Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.5% → 96.4%

Time: 10.9s

Alternatives: 14

Speedup: 1.0×

• 64.0% 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: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.5%

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

   1. associate-*l/ 81.5%

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

3. Simplified 81.5%

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

   1. associate-*r/ 98.3%

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

6. Applied egg-rr 98.3%

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

7. Final simplification 98.3%

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

Alternative 2: 90.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.5%

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

   1. associate-*l/ 81.5%

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

3. Simplified 81.5%

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

   1. expm1-log1p-u 46.1%

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

   2. expm1-udef 38.6%

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

   3. associate-*l/ 38.6%

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

   4. div-inv 38.6%

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

   5. associate-*l* 36.7%

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

   6. div-inv 36.7%

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

6. Applied egg-rr 36.7%

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

   1. expm1-def 44.2%

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

   2. expm1-log1p 77.6%

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

   3. associate-/r* 79.2%

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

   4. associate-*r/ 83.5%

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

   5. times-frac 92.5%

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

8. Simplified 92.5%

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

9. Final simplification 92.5%

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

Alternative 3: 70.7% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-39}:\\ \;\;\;\;\frac{\left(-y\right) - y \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)}{z \cdot \left(-x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{-0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}{y}}{\frac{z}{y} \cdot \left(-x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-39)
   (/ (- (- y) (* y (* x (* x 0.5)))) (* z (- x)))
   (/ (+ -1.0 (/ (* -0.5 (* y (* x x))) y)) (* (/ z y) (- x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-39) {
		tmp = (-y - (y * (x * (x * 0.5)))) / (z * -x);
	} else {
		tmp = (-1.0 + ((-0.5 * (y * (x * x))) / y)) / ((z / y) * -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 <= 2d-39) then
        tmp = (-y - (y * (x * (x * 0.5d0)))) / (z * -x)
    else
        tmp = ((-1.0d0) + (((-0.5d0) * (y * (x * x))) / y)) / ((z / y) * -x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-39) {
		tmp = (-y - (y * (x * (x * 0.5)))) / (z * -x);
	} else {
		tmp = (-1.0 + ((-0.5 * (y * (x * x))) / y)) / ((z / y) * -x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e-39:
		tmp = (-y - (y * (x * (x * 0.5)))) / (z * -x)
	else:
		tmp = (-1.0 + ((-0.5 * (y * (x * x))) / y)) / ((z / y) * -x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e-39)
		tmp = Float64(Float64(Float64(-y) - Float64(y * Float64(x * Float64(x * 0.5)))) / Float64(z * Float64(-x)));
	else
		tmp = Float64(Float64(-1.0 + Float64(Float64(-0.5 * Float64(y * Float64(x * x))) / y)) / Float64(Float64(z / y) * Float64(-x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e-39)
		tmp = (-y - (y * (x * (x * 0.5)))) / (z * -x);
	else
		tmp = (-1.0 + ((-0.5 * (y * (x * x))) / y)) / ((z / y) * -x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e-39], N[(N[((-y) - N[(y * N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * (-x)), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + N[(N[(-0.5 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / N[(N[(z / y), $MachinePrecision] * (-x)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999986e-39

   1. Initial program 79.6%

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

      1. associate-*l/ 79.6%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in x around 0 62.2%

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

      1. associate-/r* 61.7%

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

      2. *-un-lft-identity 61.7%

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

      3. associate-*l/ 61.7%

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

      4. associate-*r/ 64.8%

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

      5. div-inv 64.8%

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

      6. associate-*l/ 64.8%

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

      7. *-un-lft-identity 64.8%

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

   7. Applied egg-rr 64.8%

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{y}{z} \cdot \frac{1}{x}} \]
   8. Step-by-step derivation

      1. un-div-inv 64.8%

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

   9. Applied egg-rr 64.8%

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

       1. +-commutative 64.8%

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

       2. frac-2neg 64.8%

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

       3. associate-*r/ 64.8%

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

       4. frac-add 59.7%

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

       5. distribute-neg-frac 59.7%

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

       6. *-commutative 59.7%

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

   11. Applied egg-rr 59.7%

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

       1. associate-*l/ 53.0%

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

       2. associate-/l* 61.3%

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

       3. *-rgt-identity 61.3%

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

       4. associate-*r/ 61.3%

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

       5. rgt-mult-inverse 61.3%

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

       6. distribute-neg-frac 61.3%

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

       7. /-rgt-identity 61.3%

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

       8. distribute-lft-neg-out 61.3%

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

       9. sub-neg 61.3%

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

       10. associate-*r* 61.3%

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

       11. *-commutative 61.3%

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

       12. associate-*r* 66.4%

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

       13. distribute-lft-neg-out 66.4%

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

       14. distribute-rgt-neg-out 66.4%

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

   13. Simplified 66.4%

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

   if 1.99999999999999986e-39 < y

   1. Initial program 86.4%

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

      1. associate-*l/ 86.4%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in x around 0 76.6%

