-126)
		tmp = (y_m / x) / z;
	elseif (x <= 0.0005)
		tmp = t_0;
	elseif (x <= 1.45e+147)
		tmp = y_m * (0.5 * (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_] := Block[{t$95$0 = N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[x, 7e-213], t$95$0, If[LessEqual[x, 1.02e-126], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[x, 0.0005], t$95$0, If[LessEqual[x, 1.45e+147], N[(y$95$m * N[(0.5 * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * x), $MachinePrecision] * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if x < 7.00000000000000034e-213 or 1.02000000000000004e-126 < x < 5.0000000000000001e-4

   1. Initial program 80.9%

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

      1. associate-*l/ 80.9%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in x around 0 54.2%

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

      1. associate-*r/ 63.8%

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

      2. associate-*l/ 63.8%

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

      3. *-un-lft-identity 63.8%

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

   7. Applied egg-rr 63.8%

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

   if 7.00000000000000034e-213 < x < 1.02000000000000004e-126

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 99.8%

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

   if 5.0000000000000001e-4 < x < 1.4499999999999999e147

   1. Initial program 94.1%

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

   3. Taylor expanded in x around 0 34.7%

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

   4. Taylor expanded in x around inf 34.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z}} \]
   5. Step-by-step derivation

      1. associate-/l* 29.0%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{z}{y}}} \]

      2. associate-*r/ 29.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot x}{\frac{z}{y}}} \]

   6. Simplified 29.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot x}{\frac{z}{y}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 37.4%

         \[\leadsto \color{blue}{\frac{0.5 \cdot x}{z} \cdot y} \]

      2. *-un-lft-identity 37.4%

         \[\leadsto \frac{0.5 \cdot x}{\color{blue}{1 \cdot z}} \cdot y \]

      3. times-frac 37.4%

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

      4. metadata-eval 37.4%

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

   8. Applied egg-rr 37.4%

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

   if 1.4499999999999999e147 < x

   1. Initial program 70.4%

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

   3. Taylor expanded in x around 0 60.7%

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

   4. Taylor expanded in x around inf 60.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z}} \]
   5. Step-by-step derivation

      1. associate-/l* 39.7%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{z}{y}}} \]

      2. associate-*r/ 39.7%

         \[\leadsto \color{blue}{\frac{0.5 \cdot x}{\frac{z}{y}}} \]

   6. Simplified 39.7%

      \[\leadsto \color{blue}{\frac{0.5 \cdot x}{\frac{z}{y}}} \]
   7. Step-by-step derivation

      1. associate-/l* 39.7%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{\frac{z}{y}}{x}}} \]

      2. associate-/r* 60.7%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{z}{y \cdot x}}} \]

      3. associate-/r/ 60.7%

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

      4. *-commutative 60.7%

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

   8. Applied egg-rr 60.7%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-213}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{elif}\;x \leq 0.0005:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+147}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{0.5}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\frac{y_m}{z}}{x}\\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-213}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{y_m}{x}}{z}\\ \mathbf{elif}\;x \leq 0.0005:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{+146}:\\ \;\;\;\;y_m \cdot \frac{0.5}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(y_m \cdot x\right)}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (/ y_m z) x)))
   (*
    y_s
    (if (<= x 7e-213)
      t_0
      (if (<= x 1.05e-125)
        (/ (/ y_m x) z)
        (if (<= x 0.0005)
          t_0
          (if (<= x 6.3e+146)
            (* y_m (/ 0.5 (/ z x)))
            (/ (* 0.5 (* y_m x)) z))))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (y_m / z) / x;
	double tmp;
	if (x <= 7e-213) {
		tmp = t_0;
	} else if (x <= 1.05e-125) {
		tmp = (y_m / x) / z;
	} else if (x <= 0.0005) {
		tmp = t_0;
	} else if (x <= 6.3e+146) {
		tmp = y_m * (0.5 / (z / x));
	} else {
		tmp = (0.5 * (y_m * x)) / z;
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y_m / z) / x
    if (x <= 7d-213) then
        tmp = t_0
    else if (x <= 1.05d-125) then
        tmp = (y_m / x) / z
    else if (x <= 0.0005d0) then
        tmp = t_0
    else if (x <= 6.3d+146) then
        tmp = y_m * (0.5d0 / (z / x))
    else
        tmp = (0.5d0 * (y_m * x)) / z
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = (y_m / z) / x;
	double tmp;
	if (x <= 7e-213) {
		tmp = t_0;
	} else if (x <= 1.05e-125) {
		tmp = (y_m / x) / z;
	} else if (x <= 0.0005) {
		tmp = t_0;
	} else if (x <= 6.3e+146) {
		tmp = y_m * (0.5 / (z / x));
	} else {
		tmp = (0.5 * (y_m * x)) / z;
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = (y_m / z) / x
	tmp = 0
	if x <= 7e-213:
		tmp = t_0
	elif x <= 1.05e-125:
		tmp = (y_m / x) / z
	elif x <= 0.0005:
		tmp = t_0
	elif x <= 6.3e+146:
		tmp = y_m * (0.5 / (z / x))
	else:
		tmp = (0.5 * (y_m * x)) / z
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(y_m / z) / x)
	tmp = 0.0
	if (x <= 7e-213)
		tmp = t_0;
	elseif (x <= 1.05e-125)
		tmp = Float64(Float64(y_m / x) / z);
	elseif (x <= 0.0005)
		tmp = t_0;
	elseif (x <= 6.3e+146)
		tmp = Float64(y_m * Float64(0.5 / Float64(z / x)));
	else
		tmp = Float64(Float64(0.5 * Float64(y_m * x)) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = (y_m / z) / x;
	tmp = 0.0;
	if (x <= 7e-213)
		tmp = t_0;
	elseif (x <= 1.05e-125)
		tmp = (y_m / x) / z;
	elseif (x <= 0.0005)
		tmp = t_0;
	elseif (x <= 6.3e+146)
		tmp = y_m * (0.5 / (z / x));
	else
		tmp = (0.5 * (y_m * x)) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[x, 7e-213], t$95$0, If[LessEqual[x, 1.05e-125], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[x, 0.0005], t$95$0, If[LessEqual[x, 6.3e+146], N[(y$95$m * N[(0.5 / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if x < 7.00000000000000034e-213 or 1.05e-125 < x < 5.0000000000000001e-4

