Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.7% → 99.7%

Time: 10.3s

Alternatives: 16

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: 84.7% 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

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.85 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{\cosh x}{\frac{z\_m}{y}}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z\_m}}{x}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= z_m 1.85e-40)
    (/ (/ (cosh x) (/ z_m y)) x)
    (* 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) {
	double tmp;
	if (z_m <= 1.85e-40) {
		tmp = (cosh(x) / (z_m / y)) / x;
	} else {
		tmp = y * ((cosh(x) / 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 <= 1.85d-40) then
        tmp = (cosh(x) / (z_m / y)) / x
    else
        tmp = y * ((cosh(x) / 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 <= 1.85e-40) {
		tmp = (Math.cosh(x) / (z_m / y)) / x;
	} else {
		tmp = y * ((Math.cosh(x) / 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 <= 1.85e-40:
		tmp = (math.cosh(x) / (z_m / y)) / x
	else:
		tmp = y * ((math.cosh(x) / 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 <= 1.85e-40)
		tmp = Float64(Float64(cosh(x) / Float64(z_m / y)) / x);
	else
		tmp = Float64(y * Float64(Float64(cosh(x) / 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 <= 1.85e-40)
		tmp = (cosh(x) / (z_m / y)) / x;
	else
		tmp = y * ((cosh(x) / 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, 1.85e-40], N[(N[(N[Cosh[x], $MachinePrecision] / N[(z$95$m / y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(y * N[(N[(N[Cosh[x], $MachinePrecision] / z$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.84999999999999999e-40

   1. Initial program 91.5%

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

      1. associate-*l/ 91.5%

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

   3. Simplified 91.5%

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

      1. associate-/r/ 88.7%

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

      2. associate-/r/ 87.4%

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

      3. associate-/r* 91.3%

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

   6. Applied egg-rr 91.3%

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

   if 1.84999999999999999e-40 < z

   1. Initial program 85.0%

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

      1. associate-*l/ 85.0%

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

   3. Simplified 85.0%

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

      1. expm1-log1p-u 54.5%

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

      2. expm1-udef 41.3%

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

      3. associate-*l/ 41.3%

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

      4. div-inv 41.3%

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

      5. associate-*l* 37.4%

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

      6. div-inv 37.4%

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

   6. Applied egg-rr 37.4%

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

      1. expm1-def 49.4%

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

      2. expm1-log1p 78.6%

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

      3. associate-*r/ 85.0%

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

      4. associate-*l/ 85.0%

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

      5. *-commutative 85.0%

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

      6. associate-*l/ 98.3%

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

      7. associate-*r/ 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.85 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{\cosh x}{\frac{z}{y}}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+213}:\\ \;\;\;\;\frac{t\_0}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z\_m}}{x}\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ y x))))
   (* z_s (if (<= t_0 5e+213) (/ t_0 z_m) (* 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) {
	double t_0 = cosh(x) * (y / x);
	double tmp;
	if (t_0 <= 5e+213) {
		tmp = t_0 / z_m;
	} else {
		tmp = y * ((cosh(x) / 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) :: t_0
    real(8) :: tmp
    t_0 = cosh(x) * (y / x)
    if (t_0 <= 5d+213) then
        tmp = t_0 / z_m
    else
        tmp = y * ((cosh(x) / 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 t_0 = Math.cosh(x) * (y / x);
	double tmp;
	if (t_0 <= 5e+213) {
		tmp = t_0 / z_m;
	} else {
		tmp = y * ((Math.cosh(x) / 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):
	t_0 = math.cosh(x) * (y / x)
	tmp = 0
	if t_0 <= 5e+213:
		tmp = t_0 / z_m
	else:
		tmp = y * ((math.cosh(x) / 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)
	t_0 = Float64(cosh(x) * Float64(y / x))
	tmp = 0.0
	if (t_0 <= 5e+213)
		tmp = Float64(t_0 / z_m);
	else
		tmp = Float64(y * Float64(Float64(cosh(x) / 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)
	t_0 = cosh(x) * (y / x);
	tmp = 0.0;
	if (t_0 <= 5e+213)
		tmp = t_0 / z_m;
	else
		tmp = y * ((cosh(x) / 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_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t$95$0, 5e+213], N[(t$95$0 / z$95$m), $MachinePrecision], N[(y * N[(N[(N[Cosh[x], $MachinePrecision] / z$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 4.9999999999999998e213

