Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.8% → 98.8%

Time: 9.3s

Alternatives: 13

Speedup: 1.0×

• 62.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.8% 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: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 10^{+104}:\\ \;\;\;\;\frac{y_m \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y_m}}\\ \end{array} \end{array} \]

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 1e+104)
    (/ (* 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 <= 1e+104) {
		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 <= 1d+104) 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 <= 1e+104) {
		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 <= 1e+104:
		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 <= 1e+104)
		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 <= 1e+104)
		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, 1e+104], 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 < 1e104

   1. Initial program 85.1%

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

      1. expm1-log1p-u 50.0%

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

      2. expm1-udef 38.8%

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

   3. Applied egg-rr 38.8%

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

      1. expm1-def 50.0%

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

      2. expm1-log1p 85.1%

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

      3. associate-*r/ 97.6%

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

      4. associate-*l/ 97.5%

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

      5. *-commutative 97.5%

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

   5. Simplified 97.5%

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

   if 1e104 < y

   1. Initial program 88.0%

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

      1. associate-/l* 88.0%

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

   3. Simplified 88.0%

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

      1. associate-/r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+104}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y}}\\ \end{array} \]

Alternative 2: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 1.2 \cdot 10^{+190}:\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y_m}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y_m \cdot x}{z} + \frac{y_m}{x \cdot z}\\ \end{array} \end{array} \]

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 1.2e+190)
    (* (/ (cosh x) z) (/ y_m x))
    (+ (* 0.5 (/ (* 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 <= 1.2e+190) {
		tmp = (cosh(x) / z) * (y_m / x);
	} else {
		tmp = (0.5 * ((y_m * x) / z)) + (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 <= 1.2d+190) then
        tmp = (cosh(x) / z) * (y_m / x)
    else
        tmp = (0.5d0 * ((y_m * x) / z)) + (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 <= 1.2e+190) {
		tmp = (Math.cosh(x) / z) * (y_m / x);
	} else {
		tmp = (0.5 * ((y_m * x) / z)) + (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 <= 1.2e+190:
		tmp = (math.cosh(x) / z) * (y_m / x)
	else:
		tmp = (0.5 * ((y_m * x) / z)) + (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 <= 1.2e+190)
		tmp = Float64(Float64(cosh(x) / z) * Float64(y_m / x));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(y_m * x) / z)) + 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 <= 1.2e+190)
		tmp = (cosh(x) / z) * (y_m / x);
	else
		tmp = (0.5 * ((y_m * x) / z)) + (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, 1.2e+190], N[(N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(y$95$m * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.1999999999999999e190

   1. Initial program 85.4%

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

      1. associate-*l/ 85.3%

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

   3. Simplified 85.3%

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

   if 1.1999999999999999e190 < y

   1. Initial program 88.4%

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

      1. associate-*l/ 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in x around 0 99.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+190}:\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z} + \frac{y}{x \cdot z}\\ \end{array} \]

Alternative 3: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 1.18 \cdot 10^{+104}:\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y_m}}\\ \end{array} \end{array} \]

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 1.18e+104)
    (* (/ (cosh x) z) (/ y_m x))
    (/ (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 <= 1.18e+104) {
		tmp = (cosh(x) / z) * (y_m / x);
	} 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 <= 1.18d+104) then
        tmp = (cosh(x) / z) * (y_m / x)
    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 <= 1.18e+104) {
		tmp = (Math.cosh(x) / z) * (y_m / x);
	} 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 <= 1.18e+104:
		tmp = (math.cosh(x) / z) * (y_m / x)
	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 <= 1.18e+104)
		tmp = Float64(Float64(cosh(x) / z) * Float64(y_m / x));
	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 <= 1.18e+104)
		tmp = (cosh(x) / z) * (y_m / x);
	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, 1.18e+104], N[(N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $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 < 1.18e104

   1. Initial program 85.1%

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

      1. associate-*l/ 85.1%

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

   3. Simplified 85.1%

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

   if 1.18e104 < y

   1. Initial program 88.0%

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

      1. associate-/l* 88.0%

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

   3. Simplified 88.0%

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

      1. associate-/r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.18 \cdot 10^{+104}:\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y}}\\ \end{array} \]

