Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 85.3% → 99.7%

Time: 8.0s

Alternatives: 7

Speedup: 1.0×

• 64.3% 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: 85.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\cosh x}{z\_m}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8.8 \cdot 10^{+20}:\\ \;\;\;\;\frac{y \cdot t\_0}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t\_0}{x}\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (let* ((t_0 (/ (cosh x) z_m)))
   (* z_s (if (<= z_m 8.8e+20) (/ (* y t_0) x) (* y (/ t_0 x))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double t_0 = cosh(x) / z_m;
	double tmp;
	if (z_m <= 8.8e+20) {
		tmp = (y * t_0) / x;
	} else {
		tmp = y * (t_0 / x);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cosh(x) / z_m
    if (z_m <= 8.8d+20) then
        tmp = (y * t_0) / x
    else
        tmp = y * (t_0 / x)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double t_0 = Math.cosh(x) / z_m;
	double tmp;
	if (z_m <= 8.8e+20) {
		tmp = (y * t_0) / x;
	} else {
		tmp = y * (t_0 / x);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	t_0 = math.cosh(x) / z_m
	tmp = 0
	if z_m <= 8.8e+20:
		tmp = (y * t_0) / x
	else:
		tmp = y * (t_0 / x)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	t_0 = Float64(cosh(x) / z_m)
	tmp = 0.0
	if (z_m <= 8.8e+20)
		tmp = Float64(Float64(y * t_0) / x);
	else
		tmp = Float64(y * Float64(t_0 / x));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	t_0 = cosh(x) / z_m;
	tmp = 0.0;
	if (z_m <= 8.8e+20)
		tmp = (y * t_0) / x;
	else
		tmp = y * (t_0 / x);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] / z$95$m), $MachinePrecision]}, N[(z$95$s * If[LessEqual[z$95$m, 8.8e+20], N[(N[(y * t$95$0), $MachinePrecision] / x), $MachinePrecision], N[(y * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 8.8e20

   1. Initial program 86.3%

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

      1. associate-/l* 81.0%

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

      2. associate-/l/ 81.6%

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

   3. Simplified 81.6%

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

      1. associate-*r/ 86.8%

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

      2. frac-times 86.2%

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

      3. *-commutative 86.2%

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

      4. associate-*l/ 95.5%

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

   6. Applied egg-rr 95.5%

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

   if 8.8e20 < z

   1. Initial program 82.1%

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

      1. *-commutative 82.1%

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

      2. associate-*r/ 82.0%

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

      3. associate-*l/ 86.9%

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

      4. associate-/l* 99.8%

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

   4. Applied egg-rr 99.8%

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

Alternative 2: 95.9% accurate, 1.0× speedup

Math

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

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m) :precision binary64 (* z_s (* y (/ (/ (cosh x) z_m) x))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	return z_s * (y * ((cosh(x) / z_m) / x));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = z_s * (y * ((cosh(x) / z_m) / x))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	return z_s * (y * ((Math.cosh(x) / z_m) / x));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	return z_s * (y * ((math.cosh(x) / z_m) / x))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	return Float64(z_s * Float64(y * Float64(Float64(cosh(x) / z_m) / x)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m)
	tmp = z_s * (y * ((cosh(x) / z_m) / x));
end

Wolfram

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

Derivation

1. Initial program 85.2%

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

   1. *-commutative 85.2%

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

   2. associate-*r/ 85.1%

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

   3. associate-*l/ 93.3%

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

   4. associate-/l* 96.5%

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

4. Applied egg-rr 96.5%

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

Alternative 3: 56.3% accurate, 8.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{y}{z\_m}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z\_m}}{x}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= z_m 2.5e+20) (/ 1.0 (/ x (/ y z_m))) (* y (/ (/ 1.0 z_m) x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 2.5e+20) {
		tmp = 1.0 / (x / (y / z_m));
	} else {
		tmp = y * ((1.0 / z_m) / x);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 2.5d+20) then
        tmp = 1.0d0 / (x / (y / z_m))
    else
        tmp = y * ((1.0d0 / z_m) / x)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 2.5e+20) {
		tmp = 1.0 / (x / (y / z_m));
	} else {
		tmp = y * ((1.0 / z_m) / x);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 2.5e+20:
		tmp = 1.0 / (x / (y / z_m))
	else:
		tmp = y * ((1.0 / z_m) / x)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 2.5e+20)
		tmp = Float64(1.0 / Float64(x / Float64(y / z_m)));
	else
		tmp = Float64(y * Float64(Float64(1.0 / z_m) / x));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 2.5e+20)
		tmp = 1.0 / (x / (y / z_m));
	else
		tmp = y * ((1.0 / z_m) / x);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[z$95$m, 2.5e+20], N[(1.0 / N[(x / N[(y / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(1.0 / z$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.5e20

   1. Initial program 86.3%

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

      1. *-commutative 86.3%

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

      2. associate-/l* 86.2%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in x around 0 54.8%

