Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.5% → 96.2%

Time: 8.0s

Alternatives: 12

Speedup: 1.0×

• 63.6% 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: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

   1. associate-*r/ 95.1%

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

   2. associate-/l/ 83.0%

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

   3. associate-*l/ 82.8%

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

   4. *-commutative 82.8%

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

   5. *-commutative 82.8%

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

3. Simplified 82.8%

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

   1. clear-num 82.8%

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

   2. un-div-inv 83.0%

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

   3. *-commutative 83.0%

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

   4. associate-/l* 97.4%

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

5. Applied egg-rr 97.4%

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

6. Final simplification 97.4%

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

Alternative 2: 88.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{if}\;y \leq -1.75 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-129}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-177}:\\ \;\;\;\;y \cdot \frac{\cosh x}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ (/ y z) x))))
   (if (<= y -1.75e+62)
     t_0
     (if (<= y -4.8e-129)
       (* (/ y x) (/ (cosh x) z))
       (if (<= y 3e-177) (* y (/ (cosh x) (* z x))) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = cosh(x) * ((y / z) / x);
	double tmp;
	if (y <= -1.75e+62) {
		tmp = t_0;
	} else if (y <= -4.8e-129) {
		tmp = (y / x) * (cosh(x) / z);
	} else if (y <= 3e-177) {
		tmp = y * (cosh(x) / (z * x));
	} 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 = cosh(x) * ((y / z) / x)
    if (y <= (-1.75d+62)) then
        tmp = t_0
    else if (y <= (-4.8d-129)) then
        tmp = (y / x) * (cosh(x) / z)
    else if (y <= 3d-177) then
        tmp = y * (cosh(x) / (z * x))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.cosh(x) * ((y / z) / x);
	double tmp;
	if (y <= -1.75e+62) {
		tmp = t_0;
	} else if (y <= -4.8e-129) {
		tmp = (y / x) * (Math.cosh(x) / z);
	} else if (y <= 3e-177) {
		tmp = y * (Math.cosh(x) / (z * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cosh(x) * ((y / z) / x)
	tmp = 0
	if y <= -1.75e+62:
		tmp = t_0
	elif y <= -4.8e-129:
		tmp = (y / x) * (math.cosh(x) / z)
	elif y <= 3e-177:
		tmp = y * (math.cosh(x) / (z * x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cosh(x) * Float64(Float64(y / z) / x))
	tmp = 0.0
	if (y <= -1.75e+62)
		tmp = t_0;
	elseif (y <= -4.8e-129)
		tmp = Float64(Float64(y / x) * Float64(cosh(x) / z));
	elseif (y <= 3e-177)
		tmp = Float64(y * Float64(cosh(x) / Float64(z * x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = cosh(x) * ((y / z) / x);
	tmp = 0.0;
	if (y <= -1.75e+62)
		tmp = t_0;
	elseif (y <= -4.8e-129)
		tmp = (y / x) * (cosh(x) / z);
	elseif (y <= 3e-177)
		tmp = y * (cosh(x) / (z * x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.75e+62], t$95$0, If[LessEqual[y, -4.8e-129], N[(N[(y / x), $MachinePrecision] * N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3e-177], N[(y * N[(N[Cosh[x], $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.74999999999999992e62 or 3.00000000000000008e-177 < y