Alternative 4: 61.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{z}{y_m \cdot x}\\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 6.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{y_m}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y_m}{x} \cdot t_0 + z \cdot 0.5}{z \cdot t_0}\\ \end{array} \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ z (* y_m x))))
   (*
    y_s
    (if (<= x 6.6e-9)
      (/ (/ y_m z) x)
      (/ (+ (* (/ y_m x) t_0) (* z 0.5)) (* z t_0))))))

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 = z / (y_m * x);
	double tmp;
	if (x <= 6.6e-9) {
		tmp = (y_m / z) / x;
	} else {
		tmp = (((y_m / x) * t_0) + (z * 0.5)) / (z * t_0);
	}
	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 = z / (y_m * x)
    if (x <= 6.6d-9) then
        tmp = (y_m / z) / x
    else
        tmp = (((y_m / x) * t_0) + (z * 0.5d0)) / (z * t_0)
    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 = z / (y_m * x);
	double tmp;
	if (x <= 6.6e-9) {
		tmp = (y_m / z) / x;
	} else {
		tmp = (((y_m / x) * t_0) + (z * 0.5)) / (z * t_0);
	}
	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 = z / (y_m * x)
	tmp = 0
	if x <= 6.6e-9:
		tmp = (y_m / z) / x
	else:
		tmp = (((y_m / x) * t_0) + (z * 0.5)) / (z * t_0)
	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(z / Float64(y_m * x))
	tmp = 0.0
	if (x <= 6.6e-9)
		tmp = Float64(Float64(y_m / z) / x);
	else
		tmp = Float64(Float64(Float64(Float64(y_m / x) * t_0) + Float64(z * 0.5)) / Float64(z * t_0));
	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 = z / (y_m * x);
	tmp = 0.0;
	if (x <= 6.6e-9)
		tmp = (y_m / z) / x;
	else
		tmp = (((y_m / x) * t_0) + (z * 0.5)) / (z * t_0);
	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[(z / N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[x, 6.6e-9], N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(y$95$m / x), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(z * 0.5), $MachinePrecision]), $MachinePrecision] / N[(z * t$95$0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.60000000000000037e-9

   1. Initial program 87.5%

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

      1. associate-*l/ 87.4%

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

   3. Simplified 87.4%

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

   4. Taylor expanded in x around 0 62.2%

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

      1. *-un-lft-identity 62.2%

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

      2. times-frac 68.3%

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

   6. Applied egg-rr 68.3%

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

      1. associate-*l/ 68.3%

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

      2. *-un-lft-identity 68.3%

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

   8. Applied egg-rr 68.3%

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

   if 6.60000000000000037e-9 < x

   1. Initial program 80.6%

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

      1. associate-*l/ 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in x around 0 41.0%

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

      1. clear-num 41.0%

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

      2. un-div-inv 41.0%

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

      3. *-commutative 41.0%

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

      4. associate-/r* 36.7%

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

   6. Applied egg-rr 36.7%

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

      1. associate-/l/ 36.7%

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

      2. clear-num 36.7%

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

      3. associate-/r* 36.7%

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

      4. +-commutative 36.7%

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

      5. associate-/r* 36.7%

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

      6. clear-num 36.7%

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

      7. associate-/l/ 36.7%

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

      8. associate-/r* 36.7%

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

      9. frac-add 41.9%

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

      10. associate-/l/ 42.1%

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

      11. *-commutative 42.1%

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

      12. associate-/l/ 46.4%

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

      13. *-commutative 46.4%

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

   8. Applied egg-rr 46.4%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \frac{z}{y \cdot x} + z \cdot 0.5}{z \cdot \frac{z}{y \cdot x}}\\ \end{array} \]