   1. Initial program 95.9%

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

   if 4.9999999999999998e213 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 76.8%

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

      1. associate-*l/ 76.8%

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

   3. Simplified 76.8%

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

      1. expm1-log1p-u 42.5%

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

      2. expm1-udef 42.5%

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

      3. associate-*l/ 42.5%

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

      4. div-inv 42.5%

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

      5. associate-*l* 39.0%

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

      6. div-inv 39.0%

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

   6. Applied egg-rr 39.0%

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

      1. expm1-def 39.0%

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

      2. expm1-log1p 70.9%

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

      3. associate-*r/ 76.8%

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

      4. associate-*l/ 76.8%

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

      5. *-commutative 76.8%

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

      6. associate-*l/ 100.0%

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

      7. associate-*r/ 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 97.2%

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

Alternative 3: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{-109}:\\ \;\;\;\;\frac{\frac{y}{z\_m}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z\_m}}{x}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= x 2.65e-109) (/ (/ y z_m) x) (* 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) {
	double tmp;
	if (x <= 2.65e-109) {
		tmp = (y / z_m) / x;
	} else {
		tmp = y * ((cosh(x) / 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 (x <= 2.65d-109) then
        tmp = (y / z_m) / x
    else
        tmp = y * ((cosh(x) / 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 (x <= 2.65e-109) {
		tmp = (y / z_m) / x;
	} else {
		tmp = y * ((Math.cosh(x) / 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 x <= 2.65e-109:
		tmp = (y / z_m) / x
	else:
		tmp = y * ((math.cosh(x) / 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 (x <= 2.65e-109)
		tmp = Float64(Float64(y / z_m) / x);
	else
		tmp = Float64(y * Float64(Float64(cosh(x) / 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 (x <= 2.65e-109)
		tmp = (y / z_m) / x;
	else
		tmp = y * ((cosh(x) / 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[x, 2.65e-109], N[(N[(y / z$95$m), $MachinePrecision] / x), $MachinePrecision], N[(y * N[(N[(N[Cosh[x], $MachinePrecision] / z$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.6499999999999999e-109

   1. Initial program 91.5%

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

      1. associate-*l/ 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in x around 0 65.2%

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

      1. associate-*r/ 67.0%

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

      2. associate-*l/ 67.0%

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

      3. *-un-lft-identity 67.0%

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

   7. Applied egg-rr 67.0%

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

   if 2.6499999999999999e-109 < x

   1. Initial program 85.4%

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

      1. associate-*l/ 85.4%

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

   3. Simplified 85.4%

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

      1. expm1-log1p-u 54.6%

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

      2. expm1-udef 43.2%

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

      3. associate-*l/ 43.2%

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

      4. div-inv 43.2%

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

      5. associate-*l* 40.8%

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

      6. div-inv 40.8%

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

   6. Applied egg-rr 40.8%

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

      1. expm1-def 51.0%

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

      2. expm1-log1p 79.5%

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

      3. associate-*r/ 85.4%

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

      4. associate-*l/ 85.4%

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

      5. *-commutative 85.4%

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

      6. associate-*l/ 97.7%

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

      7. associate-*r/ 98.8%

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

   8. Simplified 98.8%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{-109}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\cosh x}{z\_m}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.45 \cdot 10^{-96}:\\ \;\;\;\;t\_0 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t\_0}{x}\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (let* ((t_0 (/ (cosh x) z_m)))
   (* z_s (if (<= z_m 1.45e-96) (* t_0 (/ y 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 <= 1.45e-96) {
		tmp = t_0 * (y / 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 <= 1.45d-96) then
        tmp = t_0 * (y / 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 <= 1.45e-96) {
		tmp = t_0 * (y / 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 <= 1.45e-96:
		tmp = t_0 * (y / 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 <= 1.45e-96)
		tmp = Float64(t_0 * Float64(y / 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 <= 1.45e-96)
		tmp = t_0 * (y / 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, 1.45e-96], N[(t$95$0 * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.44999999999999997e-96