   1. Initial program 80.9%

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

      1. associate-*l/ 80.9%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in x around 0 54.2%

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

      1. associate-*r/ 63.8%

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

      2. associate-*l/ 63.8%

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

      3. *-un-lft-identity 63.8%

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

   7. Applied egg-rr 63.8%

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

   if 7.00000000000000034e-213 < x < 1.05e-125

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 99.8%

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

   if 5.0000000000000001e-4 < x < 6.3000000000000002e146

   1. Initial program 94.1%

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

   3. Taylor expanded in x around 0 34.7%

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

   4. Taylor expanded in x around inf 34.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z}} \]
   5. Step-by-step derivation

      1. associate-/l* 29.0%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{z}{y}}} \]

      2. associate-*r/ 29.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot x}{\frac{z}{y}}} \]

   6. Simplified 29.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot x}{\frac{z}{y}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 37.4%

         \[\leadsto \color{blue}{\frac{0.5 \cdot x}{z} \cdot y} \]

      2. *-un-lft-identity 37.4%

         \[\leadsto \frac{0.5 \cdot x}{\color{blue}{1 \cdot z}} \cdot y \]

      3. times-frac 37.4%

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

      4. metadata-eval 37.4%

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

   8. Applied egg-rr 37.4%

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

      1. clear-num 37.4%

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

      2. un-div-inv 37.4%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{z}{x}}} \cdot y \]

   10. Applied egg-rr 37.4%

       \[\leadsto \color{blue}{\frac{0.5}{\frac{z}{x}}} \cdot y \]

   if 6.3000000000000002e146 < x

   1. Initial program 70.4%

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

   3. Taylor expanded in x around 0 60.7%

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

   4. Taylor expanded in x around inf 60.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z}} \]
   5. Step-by-step derivation