Alternative 5: 66.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 1.5 \cdot 10^{+180}:\\ \;\;\;\;\frac{\frac{y_m}{x} + 0.5 \cdot \left(y_m \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{x \cdot z} + 0.5 \cdot \left(y_m \cdot \frac{x}{z}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.5e+180)
    (/ (+ (/ y_m x) (* 0.5 (* y_m x))) z)
    (+ (/ y_m (* x z)) (* 0.5 (* y_m (/ x z)))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.5e+180) {
		tmp = ((y_m / x) + (0.5 * (y_m * x))) / z;
	} else {
		tmp = (y_m / (x * z)) + (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 <= 1.5d+180) then
        tmp = ((y_m / x) + (0.5d0 * (y_m * x))) / z
    else
        tmp = (y_m / (x * z)) + (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 <= 1.5e+180) {
		tmp = ((y_m / x) + (0.5 * (y_m * x))) / z;
	} else {
		tmp = (y_m / (x * z)) + (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 <= 1.5e+180:
		tmp = ((y_m / x) + (0.5 * (y_m * x))) / z
	else:
		tmp = (y_m / (x * z)) + (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 <= 1.5e+180)
		tmp = Float64(Float64(Float64(y_m / x) + Float64(0.5 * Float64(y_m * x))) / z);
	else
		tmp = Float64(Float64(y_m / Float64(x * z)) + Float64(0.5 * 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 <= 1.5e+180)
		tmp = ((y_m / x) + (0.5 * (y_m * x))) / z;
	else
		tmp = (y_m / (x * z)) + (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, 1.5e+180], N[(N[(N[(y$95$m / x), $MachinePrecision] + N[(0.5 * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(y$95$m * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.50000000000000001e180

   1. Initial program 85.4%

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

   2. Taylor expanded in x around 0 65.2%

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

   if 1.50000000000000001e180 < y

   1. Initial program 87.6%

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

      1. associate-*l/ 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in x around 0 93.7%

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

      1. associate-/l* 90.7%

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

      2. associate-/r/ 90.7%

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

   6. Applied egg-rr 90.7%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+180}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \end{array} \]

Alternative 6: 68.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 2.15 \cdot 10^{-38}:\\ \;\;\;\;\frac{\frac{y_m}{x} + 0.5 \cdot \left(y_m \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y_m \cdot x}{z} + \frac{y_m}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= z 2.15e-38)
    (/ (+ (/ y_m x) (* 0.5 (* y_m x))) z)
    (+ (* 0.5 (/ (* y_m x) z)) (/ y_m (* x z))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 2.15e-38) {
		tmp = ((y_m / x) + (0.5 * (y_m * x))) / z;
	} else {
		tmp = (0.5 * ((y_m * x) / z)) + (y_m / (x * z));
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 2.15d-38) then
        tmp = ((y_m / x) + (0.5d0 * (y_m * x))) / z
    else
        tmp = (0.5d0 * ((y_m * x) / z)) + (y_m / (x * z))
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 2.15e-38) {
		tmp = ((y_m / x) + (0.5 * (y_m * x))) / z;
	} else {
		tmp = (0.5 * ((y_m * x) / z)) + (y_m / (x * z));
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= 2.15e-38:
		tmp = ((y_m / x) + (0.5 * (y_m * x))) / z
	else:
		tmp = (0.5 * ((y_m * x) / z)) + (y_m / (x * z))
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= 2.15e-38)
		tmp = Float64(Float64(Float64(y_m / x) + Float64(0.5 * Float64(y_m * x))) / z);
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(y_m * x) / z)) + Float64(y_m / Float64(x * z)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= 2.15e-38)
		tmp = ((y_m / x) + (0.5 * (y_m * x))) / z;
	else
		tmp = (0.5 * ((y_m * x) / z)) + (y_m / (x * z));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 2.15e-38], N[(N[(N[(y$95$m / x), $MachinePrecision] + N[(0.5 * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(0.5 * N[(N[(y$95$m * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.1500000000000001e-38

   1. Initial program 87.9%

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

   2. Taylor expanded in x around 0 75.6%

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

   if 2.1500000000000001e-38 < z

   1. Initial program 79.3%

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

      1. associate-*l/ 79.2%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in x around 0 52.8%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.15 \cdot 10^{-38}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z} + \frac{y}{x \cdot z}\\ \end{array} \]