   1. Initial program 88.7%

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

      1. associate-*r/ 86.6%

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

      2. associate-/l/ 86.9%

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

      3. associate-/r* 96.4%

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

   3. Simplified 96.4%

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

   if -1.74999999999999992e62 < y < -4.79999999999999977e-129

   1. Initial program 91.4%

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

      1. *-commutative 91.4%

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

      2. associate-*r/ 91.2%

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

   3. Simplified 91.2%

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

   if -4.79999999999999977e-129 < y < 3.00000000000000008e-177

   1. Initial program 67.0%

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

      1. associate-*r/ 99.8%

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

      2. associate-/l/ 81.4%

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

      3. associate-*l/ 81.3%

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

      4. *-commutative 81.3%

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

      5. *-commutative 81.3%

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

   3. Simplified 81.3%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+62}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-129}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-177}:\\ \;\;\;\;y \cdot \frac{\cosh x}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 3: 87.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-32} \lor \neg \left(y \leq 5.4 \cdot 10^{-177}\right):\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\cosh x}{z \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.4e-32) (not (<= y 5.4e-177)))
   (* (cosh x) (/ (/ y z) x))
   (* y (/ (cosh x) (* z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.4e-32) || !(y <= 5.4e-177)) {
		tmp = cosh(x) * ((y / z) / x);
	} 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) :: tmp
    if ((y <= (-1.4d-32)) .or. (.not. (y <= 5.4d-177))) then
        tmp = cosh(x) * ((y / z) / x)
    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 tmp;
	if ((y <= -1.4e-32) || !(y <= 5.4e-177)) {
		tmp = Math.cosh(x) * ((y / z) / x);
	} else {
		tmp = y * (Math.cosh(x) / (z * x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.4e-32) or not (y <= 5.4e-177):
		tmp = math.cosh(x) * ((y / z) / x)
	else:
		tmp = y * (math.cosh(x) / (z * x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.4e-32) || !(y <= 5.4e-177))
		tmp = Float64(cosh(x) * Float64(Float64(y / z) / x));
	else
		tmp = Float64(y * Float64(cosh(x) / Float64(z * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.4e-32) || ~((y <= 5.4e-177)))
		tmp = cosh(x) * ((y / z) / x);
	else
		tmp = y * (cosh(x) / (z * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.4e-32], N[Not[LessEqual[y, 5.4e-177]], $MachinePrecision]], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[Cosh[x], $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3999999999999999e-32 or 5.4000000000000004e-177 < y

   1. Initial program 90.3%

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

      1. associate-*r/ 85.6%

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

      2. associate-/l/ 85.5%

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

      3. associate-/r* 94.0%

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

   3. Simplified 94.0%

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

   if -1.3999999999999999e-32 < y < 5.4000000000000004e-177

   1. Initial program 70.8%

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

      1. associate-*r/ 99.9%

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

      2. associate-/l/ 76.8%

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

      3. associate-*l/ 76.7%

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

      4. *-commutative 76.7%

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

      5. *-commutative 76.7%

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

   3. Simplified 76.7%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-32} \lor \neg \left(y \leq 5.4 \cdot 10^{-177}\right):\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\cosh x}{z \cdot x}\\ \end{array} \]

Alternative 4: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{+218}:\\ \;\;\;\;y \cdot \frac{\cosh x}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z \cdot z}{\left(z \cdot x\right) \cdot 0.5 + \frac{z}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.9e+218)
   (* y (/ (cosh x) (* z x)))
   (/ y (/ (* z z) (+ (* (* z x) 0.5) (/ z x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.9e+218) {
		tmp = y * (cosh(x) / (z * x));
	} else {
		tmp = y / ((z * z) / (((z * x) * 0.5) + (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 (x <= 1.9d+218) then
        tmp = y * (cosh(x) / (z * x))
    else
        tmp = y / ((z * z) / (((z * x) * 0.5d0) + (z / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.9e+218) {
		tmp = y * (Math.cosh(x) / (z * x));
	} else {
		tmp = y / ((z * z) / (((z * x) * 0.5) + (z / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1.9e+218:
		tmp = y * (math.cosh(x) / (z * x))
	else:
		tmp = y / ((z * z) / (((z * x) * 0.5) + (z / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.9e+218)
		tmp = Float64(y * Float64(cosh(x) / Float64(z * x)));
	else
		tmp = Float64(y / Float64(Float64(z * z) / Float64(Float64(Float64(z * x) * 0.5) + Float64(z / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.9e+218)
		tmp = y * (cosh(x) / (z * x));
	else
		tmp = y / ((z * z) / (((z * x) * 0.5) + (z / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.9e+218], N[(y * N[(N[Cosh[x], $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[(z * z), $MachinePrecision] / N[(N[(N[(z * x), $MachinePrecision] * 0.5), $MachinePrecision] + N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.90000000000000006e218

