Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.5% → 97.9%

Time: 8.8s

Alternatives: 10

Speedup: 1.0×

• 64.8% 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: 84.5% 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: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{+213}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* (cosh x) (/ y x)) z)))
   (if (<= t_0 4e+213) t_0 (/ (* y (/ (cosh x) z)) x))))

C

double code(double x, double y, double z) {
	double t_0 = (cosh(x) * (y / x)) / z;
	double tmp;
	if (t_0 <= 4e+213) {
		tmp = t_0;
	} else {
		tmp = (y * (cosh(x) / z)) / x;
	}
	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 = (cosh(x) * (y / x)) / z
    if (t_0 <= 4d+213) then
        tmp = t_0
    else
        tmp = (y * (cosh(x) / z)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (Math.cosh(x) * (y / x)) / z;
	double tmp;
	if (t_0 <= 4e+213) {
		tmp = t_0;
	} else {
		tmp = (y * (Math.cosh(x) / z)) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (math.cosh(x) * (y / x)) / z
	tmp = 0
	if t_0 <= 4e+213:
		tmp = t_0
	else:
		tmp = (y * (math.cosh(x) / z)) / x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(cosh(x) * Float64(y / x)) / z)
	tmp = 0.0
	if (t_0 <= 4e+213)
		tmp = t_0;
	else
		tmp = Float64(Float64(y * Float64(cosh(x) / z)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (cosh(x) * (y / x)) / z;
	tmp = 0.0;
	if (t_0 <= 4e+213)
		tmp = t_0;
	else
		tmp = (y * (cosh(x) / z)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, 4e+213], t$95$0, N[(N[(y * N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 3.99999999999999994e213

   1. Initial program 99.8%

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

   if 3.99999999999999994e213 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 75.8%

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

      1. associate-*r/ 69.0%

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

      2. associate-/r* 71.1%

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

   3. Simplified 71.1%

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

      1. associate-*r/ 79.8%

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

      2. *-commutative 79.8%

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

      3. frac-times 75.8%

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

      4. associate-*l/ 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 4 \cdot 10^{+213}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{z}}{x}\\ \end{array} \]

Alternative 2: 77.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+194}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-8} \lor \neg \left(x \leq 10^{-14}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.1e+194)
   (* 0.5 (* y (/ x z)))
   (if (or (<= x -1.5e-8) (not (<= x 1e-14)))
     (* (cosh x) (/ y (* x z)))
     (/ (/ y x) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.1e+194) {
		tmp = 0.5 * (y * (x / z));
	} else if ((x <= -1.5e-8) || !(x <= 1e-14)) {
		tmp = cosh(x) * (y / (x * z));
	} else {
		tmp = (y / x) / z;
	}
	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) :: tmp
    if (x <= (-4.1d+194)) then
        tmp = 0.5d0 * (y * (x / z))
    else if ((x <= (-1.5d-8)) .or. (.not. (x <= 1d-14))) then
        tmp = cosh(x) * (y / (x * z))
    else
        tmp = (y / x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.1e+194) {
		tmp = 0.5 * (y * (x / z));
	} else if ((x <= -1.5e-8) || !(x <= 1e-14)) {
		tmp = Math.cosh(x) * (y / (x * z));
	} else {
		tmp = (y / x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.1e+194:
		tmp = 0.5 * (y * (x / z))
	elif (x <= -1.5e-8) or not (x <= 1e-14):
		tmp = math.cosh(x) * (y / (x * z))
	else:
		tmp = (y / x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.1e+194)
		tmp = Float64(0.5 * Float64(y * Float64(x / z)));
	elseif ((x <= -1.5e-8) || !(x <= 1e-14))
		tmp = Float64(cosh(x) * Float64(y / Float64(x * z)));
	else
		tmp = Float64(Float64(y / x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.1e+194)
		tmp = 0.5 * (y * (x / z));
	elseif ((x <= -1.5e-8) || ~((x <= 1e-14)))
		tmp = cosh(x) * (y / (x * z));
	else
		tmp = (y / x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.1e+194], N[(0.5 * N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -1.5e-8], N[Not[LessEqual[x, 1e-14]], $MachinePrecision]], N[(N[Cosh[x], $MachinePrecision] * N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.1e194