      1. associate-*r/ 60.7%

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

      2. *-commutative 60.7%

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

   6. Simplified 60.7%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-213}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{elif}\;x \leq 0.0005:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{+146}:\\ \;\;\;\;y \cdot \frac{0.5}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(y \cdot x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\frac{y_m}{z}}{x}\\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-213}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{y_m}{x}}{z}\\ \mathbf{elif}\;x \leq 0.0005:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot \left(0.5 \cdot \frac{x}{z}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (/ y_m z) x)))
   (*
    y_s
    (if (<= x 6.8e-213)
      t_0
      (if (<= x 1.5e-133)
        (/ (/ y_m x) z)
        (if (<= x 0.0005) t_0 (* y_m (* 0.5 (/ 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 = (y_m / z) / x;
	double tmp;
	if (x <= 6.8e-213) {
		tmp = t_0;
	} else if (x <= 1.5e-133) {
		tmp = (y_m / x) / z;
	} else if (x <= 0.0005) {
		tmp = t_0;
	} else {
		tmp = y_m * (0.5 * (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) :: t_0
    real(8) :: tmp
    t_0 = (y_m / z) / x
    if (x <= 6.8d-213) then
        tmp = t_0
    else if (x <= 1.5d-133) then
        tmp = (y_m / x) / z
    else if (x <= 0.0005d0) then
        tmp = t_0
    else
        tmp = y_m * (0.5d0 * (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 t_0 = (y_m / z) / x;
	double tmp;
	if (x <= 6.8e-213) {
		tmp = t_0;
	} else if (x <= 1.5e-133) {
		tmp = (y_m / x) / z;
	} else if (x <= 0.0005) {
		tmp = t_0;
	} else {
		tmp = y_m * (0.5 * (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):
	t_0 = (y_m / z) / x
	tmp = 0
	if x <= 6.8e-213:
		tmp = t_0
	elif x <= 1.5e-133:
		tmp = (y_m / x) / z
	elif x <= 0.0005:
		tmp = t_0
	else:
		tmp = y_m * (0.5 * (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(Float64(y_m / z) / x)
	tmp = 0.0
	if (x <= 6.8e-213)
		tmp = t_0;
	elseif (x <= 1.5e-133)
		tmp = Float64(Float64(y_m / x) / z);
	elseif (x <= 0.0005)
		tmp = t_0;
	else
		tmp = Float64(y_m * Float64(0.5 * 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)
	t_0 = (y_m / z) / x;
	tmp = 0.0;
	if (x <= 6.8e-213)
		tmp = t_0;
	elseif (x <= 1.5e-133)
		tmp = (y_m / x) / z;
	elseif (x <= 0.0005)
		tmp = t_0;
	else
		tmp = y_m * (0.5 * (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_] := Block[{t$95$0 = N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[x, 6.8e-213], t$95$0, If[LessEqual[x, 1.5e-133], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[x, 0.0005], t$95$0, N[(y$95$m * N[(0.5 * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x < 6.8000000000000005e-213 or 1.5000000000000001e-133 < x < 5.0000000000000001e-4

   1. Initial program 80.9%

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

      1. associate-*l/ 80.9%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in x around 0 54.2%

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

      1. associate-*r/ 63.8%

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

      2. associate-*l/ 63.8%

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

      3. *-un-lft-identity 63.8%

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

   7. Applied egg-rr 63.8%

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

   if 6.8000000000000005e-213 < x < 1.5000000000000001e-133

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 99.8%

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

   if 5.0000000000000001e-4 < x

   1. Initial program 83.6%

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

   3. Taylor expanded in x around 0 46.2%

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

   4. Taylor expanded in x around inf 46.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z}} \]
   5. Step-by-step derivation

      1. associate-/l* 33.7%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{z}{y}}} \]

      2. associate-*r/ 33.7%

         \[\leadsto \color{blue}{\frac{0.5 \cdot x}{\frac{z}{y}}} \]

   6. Simplified 33.7%

      \[\leadsto \color{blue}{\frac{0.5 \cdot x}{\frac{z}{y}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 41.6%

         \[\leadsto \color{blue}{\frac{0.5 \cdot x}{z} \cdot y} \]

      2. *-un-lft-identity 41.6%

         \[\leadsto \frac{0.5 \cdot x}{\color{blue}{1 \cdot z}} \cdot y \]

      3. times-frac 41.6%

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

      4. metadata-eval 41.6%

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

   8. Applied egg-rr 41.6%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-213}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{elif}\;x \leq 0.0005:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 4.8 \cdot 10^{-250}:\\ \;\;\;\;\frac{-y_m}{\frac{z}{x \cdot -0.5 + \frac{-1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y_m}{x} + 0.5 \cdot \left(y_m \cdot x\right)}{z}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 4.8e-250)
    (/ (- y_m) (/ z (+ (* x -0.5) (/ -1.0 x))))
    (/ (+ (/ y_m x) (* 0.5 (* y_m x))) z))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 4.8e-250) {
		tmp = -y_m / (z / ((x * -0.5) + (-1.0 / x)));
	} else {
		tmp = ((y_m / x) + (0.5 * (y_m * x))) / z;
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 4.8d-250) then
        tmp = -y_m / (z / ((x * (-0.5d0)) + ((-1.0d0) / x)))
    else
        tmp = ((y_m / x) + (0.5d0 * (y_m * x))) / z
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 4.8e-250) {
		tmp = -y_m / (z / ((x * -0.5) + (-1.0 / x)));
	} else {
		tmp = ((y_m / x) + (0.5 * (y_m * x))) / z;
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 4.8e-250:
		tmp = -y_m / (z / ((x * -0.5) + (-1.0 / x)))
	else:
		tmp = ((y_m / x) + (0.5 * (y_m * x))) / z
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 4.8e-250)
		tmp = Float64(Float64(-y_m) / Float64(z / Float64(Float64(x * -0.5) + Float64(-1.0 / x))));
	else
		tmp = Float64(Float64(Float64(y_m / x) + Float64(0.5 * Float64(y_m * x))) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 4.8e-250)
		tmp = -y_m / (z / ((x * -0.5) + (-1.0 / x)));
	else
		tmp = ((y_m / x) + (0.5 * (y_m * x))) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 4.8e-250], N[((-y$95$m) / N[(z / N[(N[(x * -0.5), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m / x), $MachinePrecision] + N[(0.5 * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.7999999999999998e-250