   1. Initial program 86.7%

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

      1. associate-*r/ 94.5%

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

      2. associate-/l/ 86.6%

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

      3. associate-*l/ 86.5%

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

      4. *-commutative 86.5%

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

      5. *-commutative 86.5%

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

   3. Simplified 86.5%

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

   if 1.90000000000000006e218 < x

   1. Initial program 59.3%

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

   2. Taylor expanded in x around 0 67.9%

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

      1. +-commutative 67.9%

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

      2. associate-/l* 61.3%

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

      3. *-commutative 61.3%

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

   4. Simplified 61.3%

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

      1. +-commutative 61.3%

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

      2. *-commutative 61.3%

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

      3. frac-2neg 61.3%

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

      4. associate-*r/ 61.3%

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

      5. frac-add 51.9%

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

      6. distribute-rgt-neg-in 51.9%

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

      7. *-commutative 51.9%

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

      8. distribute-rgt-neg-in 51.9%

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

   6. Applied egg-rr 51.9%

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

      1. +-commutative 51.9%

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

      2. *-commutative 51.9%

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

      3. distribute-rgt-neg-out 51.9%

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

      4. unsub-neg 51.9%

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

      5. *-commutative 51.9%

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

      6. associate-*r* 37.2%

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

      7. *-commutative 37.2%

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

      8. associate-*l* 52.0%

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

      9. distribute-rgt-neg-in 52.0%

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

      10. *-commutative 52.0%

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

      11. associate-*r* 52.0%

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

      12. *-commutative 52.0%

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

      13. distribute-rgt-neg-in 52.0%

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

      14. metadata-eval 52.0%

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

      15. *-commutative 52.0%

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

      16. associate-*l* 52.0%

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

      17. *-commutative 52.0%

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

      18. associate-*l* 63.3%

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

   8. Simplified 63.3%

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

   9. Taylor expanded in y around -inf 63.4%

      \[\leadsto \color{blue}{\frac{y \cdot \left(0.5 \cdot \left(z \cdot x\right) - -1 \cdot \frac{z}{x}\right)}{{z}^{2}}} \]
   10. Step-by-step derivation

       1. associate-/l* 70.4%

          \[\leadsto \color{blue}{\frac{y}{\frac{{z}^{2}}{0.5 \cdot \left(z \cdot x\right) - -1 \cdot \frac{z}{x}}}} \]

       2. unpow2 70.4%

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

       3. *-commutative 70.4%

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

       4. *-commutative 70.4%

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

       5. associate-*r/ 70.4%

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

       6. neg-mul-1 70.4%

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

   11. Simplified 70.4%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{+218}:\\ \;\;\;\;y \cdot \frac{\cosh x}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z \cdot z}{\left(z \cdot x\right) \cdot 0.5 + \frac{z}{x}}}\\ \end{array} \]