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

      1. associate-/r* 64.6%

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

      2. *-un-lft-identity 64.6%

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

      3. associate-*l/ 64.6%

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

      4. associate-*r/ 76.6%

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

      5. div-inv 76.5%

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

      6. associate-*l/ 76.5%

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

      7. *-un-lft-identity 76.5%

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

   7. Applied egg-rr 76.5%

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{y}{z} \cdot \frac{1}{x}} \]
   8. Step-by-step derivation

      1. un-div-inv 76.6%

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

   9. Applied egg-rr 76.6%

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

       1. associate-*r/ 76.6%

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

       2. associate-/l/ 76.6%

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

       3. clear-num 76.6%

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

       4. associate-*l/ 66.3%

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

       5. associate-/l/ 66.3%

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

       6. frac-2neg 66.3%

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

       7. frac-add 67.5%

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

       8. *-commutative 67.5%

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

       9. distribute-neg-frac 67.5%

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

       10. distribute-neg-frac 67.5%

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

       11. metadata-eval 67.5%

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

       12. distribute-neg-frac 67.5%

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

   11. Applied egg-rr 67.5%

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

       1. +-commutative 67.5%

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

       2. *-commutative 67.5%

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

       3. associate-*l/ 67.5%

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

       4. neg-mul-1 67.5%

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

       5. distribute-frac-neg 67.5%

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

       6. *-rgt-identity 67.5%

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

       7. associate-*r/ 67.5%

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

       8. rgt-mult-inverse 67.5%

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

       9. metadata-eval 67.5%

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

       10. associate-*r/ 73.0%

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

       11. distribute-rgt-neg-out 73.0%

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

       12. associate-*l* 73.0%

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

       13. distribute-lft-neg-in 73.0%

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

       14. metadata-eval 73.0%

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

       15. associate-*l* 73.0%

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

       16. associate-*r/ 76.1%

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

       17. associate-*l/ 83.2%

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

   13. Simplified 83.2%

       \[\leadsto \color{blue}{\frac{-1 + \frac{-0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}{y}}{\frac{z}{y} \cdot \left(-x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-39}:\\ \;\;\;\;\frac{\left(-y\right) - y \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)}{z \cdot \left(-x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{-0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}{y}}{\frac{z}{y} \cdot \left(-x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.4% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+187}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 6.2e-58)
   (/ (/ y z) x)
   (if (<= x 2.6e+187)
     (/ (* y (+ (* x 0.5) (/ 1.0 x))) z)
     (* y (* x (/ 0.5 z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.2e-58) {
		tmp = (y / z) / x;
	} else if (x <= 2.6e+187) {
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z;
	} else {
		tmp = y * (x * (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 (x <= 6.2d-58) then
        tmp = (y / z) / x
    else if (x <= 2.6d+187) then
        tmp = (y * ((x * 0.5d0) + (1.0d0 / x))) / z
    else
        tmp = y * (x * (0.5d0 / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.2e-58) {
		tmp = (y / z) / x;
	} else if (x <= 2.6e+187) {
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z;
	} else {
		tmp = y * (x * (0.5 / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 6.2e-58:
		tmp = (y / z) / x
	elif x <= 2.6e+187:
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z
	else:
		tmp = y * (x * (0.5 / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 6.2e-58)
		tmp = Float64(Float64(y / z) / x);
	elseif (x <= 2.6e+187)
		tmp = Float64(Float64(y * Float64(Float64(x * 0.5) + Float64(1.0 / x))) / z);
	else
		tmp = Float64(y * Float64(x * Float64(0.5 / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 6.2e-58)
		tmp = (y / z) / x;
	elseif (x <= 2.6e+187)
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z;
	else
		tmp = y * (x * (0.5 / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 6.2e-58], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.6e+187], N[(N[(y * N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(x * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.1999999999999998e-58