   1. Initial program 90.9%

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

      1. associate-*l/ 90.8%

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

   3. Simplified 90.8%

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

   if 1.44999999999999997e-96 < z

   1. Initial program 87.0%

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

      1. associate-*l/ 87.0%

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

   3. Simplified 87.0%

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

      1. expm1-log1p-u 55.9%

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

      2. expm1-udef 43.4%

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

      3. associate-*l/ 43.4%

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

      4. div-inv 43.4%

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

      5. associate-*l* 40.0%

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

      6. div-inv 40.0%

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

   6. Applied egg-rr 40.0%

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

      1. expm1-def 51.5%

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

      2. expm1-log1p 81.5%

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

      3. associate-*r/ 87.0%

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

      4. associate-*l/ 87.0%

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

      5. *-commutative 87.0%

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

      6. associate-*l/ 98.5%

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

      7. associate-*r/ 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.45 \cdot 10^{-96}:\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \left(x \cdot y\right)\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.7 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{y}{x} + t\_0}{z\_m}\\ \mathbf{elif}\;z\_m \leq 1.45 \cdot 10^{+78}:\\ \;\;\;\;\frac{t\_0 \cdot \left(z\_m \cdot x\right) + z\_m \cdot y}{z\_m \cdot \left(z\_m \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z\_m} + \frac{y}{z\_m \cdot x}\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (* x y))))
   (*
    z_s
    (if (<= z_m 3.7e-56)
      (/ (+ (/ y x) t_0) z_m)
      (if (<= z_m 1.45e+78)
        (/ (+ (* t_0 (* z_m x)) (* z_m y)) (* z_m (* z_m x)))
        (+ (* 0.5 (/ (* x y) 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 t_0 = 0.5 * (x * y);
	double tmp;
	if (z_m <= 3.7e-56) {
		tmp = ((y / x) + t_0) / z_m;
	} else if (z_m <= 1.45e+78) {
		tmp = ((t_0 * (z_m * x)) + (z_m * y)) / (z_m * (z_m * x));
	} else {
		tmp = (0.5 * ((x * y) / z_m)) + (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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (x * y)
    if (z_m <= 3.7d-56) then
        tmp = ((y / x) + t_0) / z_m
    else if (z_m <= 1.45d+78) then
        tmp = ((t_0 * (z_m * x)) + (z_m * y)) / (z_m * (z_m * x))
    else
        tmp = (0.5d0 * ((x * y) / z_m)) + (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 t_0 = 0.5 * (x * y);
	double tmp;
	if (z_m <= 3.7e-56) {
		tmp = ((y / x) + t_0) / z_m;
	} else if (z_m <= 1.45e+78) {
		tmp = ((t_0 * (z_m * x)) + (z_m * y)) / (z_m * (z_m * x));
	} else {
		tmp = (0.5 * ((x * y) / z_m)) + (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):
	t_0 = 0.5 * (x * y)
	tmp = 0
	if z_m <= 3.7e-56:
		tmp = ((y / x) + t_0) / z_m
	elif z_m <= 1.45e+78:
		tmp = ((t_0 * (z_m * x)) + (z_m * y)) / (z_m * (z_m * x))
	else:
		tmp = (0.5 * ((x * y) / z_m)) + (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)
	t_0 = Float64(0.5 * Float64(x * y))
	tmp = 0.0
	if (z_m <= 3.7e-56)
		tmp = Float64(Float64(Float64(y / x) + t_0) / z_m);
	elseif (z_m <= 1.45e+78)
		tmp = Float64(Float64(Float64(t_0 * Float64(z_m * x)) + Float64(z_m * y)) / Float64(z_m * Float64(z_m * x)));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x * y) / z_m)) + 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)
	t_0 = 0.5 * (x * y);
	tmp = 0.0;
	if (z_m <= 3.7e-56)
		tmp = ((y / x) + t_0) / z_m;
	elseif (z_m <= 1.45e+78)
		tmp = ((t_0 * (z_m * x)) + (z_m * y)) / (z_m * (z_m * x));
	else
		tmp = (0.5 * ((x * y) / z_m)) + (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_] := Block[{t$95$0 = N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[z$95$m, 3.7e-56], N[(N[(N[(y / x), $MachinePrecision] + t$95$0), $MachinePrecision] / z$95$m), $MachinePrecision], If[LessEqual[z$95$m, 1.45e+78], N[(N[(N[(t$95$0 * N[(z$95$m * x), $MachinePrecision]), $MachinePrecision] + N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] / N[(z$95$m * N[(z$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(x * y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] + N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < 3.7000000000000002e-56