Alternative 5: 69.0% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\frac{y}{z \cdot x} + 0.5 \cdot \frac{y \cdot x}{z}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y}{x} + x \cdot \left(y \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \frac{y}{z} + x \cdot \left(0.5 \cdot \left(y \cdot x\right)\right)}{z \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e+16)
   (+ (/ y (* z x)) (* 0.5 (/ (* y x) z)))
   (if (<= y 2.5e-62)
     (/ (+ (/ y x) (* x (* y 0.5))) z)
     (/ (+ (* z (/ y z)) (* x (* 0.5 (* y x)))) (* z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e+16) {
		tmp = (y / (z * x)) + (0.5 * ((y * x) / z));
	} else if (y <= 2.5e-62) {
		tmp = ((y / x) + (x * (y * 0.5))) / z;
	} else {
		tmp = ((z * (y / z)) + (x * (0.5 * (y * 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) :: tmp
    if (y <= (-1d+16)) then
        tmp = (y / (z * x)) + (0.5d0 * ((y * x) / z))
    else if (y <= 2.5d-62) then
        tmp = ((y / x) + (x * (y * 0.5d0))) / z
    else
        tmp = ((z * (y / z)) + (x * (0.5d0 * (y * x)))) / (z * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e+16) {
		tmp = (y / (z * x)) + (0.5 * ((y * x) / z));
	} else if (y <= 2.5e-62) {
		tmp = ((y / x) + (x * (y * 0.5))) / z;
	} else {
		tmp = ((z * (y / z)) + (x * (0.5 * (y * x)))) / (z * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1e+16:
		tmp = (y / (z * x)) + (0.5 * ((y * x) / z))
	elif y <= 2.5e-62:
		tmp = ((y / x) + (x * (y * 0.5))) / z
	else:
		tmp = ((z * (y / z)) + (x * (0.5 * (y * x)))) / (z * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1e+16)
		tmp = Float64(Float64(y / Float64(z * x)) + Float64(0.5 * Float64(Float64(y * x) / z)));
	elseif (y <= 2.5e-62)
		tmp = Float64(Float64(Float64(y / x) + Float64(x * Float64(y * 0.5))) / z);
	else
		tmp = Float64(Float64(Float64(z * Float64(y / z)) + Float64(x * Float64(0.5 * Float64(y * x)))) / Float64(z * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1e+16)
		tmp = (y / (z * x)) + (0.5 * ((y * x) / z));
	elseif (y <= 2.5e-62)
		tmp = ((y / x) + (x * (y * 0.5))) / z;
	else
		tmp = ((z * (y / z)) + (x * (0.5 * (y * x)))) / (z * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1e+16], N[(N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.5e-62], N[(N[(N[(y / x), $MachinePrecision] + N[(x * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision] + N[(x * N[(0.5 * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1e16

   1. Initial program 93.1%

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

   2. Taylor expanded in x around 0 79.2%

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

   if -1e16 < y < 2.5000000000000001e-62

   1. Initial program 74.0%

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

   2. Taylor expanded in x around 0 60.2%

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

      1. associate-*r* 60.2%

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

   4. Simplified 60.2%

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

   if 2.5000000000000001e-62 < y

   1. Initial program 89.8%

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

   2. Taylor expanded in x around 0 78.4%

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

      1. associate-/r* 78.3%

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

      2. associate-*r/ 77.2%

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

      3. frac-add 80.4%

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

   4. Applied egg-rr 80.4%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\frac{y}{z \cdot x} + 0.5 \cdot \frac{y \cdot x}{z}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y}{x} + x \cdot \left(y \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \frac{y}{z} + x \cdot \left(0.5 \cdot \left(y \cdot x\right)\right)}{z \cdot x}\\ \end{array} \]

Alternative 6: 70.0% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+15} \lor \neg \left(y \leq 3 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{y}{z \cdot x} + 0.5 \cdot \frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + x \cdot \left(y \cdot 0.5\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3e+15) (not (<= y 3e-62)))
   (+ (/ y (* z x)) (* 0.5 (/ (* y x) z)))
   (/ (+ (/ y x) (* x (* y 0.5))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3e+15) || !(y <= 3e-62)) {
		tmp = (y / (z * x)) + (0.5 * ((y * x) / z));
	} else {
		tmp = ((y / x) + (x * (y * 0.5))) / 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 ((y <= (-3d+15)) .or. (.not. (y <= 3d-62))) then
        tmp = (y / (z * x)) + (0.5d0 * ((y * x) / z))
    else
        tmp = ((y / x) + (x * (y * 0.5d0))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3e+15) || !(y <= 3e-62)) {
		tmp = (y / (z * x)) + (0.5 * ((y * x) / z));
	} else {
		tmp = ((y / x) + (x * (y * 0.5))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3e+15) or not (y <= 3e-62):
		tmp = (y / (z * x)) + (0.5 * ((y * x) / z))
	else:
		tmp = ((y / x) + (x * (y * 0.5))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3e+15) || !(y <= 3e-62))
		tmp = Float64(Float64(y / Float64(z * x)) + Float64(0.5 * Float64(Float64(y * x) / z)));
	else
		tmp = Float64(Float64(Float64(y / x) + Float64(x * Float64(y * 0.5))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3e+15) || ~((y <= 3e-62)))
		tmp = (y / (z * x)) + (0.5 * ((y * x) / z));
	else
		tmp = ((y / x) + (x * (y * 0.5))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3e+15], N[Not[LessEqual[y, 3e-62]], $MachinePrecision]], N[(N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] + N[(x * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3e15 or 3.0000000000000001e-62 < y