   1. Initial program 80.4%

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

      1. associate-*l/ 80.4%

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

   3. Simplified 80.4%

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

      1. associate-*r/ 97.6%

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

   6. Applied egg-rr 97.6%

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

   7. Taylor expanded in x around 0 67.4%

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

   if 6.1999999999999998e-58 < x < 2.5999999999999999e187

   1. Initial program 92.7%

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

   3. Taylor expanded in x around 0 46.2%

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

   4. Taylor expanded in y around 0 46.3%

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

   if 2.5999999999999999e187 < x

   1. Initial program 64.0%

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

      1. associate-*l/ 64.0%

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

   3. Simplified 64.0%

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

   5. Taylor expanded in x around 0 54.3%

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

   6. Taylor expanded in x around inf 54.3%

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

      1. associate-*r/ 54.3%

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

      2. associate-*r* 54.3%

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

      3. *-commutative 54.3%

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

      4. associate-*r/ 65.3%

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

      5. associate-/l* 65.3%

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

   8. Simplified 65.3%

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

      1. associate-/r/ 65.3%

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

   10. Applied egg-rr 65.3%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+187}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.9% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+158}:\\ \;\;\;\;\frac{\left(-y\right) - y \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)}{z \cdot \left(-x\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4e+158)
   (/ (- (- y) (* y (* x (* x 0.5)))) (* z (- x)))
   (+ (* 0.5 (/ (* x y) z)) (/ y (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e+158) {
		tmp = (-y - (y * (x * (x * 0.5)))) / (z * -x);
	} else {
		tmp = (0.5 * ((x * y) / z)) + (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 (y <= 4d+158) then
        tmp = (-y - (y * (x * (x * 0.5d0)))) / (z * -x)
    else
        tmp = (0.5d0 * ((x * y) / z)) + (y / (x * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e+158) {
		tmp = (-y - (y * (x * (x * 0.5)))) / (z * -x);
	} else {
		tmp = (0.5 * ((x * y) / z)) + (y / (x * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4e+158:
		tmp = (-y - (y * (x * (x * 0.5)))) / (z * -x)
	else:
		tmp = (0.5 * ((x * y) / z)) + (y / (x * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4e+158)
		tmp = Float64(Float64(Float64(-y) - Float64(y * Float64(x * Float64(x * 0.5)))) / Float64(z * Float64(-x)));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x * y) / z)) + Float64(y / Float64(x * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4e+158)
		tmp = (-y - (y * (x * (x * 0.5)))) / (z * -x);
	else
		tmp = (0.5 * ((x * y) / z)) + (y / (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4e+158], N[(N[((-y) - N[(y * N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * (-x)), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.99999999999999981e158

   1. Initial program 80.9%

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

      1. associate-*l/ 80.9%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in x around 0 63.3%

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

      1. associate-/r* 61.2%

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

      2. *-un-lft-identity 61.2%

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

      3. associate-*l/ 61.3%

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

      4. associate-*r/ 65.4%

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

      5. div-inv 65.4%

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

      6. associate-*l/ 65.5%

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

      7. *-un-lft-identity 65.5%

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

   7. Applied egg-rr 65.5%

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{y}{z} \cdot \frac{1}{x}} \]
   8. Step-by-step derivation