   1. Initial program 91.2%

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

   3. Taylor expanded in x around 0 71.1%

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

   if 3.7000000000000002e-56 < z < 1.45000000000000008e78

   1. Initial program 94.1%

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

      1. associate-*l/ 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in x around 0 77.4%

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

      1. associate-*r/ 77.4%

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

      2. frac-add 82.6%

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

      3. *-commutative 82.6%

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

      4. *-commutative 82.6%

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

   7. Applied egg-rr 82.6%

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

   if 1.45000000000000008e78 < z

   1. Initial program 80.2%

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

      1. associate-*l/ 80.2%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in x around 0 69.6%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.7 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+78}:\\ \;\;\;\;\frac{\left(0.5 \cdot \left(x \cdot y\right)\right) \cdot \left(z \cdot x\right) + z \cdot y}{z \cdot \left(z \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \left(x \cdot y\right)\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{y}{x} + t\_0}{z\_m}\\ \mathbf{elif}\;z\_m \leq 1.35 \cdot 10^{+135}:\\ \;\;\;\;\frac{y + x \cdot t\_0}{z\_m \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z\_m} + \frac{y}{z\_m \cdot x}\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (* x y))))
   (*
    z_s
    (if (<= z_m 5e-96)
      (/ (+ (/ y x) t_0) z_m)
      (if (<= z_m 1.35e+135)
        (/ (+ y (* x t_0)) (* z_m x))
        (+ (* 0.5 (/ (* x y) 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 t_0 = 0.5 * (x * y);
	double tmp;
	if (z_m <= 5e-96) {
		tmp = ((y / x) + t_0) / z_m;
	} else if (z_m <= 1.35e+135) {
		tmp = (y + (x * t_0)) / (z_m * x);
	} else {
		tmp = (0.5 * ((x * y) / z_m)) + (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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (x * y)
    if (z_m <= 5d-96) then
        tmp = ((y / x) + t_0) / z_m
    else if (z_m <= 1.35d+135) then
        tmp = (y + (x * t_0)) / (z_m * x)
    else
        tmp = (0.5d0 * ((x * y) / z_m)) + (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 t_0 = 0.5 * (x * y);
	double tmp;
	if (z_m <= 5e-96) {
		tmp = ((y / x) + t_0) / z_m;
	} else if (z_m <= 1.35e+135) {
		tmp = (y + (x * t_0)) / (z_m * x);
	} else {
		tmp = (0.5 * ((x * y) / z_m)) + (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):
	t_0 = 0.5 * (x * y)
	tmp = 0
	if z_m <= 5e-96:
		tmp = ((y / x) + t_0) / z_m
	elif z_m <= 1.35e+135:
		tmp = (y + (x * t_0)) / (z_m * x)
	else:
		tmp = (0.5 * ((x * y) / z_m)) + (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)
	t_0 = Float64(0.5 * Float64(x * y))
	tmp = 0.0
	if (z_m <= 5e-96)
		tmp = Float64(Float64(Float64(y / x) + t_0) / z_m);
	elseif (z_m <= 1.35e+135)
		tmp = Float64(Float64(y + Float64(x * t_0)) / Float64(z_m * x));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x * y) / z_m)) + 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)
	t_0 = 0.5 * (x * y);
	tmp = 0.0;
	if (z_m <= 5e-96)
		tmp = ((y / x) + t_0) / z_m;
	elseif (z_m <= 1.35e+135)
		tmp = (y + (x * t_0)) / (z_m * x);
	else
		tmp = (0.5 * ((x * y) / z_m)) + (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_] := Block[{t$95$0 = N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[z$95$m, 5e-96], N[(N[(N[(y / x), $MachinePrecision] + t$95$0), $MachinePrecision] / z$95$m), $MachinePrecision], If[LessEqual[z$95$m, 1.35e+135], N[(N[(y + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(x * y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] + N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < 4.99999999999999995e-96