   1. Initial program 91.4%

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

   2. Taylor expanded in x around 0 78.8%

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

   if -3e15 < y < 3.0000000000000001e-62

   1. Initial program 74.0%

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

   2. Taylor expanded in x around 0 60.2%

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

      1. associate-*r* 60.2%

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

   4. Simplified 60.2%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+15} \lor \neg \left(y \leq 3 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{y}{z \cdot x} + 0.5 \cdot \frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + x \cdot \left(y \cdot 0.5\right)}{z}\\ \end{array} \]

Alternative 7: 66.4% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(\frac{1}{z \cdot x} + 0.5 \cdot \frac{x}{z}\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* y (+ (/ 1.0 (* z x)) (* 0.5 (/ x z)))))

C

double code(double x, double y, double z) {
	return y * ((1.0 / (z * x)) + (0.5 * (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 * ((1.0d0 / (z * x)) + (0.5d0 * (x / z)))
end function

Java

public static double code(double x, double y, double z) {
	return y * ((1.0 / (z * x)) + (0.5 * (x / z)));
}

Python

def code(x, y, z):
	return y * ((1.0 / (z * x)) + (0.5 * (x / z)))

Julia

function code(x, y, z)
	return Float64(y * Float64(Float64(1.0 / Float64(z * x)) + Float64(0.5 * Float64(x / z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = y * ((1.0 / (z * x)) + (0.5 * (x / z)));
end

Wolfram

code[x_, y_, z_] := N[(y * N[(N[(1.0 / N[(z * x), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.8%

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

   1. associate-*r/ 95.1%

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

   2. associate-/l/ 83.0%

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

   3. associate-*l/ 82.8%

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

   4. *-commutative 82.8%

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

   5. *-commutative 82.8%

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

3. Simplified 82.8%

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

4. Taylor expanded in x around 0 66.7%

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

5. Final simplification 66.7%

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

Alternative 8: 62.5% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.42 \lor \neg \left(x \leq 40000\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.42) (not (<= x 40000.0)))
   (* 0.5 (* x (/ y z)))
   (/ y (* z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.42) || !(x <= 40000.0)) {
		tmp = 0.5 * (x * (y / 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 ((x <= (-1.42d0)) .or. (.not. (x <= 40000.0d0))) then
        tmp = 0.5d0 * (x * (y / 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 ((x <= -1.42) || !(x <= 40000.0)) {
		tmp = 0.5 * (x * (y / z));
	} else {
		tmp = y / (z * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.42) or not (x <= 40000.0):
		tmp = 0.5 * (x * (y / z))
	else:
		tmp = y / (z * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.42) || !(x <= 40000.0))
		tmp = Float64(0.5 * Float64(x * Float64(y / z)));
	else
		tmp = Float64(y / Float64(z * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.42) || ~((x <= 40000.0)))
		tmp = 0.5 * (x * (y / z));
	else
		tmp = y / (z * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.42], N[Not[LessEqual[x, 40000.0]], $MachinePrecision]], N[(0.5 * N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4199999999999999 or 4e4 < x