      1. un-div-inv 65.5%

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

   9. Applied egg-rr 65.5%

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

       1. +-commutative 65.5%

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

       2. frac-2neg 65.5%

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

       3. associate-*r/ 65.5%

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

       4. frac-add 62.4%

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

       5. distribute-neg-frac 62.4%

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

       6. *-commutative 62.4%

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

   11. Applied egg-rr 62.4%

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

       1. associate-*l/ 57.2%

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

       2. associate-/l* 63.8%

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

       3. *-rgt-identity 63.8%

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

       4. associate-*r/ 63.7%

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

       5. rgt-mult-inverse 63.8%

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

       6. distribute-neg-frac 63.8%

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

       7. /-rgt-identity 63.8%

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

       8. distribute-lft-neg-out 63.8%

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

       9. sub-neg 63.8%

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

       10. associate-*r* 63.8%

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

       11. *-commutative 63.8%

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

       12. associate-*r* 68.0%

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

       13. distribute-lft-neg-out 68.0%

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

       14. distribute-rgt-neg-out 68.0%

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

   13. Simplified 68.0%

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

   if 3.99999999999999981e158 < y

   1. Initial program 85.3%

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

      1. associate-*l/ 85.2%

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

   3. Simplified 85.2%

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

   5. Taylor expanded in x around 0 86.0%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+158}:\\ \;\;\;\;\frac{\left(-y\right) - y \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)}{z \cdot \left(-x\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{1}{x \cdot z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 7e-151) (/ (/ y z) x) (* y (+ (* 0.5 (/ x z)) (/ 1.0 (* x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 7e-151) {
		tmp = (y / z) / x;
	} else {
		tmp = y * ((0.5 * (x / z)) + (1.0 / (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 <= 7d-151) then
        tmp = (y / z) / x
    else
        tmp = y * ((0.5d0 * (x / z)) + (1.0d0 / (x * z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 7e-151) {
		tmp = (y / z) / x;
	} else {
		tmp = y * ((0.5 * (x / z)) + (1.0 / (x * z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 7e-151:
		tmp = (y / z) / x
	else:
		tmp = y * ((0.5 * (x / z)) + (1.0 / (x * z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 7e-151)
		tmp = Float64(Float64(y / z) / x);
	else
		tmp = Float64(y * Float64(Float64(0.5 * Float64(x / z)) + Float64(1.0 / Float64(x * z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 7e-151)
		tmp = (y / z) / x;
	else
		tmp = y * ((0.5 * (x / z)) + (1.0 / (x * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 7e-151], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], N[(y * N[(N[(0.5 * N[(x / z), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.99999999999999991e-151

   1. Initial program 80.1%

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

      1. associate-*l/ 80.1%

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

   3. Simplified 80.1%

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

      1. associate-*r/ 97.9%

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

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in x around 0 63.4%

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

   if 6.99999999999999991e-151 < x

   1. Initial program 83.5%

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

      1. associate-*l/ 83.5%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in x around 0 59.3%

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

   6. Taylor expanded in y around 0 60.3%

      \[\leadsto \color{blue}{y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{1}{x \cdot z}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{1}{x \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.5% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-72}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{1}{x \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2e-72)
   (* y (+ (* 0.5 (/ x z)) (/ 1.0 (* x z))))
   (+ (* 0.5 (/ (* x y) z)) (/ y (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2e-72) {
		tmp = y * ((0.5 * (x / z)) + (1.0 / (x * z)));
	} else {
		tmp = (0.5 * ((x * y) / z)) + (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 (z <= 2d-72) then
        tmp = y * ((0.5d0 * (x / z)) + (1.0d0 / (x * z)))
    else
        tmp = (0.5d0 * ((x * y) / z)) + (y / (x * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 2e-72) {
		tmp = y * ((0.5 * (x / z)) + (1.0 / (x * z)));
	} else {
		tmp = (0.5 * ((x * y) / z)) + (y / (x * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2e-72:
		tmp = y * ((0.5 * (x / z)) + (1.0 / (x * z)))
	else:
		tmp = (0.5 * ((x * y) / z)) + (y / (x * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2e-72)
		tmp = Float64(y * Float64(Float64(0.5 * Float64(x / z)) + Float64(1.0 / Float64(x * z))));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x * y) / z)) + Float64(y / Float64(x * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2e-72)
		tmp = y * ((0.5 * (x / z)) + (1.0 / (x * z)));
	else
		tmp = (0.5 * ((x * y) / z)) + (y / (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2e-72], N[(y * N[(N[(0.5 * N[(x / z), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.9999999999999999e-72

   1. Initial program 81.4%

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

      1. associate-*l/ 81.4%

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

   3. Simplified 81.4%

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

   5. Taylor expanded in x around 0 68.8%

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

   6. Taylor expanded in y around 0 72.8%

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

   if 1.9999999999999999e-72 < z

   1. Initial program 81.6%

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

      1. associate-*l/ 81.6%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in x around 0 61.2%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-72}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{1}{x \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+187}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 7.5e+187)
   (+ (* 0.5 (/ (* x y) z)) (/ (/ y z) x))
   (* y (* x (/ 0.5 z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 7.5e+187) {
		tmp = (0.5 * ((x * y) / z)) + ((y / z) / x);
	} else {
		tmp = y * (x * (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 (x <= 7.5d+187) then
        tmp = (0.5d0 * ((x * y) / z)) + ((y / z) / x)
    else
        tmp = y * (x * (0.5d0 / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 7.5e+187) {
		tmp = (0.5 * ((x * y) / z)) + ((y / z) / x);
	} else {
		tmp = y * (x * (0.5 / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 7.5e+187:
		tmp = (0.5 * ((x * y) / z)) + ((y / z) / x)
	else:
		tmp = y * (x * (0.5 / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 7.5e+187)
		tmp = Float64(Float64(0.5 * Float64(Float64(x * y) / z)) + Float64(Float64(y / z) / x));
	else
		tmp = Float64(y * Float64(x * Float64(0.5 / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 7.5e+187)
		tmp = (0.5 * ((x * y) / z)) + ((y / z) / x);
	else
		tmp = y * (x * (0.5 / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 7.5e+187], N[(N[(0.5 * N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.5000000000000002e187