   1. Initial program 90.9%

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

   3. Taylor expanded in x around 0 71.6%

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

   if 4.99999999999999995e-96 < z < 1.34999999999999992e135

   1. Initial program 93.3%

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

      1. associate-*l/ 93.3%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in x around 0 70.1%

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

      1. +-commutative 70.1%

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

      2. associate-/r* 67.0%

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

      3. *-un-lft-identity 67.0%

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

      4. associate-*l/ 67.0%

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

      5. associate-*r/ 70.1%

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

      6. associate-*r/ 70.1%

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

      7. frac-add 76.5%

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

      8. associate-*l/ 76.5%

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

      9. *-un-lft-identity 76.5%

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

      10. *-commutative 76.5%

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

      11. *-commutative 76.5%

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

   7. Applied egg-rr 76.5%

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

   8. Taylor expanded in y around 0 76.6%

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

   if 1.34999999999999992e135 < z

   1. Initial program 75.3%

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

      1. associate-*l/ 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in x around 0 74.8%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+135}:\\ \;\;\;\;\frac{y + x \cdot \left(0.5 \cdot \left(x \cdot y\right)\right)}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z\_m \cdot x} + 0.5 \cdot \left(x \cdot \frac{y}{z\_m}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= y 4.2e+181)
    (/ (+ (/ y x) (* 0.5 (* x y))) z_m)
    (+ (/ y (* z_m x)) (* 0.5 (* x (/ y z_m)))))))

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 (y <= 4.2e+181) {
		tmp = ((y / x) + (0.5 * (x * y))) / z_m;
	} else {
		tmp = (y / (z_m * x)) + (0.5 * (x * (y / z_m)));
	}
	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 (y <= 4.2d+181) then
        tmp = ((y / x) + (0.5d0 * (x * y))) / z_m
    else
        tmp = (y / (z_m * x)) + (0.5d0 * (x * (y / z_m)))
    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 (y <= 4.2e+181) {
		tmp = ((y / x) + (0.5 * (x * y))) / z_m;
	} else {
		tmp = (y / (z_m * x)) + (0.5 * (x * (y / z_m)));
	}
	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 y <= 4.2e+181:
		tmp = ((y / x) + (0.5 * (x * y))) / z_m
	else:
		tmp = (y / (z_m * x)) + (0.5 * (x * (y / z_m)))
	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 (y <= 4.2e+181)
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(x * y))) / z_m);
	else
		tmp = Float64(Float64(y / Float64(z_m * x)) + Float64(0.5 * Float64(x * Float64(y / z_m))));
	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 (y <= 4.2e+181)
		tmp = ((y / x) + (0.5 * (x * y))) / z_m;
	else
		tmp = (y / (z_m * x)) + (0.5 * (x * (y / z_m)));
	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[y, 4.2e+181], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(x * N[(y / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.19999999999999995e181

   1. Initial program 90.0%

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

   3. Taylor expanded in x around 0 66.9%

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

   if 4.19999999999999995e181 < y

   1. Initial program 85.5%

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

      1. associate-*l/ 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in x around 0 96.4%