   1. Initial program 76.6%

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

   2. Taylor expanded in x around 0 40.3%

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

      1. associate-*r* 40.3%

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

   4. Simplified 40.3%

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

   5. Taylor expanded in x around inf 41.0%

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

      1. associate-*l/ 34.2%

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

      2. *-commutative 34.2%

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

   7. Simplified 34.2%

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

   if -1.4199999999999999 < x < 4e4

   1. Initial program 90.5%

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

   2. Taylor expanded in x around 0 93.7%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.42 \lor \neg \left(x \leq 40000\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]

Alternative 9: 66.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.42 \lor \neg \left(x \leq 40000\right):\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.42) (not (<= x 40000.0)))
   (* 0.5 (/ (* y x) z))
   (/ y (* z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.42) || !(x <= 40000.0)) {
		tmp = 0.5 * ((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 ((x <= (-1.42d0)) .or. (.not. (x <= 40000.0d0))) then
        tmp = 0.5d0 * ((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 ((x <= -1.42) || !(x <= 40000.0)) {
		tmp = 0.5 * ((y * x) / z);
	} else {
		tmp = y / (z * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.42) or not (x <= 40000.0):
		tmp = 0.5 * ((y * x) / z)
	else:
		tmp = y / (z * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.42) || !(x <= 40000.0))
		tmp = Float64(0.5 * Float64(Float64(y * x) / z));
	else
		tmp = Float64(y / Float64(z * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.42) || ~((x <= 40000.0)))
		tmp = 0.5 * ((y * x) / z);
	else
		tmp = y / (z * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.42], N[Not[LessEqual[x, 40000.0]], $MachinePrecision]], N[(0.5 * N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4199999999999999 or 4e4 < x

   1. Initial program 76.6%

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

   2. Taylor expanded in x around 0 40.3%

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

      1. associate-*r* 40.3%

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

   4. Simplified 40.3%

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

   5. Taylor expanded in x around inf 41.0%

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

   if -1.4199999999999999 < x < 4e4

   1. Initial program 90.5%

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

   2. Taylor expanded in x around 0 93.7%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.42 \lor \neg \left(x \leq 40000\right):\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]

Alternative 10: 65.7% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

2. Taylor expanded in x around 0 65.5%

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

   1. associate-*r* 65.5%

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

4. Simplified 65.5%

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

5. Final simplification 65.5%

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

Alternative 11: 54.9% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-20} \lor \neg \left(z \leq 1.12 \cdot 10^{-251}\right):\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1e-20) (not (<= z 1.12e-251))) (/ y (* z x)) (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1e-20) || !(z <= 1.12e-251)) {
		tmp = y / (z * x);
	} 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 <= (-1d-20)) .or. (.not. (z <= 1.12d-251))) then
        tmp = y / (z * x)
    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 <= -1e-20) || !(z <= 1.12e-251)) {
		tmp = y / (z * x);
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1e-20) or not (z <= 1.12e-251):
		tmp = y / (z * x)
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1e-20) || !(z <= 1.12e-251))
		tmp = Float64(y / Float64(z * x));
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1e-20) || ~((z <= 1.12e-251)))
		tmp = y / (z * x);
	else
		tmp = (y / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1e-20], N[Not[LessEqual[z, 1.12e-251]], $MachinePrecision]], N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.99999999999999945e-21 or 1.12000000000000007e-251 < z

   1. Initial program 82.3%

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

   2. Taylor expanded in x around 0 52.4%

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

   if -9.99999999999999945e-21 < z < 1.12000000000000007e-251

   1. Initial program 88.0%

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

      1. associate-*r/ 99.9%

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

      2. associate-/l/ 90.5%

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

      3. associate-*l/ 90.2%

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

      4. *-commutative 90.2%

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

      5. *-commutative 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in x around 0 51.5%

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

      1. div-inv 51.8%

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

      2. associate-/r* 69.6%

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

   6. Applied egg-rr 69.6%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-20} \lor \neg \left(z \leq 1.12 \cdot 10^{-251}\right):\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 12: 50.1% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

2. Taylor expanded in x around 0 52.2%

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

3. Final simplification 52.2%

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

Developer target: 97.1% 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 2023195 
(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))