   1. Initial program 75.8%

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

   3. Taylor expanded in x around 0 63.5%

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

   4. Taylor expanded in y around -inf 63.5%

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

      1. mul-1-neg 63.5%

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

      2. associate-/l* 63.6%

         \[\leadsto -\color{blue}{\frac{y}{\frac{z}{-0.5 \cdot x - \frac{1}{x}}}} \]

      3. *-commutative 63.6%

         \[\leadsto -\frac{y}{\frac{z}{\color{blue}{x \cdot -0.5} - \frac{1}{x}}} \]

   6. Simplified 63.6%

      \[\leadsto \color{blue}{-\frac{y}{\frac{z}{x \cdot -0.5 - \frac{1}{x}}}} \]

   if 4.7999999999999998e-250 < y

   1. Initial program 92.7%

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

   3. Taylor expanded in x around 0 70.6%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-250}:\\ \;\;\;\;\frac{-y}{\frac{z}{x \cdot -0.5 + \frac{-1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \frac{\frac{y_m}{x} + 0.5 \cdot \left(y_m \cdot x\right)}{z} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (/ (+ (/ y_m x) (* 0.5 (* y_m x))) z)))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (((y_m / x) + (0.5 * (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) + (0.5d0 * (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) + (0.5 * (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) + (0.5 * (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(Float64(Float64(y_m / x) + Float64(0.5 * Float64(y_m * 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) + (0.5 * (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[(N[(N[(y$95$m / x), $MachinePrecision] + N[(0.5 * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $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 66.5%

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

4. Final simplification 66.5%

   \[\leadsto \frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z} \]
5. Add Preprocessing

Alternative 10: 52.9% accurate, 10.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{y_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= y_m 5e-20) (/ (/ 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 (y_m <= 5e-20) {
		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 (y_m <= 5d-20) 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 (y_m <= 5e-20) {
		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 y_m <= 5e-20:
		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 (y_m <= 5e-20)
		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 (y_m <= 5e-20)
		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[y$95$m, 5e-20], 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 y < 4.9999999999999999e-20

   1. Initial program 79.5%

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

   3. Taylor expanded in x around 0 48.8%

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

   if 4.9999999999999999e-20 < y

   1. Initial program 93.9%

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

      1. associate-*l/ 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in x around 0 45.3%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.9% accurate, 10.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 10^{-40}:\\ \;\;\;\;\frac{\frac{y_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y_m}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= y_m 1e-40) (/ (/ y_m x) z) (/ (/ 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 <= 1e-40) {
		tmp = (y_m / x) / z;
	} else {
		tmp = (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 <= 1d-40) then
        tmp = (y_m / x) / z
    else
        tmp = (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 <= 1e-40) {
		tmp = (y_m / x) / z;
	} else {
		tmp = (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 <= 1e-40:
		tmp = (y_m / x) / z
	else:
		tmp = (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 <= 1e-40)
		tmp = Float64(Float64(y_m / x) / z);
	else
		tmp = 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 <= 1e-40)
		tmp = (y_m / x) / z;
	else
		tmp = (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, 1e-40], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 9.9999999999999993e-41

   1. Initial program 78.9%

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

   3. Taylor expanded in x around 0 47.5%

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

   if 9.9999999999999993e-41 < y

   1. Initial program 94.3%

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

      1. associate-*l/ 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in x around 0 43.9%

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

      1. associate-*r/ 60.7%

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

      2. associate-*l/ 60.8%

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

      3. *-un-lft-identity 60.8%

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

   7. Applied egg-rr 60.8%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-40}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.1% 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 1 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. Step-by-step derivation

   1. associate-*l/ 82.9%

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

3. Simplified 82.9%

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

5. Taylor expanded in x around 0 45.2%

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

6. Final simplification 45.2%

   \[\leadsto \frac{y}{x \cdot z} \]
7. Add Preprocessing

Developer target: 97.5% 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 2024014 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (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))