Alternative 7: 65.9% accurate, 8.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 7.8 \cdot 10^{+210}:\\ \;\;\;\;\frac{y_m \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+299}:\\ \;\;\;\;\frac{y_m}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(y_m \cdot x\right)}{z}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= z 7.8e+210)
    (/ (* y_m (+ (* x 0.5) (/ 1.0 x))) z)
    (if (<= z 1.55e+299) (/ y_m (* x z)) (/ (* 0.5 (* y_m x)) z)))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 7.8e+210) {
		tmp = (y_m * ((x * 0.5) + (1.0 / x))) / z;
	} else if (z <= 1.55e+299) {
		tmp = y_m / (x * z);
	} else {
		tmp = (0.5 * (y_m * x)) / z;
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 7.8d+210) then
        tmp = (y_m * ((x * 0.5d0) + (1.0d0 / x))) / z
    else if (z <= 1.55d+299) then
        tmp = y_m / (x * z)
    else
        tmp = (0.5d0 * (y_m * x)) / z
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 7.8e+210) {
		tmp = (y_m * ((x * 0.5) + (1.0 / x))) / z;
	} else if (z <= 1.55e+299) {
		tmp = y_m / (x * z);
	} else {
		tmp = (0.5 * (y_m * x)) / z;
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= 7.8e+210:
		tmp = (y_m * ((x * 0.5) + (1.0 / x))) / z
	elif z <= 1.55e+299:
		tmp = y_m / (x * z)
	else:
		tmp = (0.5 * (y_m * x)) / z
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= 7.8e+210)
		tmp = Float64(Float64(y_m * Float64(Float64(x * 0.5) + Float64(1.0 / x))) / z);
	elseif (z <= 1.55e+299)
		tmp = Float64(y_m / Float64(x * z));
	else
		tmp = Float64(Float64(0.5 * Float64(y_m * x)) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= 7.8e+210)
		tmp = (y_m * ((x * 0.5) + (1.0 / x))) / z;
	elseif (z <= 1.55e+299)
		tmp = y_m / (x * z);
	else
		tmp = (0.5 * (y_m * x)) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 7.8e+210], N[(N[(y$95$m * N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.55e+299], N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 7.8e210

   1. Initial program 88.0%

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

      1. expm1-log1p-u 50.7%

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

      2. expm1-udef 40.6%

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

   3. Applied egg-rr 40.6%

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

      1. expm1-def 50.7%

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

      2. expm1-log1p 88.0%

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

      3. associate-*r/ 97.5%

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

      4. associate-*l/ 97.4%

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

      5. *-commutative 97.4%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in x around 0 71.0%

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

   if 7.8e210 < z < 1.55e299

   1. Initial program 58.2%

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

      1. associate-*l/ 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in x around 0 48.2%

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

   if 1.55e299 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 51.3%

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

   3. Taylor expanded in x around inf 51.3%

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

      1. *-commutative 51.3%

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

   5. Simplified 51.3%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.8 \cdot 10^{+210}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+299}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(y \cdot x\right)}{z}\\ \end{array} \]

