Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.3% → 99.6%

Time: 8.4s

Alternatives: 8

Speedup: 1.0×

• 64.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 4 \cdot 10^{+43}:\\ \;\;\;\;\frac{y_m \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x}{\frac{z}{y_m}}}{x}\\ \end{array} \end{array} \]

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 4e+43)
    (/ (* 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 <= 4e+43) {
		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 <= 4d+43) 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 <= 4e+43) {
		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 <= 4e+43:
		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 <= 4e+43)
		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 <= 4e+43)
		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, 4e+43], 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 < 4.00000000000000006e43

   1. Initial program 84.1%

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

      1. expm1-log1p-u 52.8%

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

      2. expm1-udef 43.4%

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

   4. Applied egg-rr 43.4%

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

      1. expm1-def 52.8%

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

      2. expm1-log1p 84.1%

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

      3. associate-*r/ 97.6%

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

      4. associate-*l/ 97.6%

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

      5. *-commutative 97.6%

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

   6. Simplified 97.6%

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

   if 4.00000000000000006e43 < y

   1. Initial program 90.3%

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

      1. associate-*l/ 90.2%

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

   3. Simplified 90.2%

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

      1. associate-/r/ 88.2%

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

      2. associate-/r/ 97.8%

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

      3. associate-/r* 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+43}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x}{\frac{z}{y}}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{y_m}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y_m}{x}\\ \end{array} \end{array} \]

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 2e-21) (/ y_m (* 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 (x <= 2e-21) {
		tmp = y_m / (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 (x <= 2d-21) then
        tmp = y_m / (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 (x <= 2e-21) {
		tmp = y_m / (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 x <= 2e-21:
		tmp = y_m / (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 (x <= 2e-21)
		tmp = Float64(y_m / Float64(x * z));
	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 <= 2e-21)
		tmp = y_m / (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[x, 2e-21], N[(y$95$m / N[(x * z), $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 x < 1.99999999999999982e-21

   1. Initial program 87.1%

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

      1. associate-*l/ 87.0%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in x around 0 66.4%

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

   if 1.99999999999999982e-21 < x

   1. Initial program 78.9%

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

      1. associate-*l/ 78.9%

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

   3. Simplified 78.9%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 2.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{\cosh x}{\frac{x \cdot z}{y_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y_m}{x}\\ \end{array} \end{array} \]

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 2.2e-88)
    (/ (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 <= 2.2e-88) {
		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 <= 2.2d-88) 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 <= 2.2e-88) {
		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 <= 2.2e-88:
		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 <= 2.2e-88)
		tmp = Float64(cosh(x) / Float64(Float64(x * 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 <= 2.2e-88)
		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, 2.2e-88], N[(N[Cosh[x], $MachinePrecision] / N[(N[(x * z), $MachinePrecision] / y$95$m), $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 < 2.20000000000000005e-88

   1. Initial program 84.3%

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

      1. associate-/l* 78.1%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in z around 0 82.8%

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

   if 2.20000000000000005e-88 < z

   1. Initial program 87.0%

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

      1. associate-*l/ 86.8%

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

   3. Simplified 86.8%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{\cosh x}{\frac{x \cdot z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 10^{-26}:\\ \;\;\;\;\frac{y_m}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m \cdot \frac{\cosh x}{x}}{z}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 1e-26) (/ 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 (x <= 1e-26) {
		tmp = 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 (x <= 1d-26) then
        tmp = 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 (x <= 1e-26) {
		tmp = 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 x <= 1e-26:
		tmp = 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 (x <= 1e-26)
		tmp = Float64(y_m / Float64(x * z));
	else
		tmp = Float64(Float64(y_m * Float64(cosh(x) / x)) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 1e-26)
		tmp = 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[x, 1e-26], N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1e-26

   1. Initial program 87.0%

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

      1. associate-*l/ 86.9%

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

   3. Simplified 86.9%

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

   5. Taylor expanded in x around 0 66.0%

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

   if 1e-26 < x

   1. Initial program 79.6%

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

      1. expm1-log1p-u 45.8%

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

      2. expm1-udef 41.2%

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

   4. Applied egg-rr 41.2%

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

      1. expm1-def 45.8%

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

      2. expm1-log1p 79.6%

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

      3. associate-*r/ 99.9%

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

      4. associate-*l/ 100.0%

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

      5. *-commutative 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-26}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.0% accurate, 6.7× speedup

Math

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

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 1.4) (/ y_m (* x z)) (/ 1.0 (/ 1.0 (* y_m (* (/ x z) 0.5)))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.4) {
		tmp = y_m / (x * z);
	} else {
		tmp = 1.0 / (1.0 / (y_m * ((x / z) * 0.5)));
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = y_m / (x * z)
    else
        tmp = 1.0d0 / (1.0d0 / (y_m * ((x / z) * 0.5d0)))
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.4) {
		tmp = y_m / (x * z);
	} else {
		tmp = 1.0 / (1.0 / (y_m * ((x / z) * 0.5)));
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 1.4:
		tmp = y_m / (x * z)
	else:
		tmp = 1.0 / (1.0 / (y_m * ((x / z) * 0.5)))
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 1.4)
		tmp = Float64(y_m / Float64(x * z));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(y_m * Float64(Float64(x / z) * 0.5))));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = y_m / (x * z);
	else
		tmp = 1.0 / (1.0 / (y_m * ((x / z) * 0.5)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 1.4], N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(y$95$m * N[(N[(x / z), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 87.2%