   1. Initial program 64.7%

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

   2. Taylor expanded in x around 0 54.5%

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

   3. Taylor expanded in x around inf 54.5%

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

      1. associate-/l* 43.8%

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

      2. associate-/r/ 55.1%

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

   5. Simplified 55.1%

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

   if -4.1e194 < x < -1.49999999999999987e-8 or 9.99999999999999999e-15 < x

   1. Initial program 85.1%

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

      1. associate-*r/ 76.3%

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

      2. associate-/r* 78.9%

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

   3. Simplified 78.9%

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

   if -1.49999999999999987e-8 < x < 9.99999999999999999e-15

   1. Initial program 98.2%

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

   2. Taylor expanded in x around 0 98.2%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+194}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-8} \lor \neg \left(x \leq 10^{-14}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 3: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+35} \lor \neg \left(z \leq 5 \cdot 10^{-126}\right):\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.5e+35) (not (<= z 5e-126)))
   (* (/ y x) (/ (cosh x) z))
   (/ (cosh x) (* x (/ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.5e+35) || !(z <= 5e-126)) {
		tmp = (y / x) * (cosh(x) / z);
	} else {
		tmp = cosh(x) / (x * (z / y));
	}
	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) :: tmp
    if ((z <= (-1.5d+35)) .or. (.not. (z <= 5d-126))) then
        tmp = (y / x) * (cosh(x) / z)
    else
        tmp = cosh(x) / (x * (z / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.5e+35) || !(z <= 5e-126)) {
		tmp = (y / x) * (Math.cosh(x) / z);
	} else {
		tmp = Math.cosh(x) / (x * (z / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.5e+35) or not (z <= 5e-126):
		tmp = (y / x) * (math.cosh(x) / z)
	else:
		tmp = math.cosh(x) / (x * (z / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.5e+35) || !(z <= 5e-126))
		tmp = Float64(Float64(y / x) * Float64(cosh(x) / z));
	else
		tmp = Float64(cosh(x) / Float64(x * Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.5e+35) || ~((z <= 5e-126)))
		tmp = (y / x) * (cosh(x) / z);
	else
		tmp = cosh(x) / (x * (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.5e+35], N[Not[LessEqual[z, 5e-126]], $MachinePrecision]], N[(N[(y / x), $MachinePrecision] * N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] / N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.49999999999999995e35 or 5.00000000000000006e-126 < z

   1. Initial program 90.7%

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

      1. associate-*l/ 90.7%

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

   3. Simplified 90.7%

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

   if -1.49999999999999995e35 < z < 5.00000000000000006e-126

   1. Initial program 89.6%

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

      1. associate-/l* 89.5%

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

   3. Simplified 89.5%

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

      1. associate-/r/ 96.7%

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

   5. Applied egg-rr 96.7%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+35} \lor \neg \left(z \leq 5 \cdot 10^{-126}\right):\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y}}\\ \end{array} \]

Alternative 4: 84.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (y / x) * (cosh(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 = (y / x) * (cosh(x) / z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.1%