Alternative 8: 65.9% accurate, 8.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \left(y_m \cdot x\right)\\ y_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{+211}:\\ \;\;\;\;\frac{\frac{y_m}{x} + t_0}{z}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+299}:\\ \;\;\;\;\frac{y_m}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* 0.5 (* y_m x))))
   (*
    y_s
    (if (<= z 3.5e+211)
      (/ (+ (/ y_m x) t_0) z)
      (if (<= z 1.8e+299) (/ y_m (* x z)) (/ t_0 z))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = 0.5 * (y_m * x);
	double tmp;
	if (z <= 3.5e+211) {
		tmp = ((y_m / x) + t_0) / z;
	} else if (z <= 1.8e+299) {
		tmp = y_m / (x * z);
	} else {
		tmp = t_0 / 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 = 0.5d0 * (y_m * x)
    if (z <= 3.5d+211) then
        tmp = ((y_m / x) + t_0) / z
    else if (z <= 1.8d+299) then
        tmp = y_m / (x * z)
    else
        tmp = t_0 / 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 = 0.5 * (y_m * x);
	double tmp;
	if (z <= 3.5e+211) {
		tmp = ((y_m / x) + t_0) / z;
	} else if (z <= 1.8e+299) {
		tmp = y_m / (x * z);
	} else {
		tmp = t_0 / 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 = 0.5 * (y_m * x)
	tmp = 0
	if z <= 3.5e+211:
		tmp = ((y_m / x) + t_0) / z
	elif z <= 1.8e+299:
		tmp = y_m / (x * z)
	else:
		tmp = t_0 / z
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(0.5 * Float64(y_m * x))
	tmp = 0.0
	if (z <= 3.5e+211)
		tmp = Float64(Float64(Float64(y_m / x) + t_0) / z);
	elseif (z <= 1.8e+299)
		tmp = Float64(y_m / Float64(x * z));
	else
		tmp = Float64(t_0 / 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 = 0.5 * (y_m * x);
	tmp = 0.0;
	if (z <= 3.5e+211)
		tmp = ((y_m / x) + t_0) / z;
	elseif (z <= 1.8e+299)
		tmp = y_m / (x * z);
	else
		tmp = t_0 / 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[(0.5 * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[z, 3.5e+211], N[(N[(N[(y$95$m / x), $MachinePrecision] + t$95$0), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.8e+299], N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < 3.49999999999999996e211

   1. Initial program 88.0%

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

   2. Taylor expanded in x around 0 71.0%

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

   if 3.49999999999999996e211 < z < 1.80000000000000002e299

   1. Initial program 58.2%

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

      1. associate-*l/ 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in x around 0 48.2%

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

   if 1.80000000000000002e299 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 51.3%

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

   3. Taylor expanded in x around inf 51.3%

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

      1. *-commutative 51.3%

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

   5. Simplified 51.3%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{+211}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+299}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(y \cdot x\right)}{z}\\ \end{array} \]

Alternative 9: 60.0% accurate, 11.8× speedup

Math

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

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 1.45) (/ (/ y_m z) x) (* 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 tmp;
	if (x <= 1.45) {
		tmp = (y_m / z) / x;
	} 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) :: tmp
    if (x <= 1.45d0) then
        tmp = (y_m / z) / x
    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 tmp;
	if (x <= 1.45) {
		tmp = (y_m / z) / x;
	} 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):
	tmp = 0
	if x <= 1.45:
		tmp = (y_m / z) / x
	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)
	tmp = 0.0
	if (x <= 1.45)
		tmp = Float64(Float64(y_m / z) / x);
	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)
	tmp = 0.0;
	if (x <= 1.45)
		tmp = (y_m / z) / x;
	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_] := N[(y$95$s * If[LessEqual[x, 1.45], N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision], N[(y$95$m * N[(0.5 * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 87.6%

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

      1. associate-*l/ 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in x around 0 62.5%