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

      1. expm1-log1p-u 57.7%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{z}\right)\right)} + \frac{y}{x \cdot z} \]

      2. expm1-udef 57.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x \cdot y}{z}\right)} - 1\right)} + \frac{y}{x \cdot z} \]

      3. div-inv 57.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot y\right) \cdot \frac{1}{z}}\right)} - 1\right) + \frac{y}{x \cdot z} \]

      4. associate-*l* 57.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{x \cdot \left(y \cdot \frac{1}{z}\right)}\right)} - 1\right) + \frac{y}{x \cdot z} \]

      5. *-commutative 57.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(x \cdot \color{blue}{\left(\frac{1}{z} \cdot y\right)}\right)} - 1\right) + \frac{y}{x \cdot z} \]

      6. associate-*l/ 57.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{1 \cdot y}{z}}\right)} - 1\right) + \frac{y}{x \cdot z} \]

      7. *-un-lft-identity 57.7%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(x \cdot \frac{\color{blue}{y}}{z}\right)} - 1\right) + \frac{y}{x \cdot z} \]

   7. Applied egg-rr 57.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot \frac{y}{z}\right)} - 1\right)} + \frac{y}{x \cdot z} \]
   8. Step-by-step derivation

      1. expm1-def 57.7%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{y}{z}\right)\right)} + \frac{y}{x \cdot z} \]

      2. expm1-log1p 92.7%

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

   9. Simplified 92.7%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x} + 0.5 \cdot \left(x \cdot \frac{y}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z\_m} + \frac{y}{z\_m \cdot x}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= y 6e+32)
    (/ (+ (/ y x) (* 0.5 (* x y))) z_m)
    (+ (* 0.5 (/ (* x y) 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 (y <= 6e+32) {
		tmp = ((y / x) + (0.5 * (x * y))) / z_m;
	} else {
		tmp = (0.5 * ((x * y) / z_m)) + (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 (y <= 6d+32) then
        tmp = ((y / x) + (0.5d0 * (x * y))) / z_m
    else
        tmp = (0.5d0 * ((x * y) / z_m)) + (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 (y <= 6e+32) {
		tmp = ((y / x) + (0.5 * (x * y))) / z_m;
	} else {
		tmp = (0.5 * ((x * y) / z_m)) + (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 y <= 6e+32:
		tmp = ((y / x) + (0.5 * (x * y))) / z_m
	else:
		tmp = (0.5 * ((x * y) / z_m)) + (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 (y <= 6e+32)
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(x * y))) / z_m);
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x * y) / z_m)) + 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 (y <= 6e+32)
		tmp = ((y / x) + (0.5 * (x * y))) / z_m;
	else
		tmp = (0.5 * ((x * y) / z_m)) + (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[y, 6e+32], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(0.5 * N[(N[(x * y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] + N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 6e32

   1. Initial program 88.7%

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

   3. Taylor expanded in x around 0 66.9%

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

   if 6e32 < y

   1. Initial program 92.2%

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

      1. associate-*l/ 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in x around 0 80.9%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.6% accurate, 8.9× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= x 1.4) (/ (/ y z_m) x) (* 0.5 (* x (/ y z_m))))))

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 (x <= 1.4) {
		tmp = (y / z_m) / x;
	} else {
		tmp = 0.5 * (x * (y / z_m));
	}
	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 (x <= 1.4d0) then
        tmp = (y / z_m) / x
    else
        tmp = 0.5d0 * (x * (y / z_m))
    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 (x <= 1.4) {
		tmp = (y / z_m) / x;
	} else {
		tmp = 0.5 * (x * (y / z_m));
	}
	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 x <= 1.4:
		tmp = (y / z_m) / x
	else:
		tmp = 0.5 * (x * (y / z_m))
	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 (x <= 1.4)
		tmp = Float64(Float64(y / z_m) / x);
	else
		tmp = Float64(0.5 * Float64(x * Float64(y / z_m)));
	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 (x <= 1.4)
		tmp = (y / z_m) / x;
	else
		tmp = 0.5 * (x * (y / z_m));
	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[x, 1.4], N[(N[(y / z$95$m), $MachinePrecision] / x), $MachinePrecision], N[(0.5 * N[(x * N[(y / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 92.6%