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

      1. associate-*l/ 54.4%

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

      2. clear-num 54.4%

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

      3. un-div-inv 54.4%

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

   7. Applied egg-rr 54.4%

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

   if 2.5e20 < z

   1. Initial program 82.1%

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

      1. *-commutative 82.1%

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

      2. associate-*r/ 82.0%

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

      3. associate-*l/ 86.9%

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

      4. associate-/l* 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 50.9%

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

      1. associate-/l/ 50.9%

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

   7. Simplified 50.9%

      \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{z}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 56.3% accurate, 8.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.3 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{y}{z\_m}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z\_m}}{x}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= z_m 1.3e+21) (/ (/ y z_m) x) (* y (/ (/ 1.0 z_m) x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 1.3e+21) {
		tmp = (y / z_m) / x;
	} else {
		tmp = y * ((1.0 / z_m) / x);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 1.3d+21) then
        tmp = (y / z_m) / x
    else
        tmp = y * ((1.0d0 / z_m) / x)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 1.3e+21) {
		tmp = (y / z_m) / x;
	} else {
		tmp = y * ((1.0 / z_m) / x);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 1.3e+21:
		tmp = (y / z_m) / x
	else:
		tmp = y * ((1.0 / z_m) / x)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 1.3e+21)
		tmp = Float64(Float64(y / z_m) / x);
	else
		tmp = Float64(y * Float64(Float64(1.0 / z_m) / x));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 1.3e+21)
		tmp = (y / z_m) / x;
	else
		tmp = y * ((1.0 / z_m) / x);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[z$95$m, 1.3e+21], N[(N[(y / z$95$m), $MachinePrecision] / x), $MachinePrecision], N[(y * N[(N[(1.0 / z$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.3e21

   1. Initial program 86.3%

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

      1. *-commutative 86.3%

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

      2. associate-*r/ 86.2%

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

      3. associate-*l/ 95.5%

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

      4. associate-/l* 95.4%

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

   4. Applied egg-rr 95.4%

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

   5. Taylor expanded in x around 0 52.2%

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

      1. associate-/l/ 54.4%

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

   7. Simplified 54.4%

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

   if 1.3e21 < z

   1. Initial program 82.1%

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

      1. *-commutative 82.1%

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

      2. associate-*r/ 82.0%

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

      3. associate-*l/ 86.9%

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

      4. associate-/l* 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 50.9%

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

      1. associate-/l/ 50.9%

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

   7. Simplified 50.9%

      \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{z}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 56.4% accurate, 10.7× speedup

Math

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

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= z_m 1e+16) (/ (/ y z_m) x) (/ y (* z_m x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 1e+16) {
		tmp = (y / z_m) / x;
	} else {
		tmp = y / (z_m * x);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 1d+16) then
        tmp = (y / z_m) / x
    else
        tmp = y / (z_m * x)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 1e+16) {
		tmp = (y / z_m) / x;
	} else {
		tmp = y / (z_m * x);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 1e+16:
		tmp = (y / z_m) / x
	else:
		tmp = y / (z_m * x)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 1e+16)
		tmp = Float64(Float64(y / z_m) / x);
	else
		tmp = Float64(y / Float64(z_m * x));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 1e+16)
		tmp = (y / z_m) / x;
	else
		tmp = y / (z_m * x);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1e16

   1. Initial program 86.1%

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

      1. *-commutative 86.1%

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

      2. associate-*r/ 86.1%

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

      3. associate-*l/ 95.4%

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

      4. associate-/l* 95.4%

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

   4. Applied egg-rr 95.4%

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

   5. Taylor expanded in x around 0 51.7%

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

      1. associate-/l/ 53.9%

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

   7. Simplified 53.9%

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

   if 1e16 < z

   1. Initial program 82.6%

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

      1. associate-/l* 69.4%

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

      2. associate-/l/ 63.1%

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

   3. Simplified 63.1%

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

   5. Taylor expanded in x around 0 52.4%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{+16}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.8% accurate, 10.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.7 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{y}{x}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z\_m \cdot x}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= z_m 1.7e+17) (/ (/ y x) z_m) (/ y (* z_m x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 1.7e+17) {
		tmp = (y / x) / z_m;
	} else {
		tmp = y / (z_m * x);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 1.7d+17) then
        tmp = (y / x) / z_m
    else
        tmp = y / (z_m * x)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 1.7e+17) {
		tmp = (y / x) / z_m;
	} else {
		tmp = y / (z_m * x);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 1.7e+17:
		tmp = (y / x) / z_m
	else:
		tmp = y / (z_m * x)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 1.7e+17)
		tmp = Float64(Float64(y / x) / z_m);
	else
		tmp = Float64(y / Float64(z_m * x));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 1.7e+17)
		tmp = (y / x) / z_m;
	else
		tmp = y / (z_m * x);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[z$95$m, 1.7e+17], N[(N[(y / x), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.7e17

   1. Initial program 86.1%

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

   3. Taylor expanded in x around 0 54.3%

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

   if 1.7e17 < z

   1. Initial program 82.6%

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

      1. associate-/l* 69.4%

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

      2. associate-/l/ 63.1%

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

   3. Simplified 63.1%

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

   5. Taylor expanded in x around 0 52.4%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.3% accurate, 21.4× speedup

Math

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

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m) :precision binary64 (* z_s (/ y (* z_m x))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	return z_s * (y / (z_m * x));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = z_s * (y / (z_m * x))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	return z_s * (y / (z_m * x));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	return z_s * (y / (z_m * x))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	return Float64(z_s * Float64(y / Float64(z_m * x)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m)
	tmp = z_s * (y / (z_m * x));
end

Wolfram

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

Derivation

1. Initial program 85.2%

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

   1. associate-/l* 77.8%

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

   2. associate-/l/ 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in x around 0 51.9%

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

6. Final simplification 51.9%

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

Developer Target 1: 96.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024150 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -2309451133843521/5000000000000000000000000000000000000000000000000000000000000000000) (* (/ (/ y z) x) (cosh x)) (if (< y 1038530535935153/1000000000000000000000000000000000000000000000000000000) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x)))))

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