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

      1. *-un-lft-identity 62.5%

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

      2. times-frac 68.4%

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

   6. Applied egg-rr 68.4%

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

      1. associate-*l/ 68.5%

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

      2. *-un-lft-identity 68.5%

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

   8. Applied egg-rr 68.5%

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

   if 1.44999999999999996 < x

   1. Initial program 80.0%

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

      1. expm1-log1p-u 38.5%

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

      2. expm1-udef 37.0%

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

   3. Applied egg-rr 37.0%

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

      1. expm1-def 38.5%

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

      2. expm1-log1p 80.0%

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

      3. associate-*r/ 100.0%

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

      4. associate-*l/ 100.0%

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

      5. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 39.2%

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

   7. Taylor expanded in x around inf 39.2%

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

      1. associate-*l/ 37.7%

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

      2. associate-*l* 37.7%

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

      3. *-commutative 37.7%

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

   9. Simplified 37.7%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z}\right)\\ \end{array} \]

Alternative 10: 60.0% accurate, 11.8× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 1.45) (/ (/ y_m z) x) (/ (* 0.5 (* y_m x)) z))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.45) {
		tmp = (y_m / z) / x;
	} else {
		tmp = (0.5 * (y_m * x)) / z;
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.45d0) then
        tmp = (y_m / z) / x
    else
        tmp = (0.5d0 * (y_m * x)) / z
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.45) {
		tmp = (y_m / z) / x;
	} else {
		tmp = (0.5 * (y_m * x)) / z;
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 1.45:
		tmp = (y_m / z) / x
	else:
		tmp = (0.5 * (y_m * x)) / z
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 1.45)
		tmp = Float64(Float64(y_m / z) / x);
	else
		tmp = Float64(Float64(0.5 * Float64(y_m * x)) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 1.45)
		tmp = (y_m / z) / x;
	else
		tmp = (0.5 * (y_m * x)) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 1.45], N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision], N[(N[(0.5 * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 87.6%

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

      1. associate-*l/ 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in x around 0 62.5%

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

      1. *-un-lft-identity 62.5%

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

      2. times-frac 68.4%

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

   6. Applied egg-rr 68.4%

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

      1. associate-*l/ 68.5%

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

      2. *-un-lft-identity 68.5%

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

   8. Applied egg-rr 68.5%

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

   if 1.44999999999999996 < x

   1. Initial program 80.0%

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

   2. Taylor expanded in x around 0 39.2%

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

   3. Taylor expanded in x around inf 39.2%

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

      1. *-commutative 39.2%

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

   5. Simplified 39.2%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(y \cdot x\right)}{z}\\ \end{array} \]

Alternative 11: 53.9% accurate, 15.2× speedup

Math

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

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 1e+29) (/ (/ 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 <= 1e+29) {
		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 <= 1d+29) 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 <= 1e+29) {
		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 <= 1e+29:
		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 <= 1e+29)
		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 <= 1e+29)
		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, 1e+29], 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 < 9.99999999999999914e28

   1. Initial program 83.7%

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

   2. Taylor expanded in x around 0 50.0%

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

   if 9.99999999999999914e28 < y

   1. Initial program 90.7%

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

      1. associate-*l/ 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in x around 0 56.7%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+29}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 12: 56.0% accurate, 15.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{y_m}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= z 2e-25) (/ (/ y_m z) x) (/ y_m (* x z)))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 2e-25) {
		tmp = (y_m / z) / x;
	} else {
		tmp = y_m / (x * z);
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 2d-25) then
        tmp = (y_m / z) / x
    else
        tmp = y_m / (x * z)
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 2e-25) {
		tmp = (y_m / z) / x;
	} else {
		tmp = y_m / (x * z);
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= 2e-25:
		tmp = (y_m / z) / x
	else:
		tmp = y_m / (x * z)
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= 2e-25)
		tmp = Float64(Float64(y_m / z) / x);
	else
		tmp = Float64(y_m / Float64(x * z));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= 2e-25)
		tmp = (y_m / z) / x;
	else
		tmp = y_m / (x * z);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 2e-25], N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision], N[(y$95$m / N[(x * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.00000000000000008e-25

   1. Initial program 88.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. Taylor expanded in x around 0 49.3%

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

      1. *-un-lft-identity 49.3%

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

      2. times-frac 58.1%

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

   6. Applied egg-rr 58.1%

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

      1. associate-*l/ 58.2%

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

      2. *-un-lft-identity 58.2%

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

   8. Applied egg-rr 58.2%

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

   if 2.00000000000000008e-25 < z

   1. Initial program 78.0%

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

      1. associate-*l/ 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in x around 0 47.2%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 13: 50.2% 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 85.7%

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

   1. associate-*l/ 85.6%

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

3. Simplified 85.6%

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

4. Taylor expanded in x around 0 48.8%

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

5. Final simplification 48.8%

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

Developer target: 97.4% accurate, 1.0× 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 2023332 
(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))