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

      1. associate-*l/ 87.1%

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

   3. Simplified 87.1%

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

   5. Taylor expanded in x around 0 66.6%

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

   if 1.3999999999999999 < x

   1. Initial program 78.2%

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

   3. Taylor expanded in x around 0 37.2%

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

   4. Taylor expanded in x around inf 37.2%

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

      1. associate-*r/ 37.2%

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

      2. associate-*r* 37.2%

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

      3. *-commutative 37.2%

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

      4. *-commutative 37.2%

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

      5. associate-/l* 37.3%

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

      6. *-commutative 37.3%

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

   6. Simplified 37.3%

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

      1. associate-/r* 37.3%

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

      2. associate-/r/ 33.8%

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

      3. div-inv 33.8%

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

      4. metadata-eval 33.8%

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

   8. Applied egg-rr 33.8%

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

      1. associate-/r* 33.8%

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

      2. associate-/r/ 33.8%

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

      3. *-un-lft-identity 33.8%

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

      4. associate-*l/ 33.8%

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

      5. associate-*r/ 37.2%

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

      6. /-rgt-identity 37.2%

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

      7. clear-num 37.2%

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

      8. associate-*l/ 37.2%

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

      9. *-un-lft-identity 37.2%

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

      10. associate-/r/ 37.2%

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

      11. div-inv 37.2%

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

      12. metadata-eval 37.2%

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

      13. associate-*r/ 39.0%

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

      14. *-commutative 39.0%

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

      15. *-commutative 39.0%

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

      16. associate-*r* 39.0%

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

   10. Applied egg-rr 39.0%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{y \cdot \left(\frac{x}{z} \cdot 0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.0% accurate, 8.9× speedup

Math

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

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 1.4) (/ y_m (* x z)) (* y_m (* x (/ 0.5 z))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.4) {
		tmp = y_m / (x * z);
	} else {
		tmp = y_m * (x * (0.5 / z));
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = y_m / (x * z)
    else
        tmp = y_m * (x * (0.5d0 / z))
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.4) {
		tmp = y_m / (x * z);
	} else {
		tmp = y_m * (x * (0.5 / z));
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 1.4:
		tmp = y_m / (x * z)
	else:
		tmp = y_m * (x * (0.5 / z))
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 1.4)
		tmp = Float64(y_m / Float64(x * z));
	else
		tmp = Float64(y_m * Float64(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)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = y_m / (x * z);
	else
		tmp = y_m * (x * (0.5 / z));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 1.4], N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 87.2%

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

      1. associate-*l/ 87.1%

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

   3. Simplified 87.1%

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

   5. Taylor expanded in x around 0 66.6%

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

   if 1.3999999999999999 < x

   1. Initial program 78.2%

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

   3. Taylor expanded in x around 0 37.2%

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

   4. Taylor expanded in x around inf 37.2%

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

      1. associate-*r/ 37.2%

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

      2. associate-*r* 37.2%

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

      3. *-commutative 37.2%

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

      4. *-commutative 37.2%

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

      5. associate-/l* 37.3%

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

      6. *-commutative 37.3%

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

   6. Simplified 37.3%

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

      1. clear-num 37.3%

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

      2. associate-/r/ 39.0%

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

      3. clear-num 39.0%

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

      4. *-commutative 39.0%

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

      5. *-un-lft-identity 39.0%

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

      6. times-frac 39.0%

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

      7. /-rgt-identity 39.0%

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

   8. Applied egg-rr 39.0%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.1% accurate, 10.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 7.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{y_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y_m}{z}}{x}\\ \end{array} \end{array} \]

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 7.5e+21) (/ (/ 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 <= 7.5e+21) {
		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 <= 7.5d+21) 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 <= 7.5e+21) {
		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 <= 7.5e+21:
		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 <= 7.5e+21)
		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 <= 7.5e+21)
		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, 7.5e+21], 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 < 7.5e21

   1. Initial program 83.9%

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

   3. Taylor expanded in x around 0 52.5%

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

   if 7.5e21 < y

   1. Initial program 90.8%

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

      1. associate-*l/ 90.8%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in x around 0 44.6%

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

      1. associate-*r/ 64.4%

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

      2. associate-*l/ 64.5%

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

      3. *-un-lft-identity 64.5%

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

   7. Applied egg-rr 64.5%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.4% 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 85.3%

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

   1. associate-*l/ 85.2%

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

3. Simplified 85.2%

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

5. Taylor expanded in x around 0 53.3%

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

6. Final simplification 53.3%

   \[\leadsto \frac{y}{x \cdot z} \]
7. 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 2024019 
(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))