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

      1. associate-*l/ 92.5%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in x around 0 69.3%

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

      1. associate-*r/ 69.9%

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

      2. associate-*l/ 69.9%

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

      3. *-un-lft-identity 69.9%

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

   7. Applied egg-rr 69.9%

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

   if 1.3999999999999999 < x

   1. Initial program 79.3%

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

      1. associate-*l/ 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in x around 0 32.1%

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

   6. Taylor expanded in x around inf 32.1%

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

      1. associate-*r/ 22.4%

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

   8. Simplified 22.4%

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

4. Final simplification 59.2%

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

Alternative 10: 59.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 190:\\ \;\;\;\;\frac{\frac{y}{z\_m}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z\_m}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= x 190.0) (/ (/ y z_m) x) (* 0.5 (* y (/ x z_m))))))

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 (x <= 190.0) {
		tmp = (y / z_m) / x;
	} else {
		tmp = 0.5 * (y * (x / z_m));
	}
	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 (x <= 190.0d0) then
        tmp = (y / z_m) / x
    else
        tmp = 0.5d0 * (y * (x / z_m))
    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 (x <= 190.0) {
		tmp = (y / z_m) / x;
	} else {
		tmp = 0.5 * (y * (x / z_m));
	}
	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 x <= 190.0:
		tmp = (y / z_m) / x
	else:
		tmp = 0.5 * (y * (x / z_m))
	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 (x <= 190.0)
		tmp = Float64(Float64(y / z_m) / x);
	else
		tmp = Float64(0.5 * Float64(y * Float64(x / z_m)));
	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 (x <= 190.0)
		tmp = (y / z_m) / x;
	else
		tmp = 0.5 * (y * (x / z_m));
	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[x, 190.0], N[(N[(y / z$95$m), $MachinePrecision] / x), $MachinePrecision], N[(0.5 * N[(y * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 190

   1. Initial program 92.6%

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

      1. associate-*l/ 92.6%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in x around 0 69.0%

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

      1. associate-*r/ 69.6%

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

      2. associate-*l/ 69.6%

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

      3. *-un-lft-identity 69.6%

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

   7. Applied egg-rr 69.6%

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

   if 190 < x

   1. Initial program 78.9%

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

      1. associate-*l/ 78.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in x around 0 32.5%

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

   6. Taylor expanded in x around inf 32.5%

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

      1. associate-*l/ 31.0%

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

      2. *-commutative 31.0%

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

   8. Simplified 31.0%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 190:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 190:\\ \;\;\;\;\frac{\frac{y}{z\_m}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z\_m}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= x 190.0) (/ (/ y z_m) x) (* 0.5 (/ (* x y) z_m)))))

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 (x <= 190.0) {
		tmp = (y / z_m) / x;
	} else {
		tmp = 0.5 * ((x * y) / z_m);
	}
	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 (x <= 190.0d0) then
        tmp = (y / z_m) / x
    else
        tmp = 0.5d0 * ((x * y) / z_m)
    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 (x <= 190.0) {
		tmp = (y / z_m) / x;
	} else {
		tmp = 0.5 * ((x * y) / z_m);
	}
	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 x <= 190.0:
		tmp = (y / z_m) / x
	else:
		tmp = 0.5 * ((x * y) / z_m)
	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 (x <= 190.0)
		tmp = Float64(Float64(y / z_m) / x);
	else
		tmp = Float64(0.5 * Float64(Float64(x * y) / z_m));
	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 (x <= 190.0)
		tmp = (y / z_m) / x;
	else
		tmp = 0.5 * ((x * y) / z_m);
	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[x, 190.0], N[(N[(y / z$95$m), $MachinePrecision] / x), $MachinePrecision], N[(0.5 * N[(N[(x * y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 190