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

   1. associate-*l/ 90.1%

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

3. Simplified 90.1%

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

4. Final simplification 90.1%

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

Alternative 5: 95.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.1%

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

   1. associate-*r/ 85.4%

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

   2. associate-/r* 79.2%

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

3. Simplified 79.2%

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

   1. associate-*r/ 83.5%

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

   2. *-commutative 83.5%

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

   3. frac-times 90.1%

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

   4. associate-*l/ 97.0%

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

5. Applied egg-rr 97.0%

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

6. Final simplification 97.0%

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

Alternative 6: 64.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \lor \neg \left(x \leq 1.42\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.4) (not (<= x 1.42))) (* 0.5 (* y (/ x z))) (/ (/ y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4) || !(x <= 1.42)) {
		tmp = 0.5 * (y * (x / z));
	} else {
		tmp = (y / x) / z;
	}
	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) :: tmp
    if ((x <= (-1.4d0)) .or. (.not. (x <= 1.42d0))) then
        tmp = 0.5d0 * (y * (x / z))
    else
        tmp = (y / x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4) || !(x <= 1.42)) {
		tmp = 0.5 * (y * (x / z));
	} else {
		tmp = (y / x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.4) or not (x <= 1.42):
		tmp = 0.5 * (y * (x / z))
	else:
		tmp = (y / x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.4) || !(x <= 1.42))
		tmp = Float64(0.5 * Float64(y * Float64(x / z)));
	else
		tmp = Float64(Float64(y / x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.4) || ~((x <= 1.42)))
		tmp = 0.5 * (y * (x / z));
	else
		tmp = (y / x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.4], N[Not[LessEqual[x, 1.42]], $MachinePrecision]], N[(0.5 * N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999 or 1.4199999999999999 < x

   1. Initial program 81.7%

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

   2. Taylor expanded in x around 0 41.3%

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

   3. Taylor expanded in x around inf 41.3%

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

      1. associate-/l* 35.5%

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

      2. associate-/r/ 43.0%

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

   5. Simplified 43.0%

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

   if -1.3999999999999999 < x < 1.4199999999999999

   1. Initial program 98.3%

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

   2. Taylor expanded in x around 0 97.0%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \lor \neg \left(x \leq 1.42\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 7: 65.1% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (+ (/ y x) (* 0.5 (* x y))) z))

C

double code(double x, double y, double z) {
	return ((y / x) + (0.5 * (x * y))) / 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 = ((y / x) + (0.5d0 * (x * y))) / z
end function

Java

public static double code(double x, double y, double z) {
	return ((y / x) + (0.5 * (x * y))) / z;
}

Python

def code(x, y, z):
	return ((y / x) + (0.5 * (x * y))) / z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(x * y))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((y / x) + (0.5 * (x * y))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Derivation

1. Initial program 90.1%

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

2. Taylor expanded in x around 0 69.8%

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

3. Final simplification 69.8%

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

Alternative 8: 53.7% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+63}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5e+63) (/ y (* x z)) (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e+63) {
		tmp = y / (x * z);
	} else {
		tmp = (y / z) / x;
	}
	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) :: tmp
    if (z <= (-5d+63)) then
        tmp = y / (x * z)
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e+63) {
		tmp = y / (x * z);
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5e+63:
		tmp = y / (x * z)
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5e+63)
		tmp = Float64(y / Float64(x * z));
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5e+63)
		tmp = y / (x * z);
	else
		tmp = (y / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5e+63], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.00000000000000011e63

   1. Initial program 95.4%

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

      1. associate-*r/ 81.8%

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

      2. associate-/r* 72.5%

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

   3. Simplified 72.5%

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

   4. Taylor expanded in x around 0 53.5%

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

   if -5.00000000000000011e63 < z

   1. Initial program 89.0%

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

      1. associate-*r/ 86.2%

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

      2. associate-/r* 80.6%

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

   3. Simplified 80.6%

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

      1. associate-*r/ 84.9%

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

      2. *-commutative 84.9%

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

      3. frac-times 89.0%

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

      4. associate-*l/ 98.9%

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

   5. Applied egg-rr 98.9%

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

   6. Taylor expanded in x around 0 59.0%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+63}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 9: 48.4% accurate, 21.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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 = y / (x * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.1%

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

   1. associate-*r/ 85.4%

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

   2. associate-/r* 79.2%

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

3. Simplified 79.2%

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

4. Taylor expanded in x around 0 47.1%

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

5. Final simplification 47.1%

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

Alternative 10: 48.5% accurate, 21.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (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 = (y / x) / z
end function

Java

public static double code(double x, double y, double z) {
	return (y / x) / z;
}

Python

def code(x, y, z):
	return (y / x) / z

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.1%

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

2. Taylor expanded in x around 0 53.0%

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

3. Final simplification 53.0%

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

Developer target: 96.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023287 
(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))