   1. Initial program 92.6%

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

      1. associate-*l/ 92.6%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in x around 0 69.0%

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

      1. associate-*r/ 69.6%

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

      2. associate-*l/ 69.6%

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

      3. *-un-lft-identity 69.6%

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

   7. Applied egg-rr 69.6%

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

   if 190 < x

   1. Initial program 78.9%

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

      1. associate-*l/ 78.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in x around 0 32.5%

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

   6. Taylor expanded in x around inf 32.5%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 190:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.1% accurate, 9.7× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (/ (* y (+ (* x 0.5) (/ 1.0 x))) z_m)))

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 * ((x * 0.5) + (1.0 / x))) / z_m);
}

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 * ((x * 0.5d0) + (1.0d0 / x))) / z_m)
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 * ((x * 0.5) + (1.0 / x))) / z_m);
}

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 * ((x * 0.5) + (1.0 / x))) / z_m)

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	return Float64(z_s * Float64(Float64(y * Float64(Float64(x * 0.5) + Float64(1.0 / x))) / z_m))
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 * ((x * 0.5) + (1.0 / x))) / z_m);
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[(N[(y * N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.6%

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

3. Taylor expanded in x around 0 68.4%

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

4. Taylor expanded in y around 0 68.3%

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

5. Final simplification 68.3%

   \[\leadsto \frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z} \]
6. Add Preprocessing

Alternative 13: 66.2% accurate, 9.7× speedup

Math

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

FPCore

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

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 / x) + (0.5 * (x * y))) / z_m);
}

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 / x) + (0.5d0 * (x * y))) / z_m)
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 / x) + (0.5 * (x * y))) / z_m);
}

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 / x) + (0.5 * (x * y))) / z_m)

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	return Float64(z_s * Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(x * y))) / z_m))
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 / x) + (0.5 * (x * y))) / z_m);
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[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.6%

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

3. Taylor expanded in x around 0 68.4%

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

4. Final simplification 68.4%

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

Alternative 14: 51.0% accurate, 10.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{y}{x}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z\_m \cdot x}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= y 4.5e+32) (/ (/ 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 (y <= 4.5e+32) {
		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 (y <= 4.5d+32) 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 (y <= 4.5e+32) {
		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 y <= 4.5e+32:
		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 (y <= 4.5e+32)
		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 (y <= 4.5e+32)
		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[y, 4.5e+32], 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 y < 4.5000000000000003e32

   1. Initial program 88.7%

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

   3. Taylor expanded in x around 0 56.1%

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

   if 4.5000000000000003e32 < y

   1. Initial program 92.2%

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

      1. associate-*l/ 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in x around 0 57.9%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.9% accurate, 10.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{y}{x}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z\_m}}{x}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= y 7e+15) (/ (/ 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 (y <= 7e+15) {
		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 (y <= 7d+15) 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 (y <= 7e+15) {
		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 y <= 7e+15:
		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 (y <= 7e+15)
		tmp = Float64(Float64(y / x) / z_m);
	else
		tmp = Float64(Float64(y / 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 (y <= 7e+15)
		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[y, 7e+15], N[(N[(y / x), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(y / z$95$m), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 7e15

   1. Initial program 88.4%

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

   3. Taylor expanded in x around 0 55.7%

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

   if 7e15 < y

   1. Initial program 92.8%

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

      1. associate-*l/ 92.8%

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

   3. Simplified 92.8%

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

   5. Taylor expanded in x around 0 51.7%

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

      1. associate-*r/ 61.6%

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

      2. associate-*l/ 61.6%

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

      3. *-un-lft-identity 61.6%

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

   7. Applied egg-rr 61.6%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 49.6% 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 1 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 89.6%

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

   1. associate-*l/ 89.5%

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

3. Simplified 89.5%

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

5. Taylor expanded in x around 0 53.0%

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

6. Final simplification 53.0%

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

Developer target: 97.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024026 
(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))