   1. Initial program 83.4%

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

      1. associate-*l/ 83.4%

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

   3. Simplified 83.4%

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

   5. Taylor expanded in x around 0 67.4%

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

      1. associate-/r* 63.4%

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

      2. *-un-lft-identity 63.4%

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

      3. associate-*l/ 63.4%

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

      4. associate-*r/ 69.5%

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

      5. div-inv 69.5%

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

      6. associate-*l/ 69.5%

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

      7. *-un-lft-identity 69.5%

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

   7. Applied egg-rr 69.5%

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{y}{z} \cdot \frac{1}{x}} \]
   8. Step-by-step derivation

      1. un-div-inv 69.6%

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

   9. Applied egg-rr 69.6%

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

   if 7.5000000000000002e187 < x

   1. Initial program 64.0%

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

      1. associate-*l/ 64.0%

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

   3. Simplified 64.0%

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

   5. Taylor expanded in x around 0 54.3%

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

   6. Taylor expanded in x around inf 54.3%

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

      1. associate-*r/ 54.3%

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

      2. associate-*r* 54.3%

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

      3. *-commutative 54.3%

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

      4. associate-*r/ 65.3%

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

      5. associate-/l* 65.3%

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

   8. Simplified 65.3%

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

      1. associate-/r/ 65.3%

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

   10. Applied egg-rr 65.3%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+187}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.4) (/ (/ y z) x) (* 0.5 (/ x (/ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.4) {
		tmp = (y / z) / x;
	} else {
		tmp = 0.5 * (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 (x <= 1.4d0) then
        tmp = (y / z) / x
    else
        tmp = 0.5d0 * (x / (z / y))
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z):
	tmp = 0
	if x <= 1.4:
		tmp = (y / z) / x
	else:
		tmp = 0.5 * (x / (z / y))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.4], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], N[(0.5 * N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 81.7%

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

      1. associate-*l/ 81.7%

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

   3. Simplified 81.7%

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

      1. associate-*r/ 97.7%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in x around 0 69.1%

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

   if 1.3999999999999999 < x

   1. Initial program 80.9%

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

      1. associate-*l/ 80.9%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in x around 0 39.0%

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

   6. Taylor expanded in x around inf 39.0%

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

      1. associate-/l* 34.8%

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

   8. Simplified 34.8%

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

4. Final simplification 60.0%

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

Alternative 10: 59.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.4) (/ (/ y z) x) (* 0.5 (/ (* x y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.4) {
		tmp = (y / z) / x;
	} else {
		tmp = 0.5 * ((x * y) / 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) then
        tmp = (y / z) / x
    else
        tmp = 0.5d0 * ((x * y) / z)
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z):
	tmp = 0
	if x <= 1.4:
		tmp = (y / z) / x
	else:
		tmp = 0.5 * ((x * y) / z)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.4], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], N[(0.5 * N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 81.7%

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

      1. associate-*l/ 81.7%

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

   3. Simplified 81.7%

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

      1. associate-*r/ 97.7%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in x around 0 69.1%

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

   if 1.3999999999999999 < x

   1. Initial program 80.9%

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

      1. associate-*l/ 80.9%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in x around 0 39.0%

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

   6. Taylor expanded in x around inf 39.0%

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

4. Final simplification 61.1%

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

Alternative 11: 59.3% accurate, 8.9× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.4) (/ (/ y z) x) (* y (* x (/ 0.5 z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.4) {
		tmp = (y / z) / x;
	} else {
		tmp = y * (x * (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 (x <= 1.4d0) then
        tmp = (y / z) / x
    else
        tmp = y * (x * (0.5d0 / z))
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z):
	tmp = 0
	if x <= 1.4:
		tmp = (y / z) / x
	else:
		tmp = y * (x * (0.5 / z))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.4], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], N[(y * N[(x * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 81.7%

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

      1. associate-*l/ 81.7%

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

   3. Simplified 81.7%

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

      1. associate-*r/ 97.7%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in x around 0 69.1%

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

   if 1.3999999999999999 < x

   1. Initial program 80.9%

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

      1. associate-*l/ 80.9%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in x around 0 39.0%

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

   6. Taylor expanded in x around inf 39.0%

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

      1. associate-*r/ 39.0%

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

      2. associate-*r* 39.0%

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

      3. *-commutative 39.0%

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

      4. associate-*r/ 41.7%

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

      5. associate-/l* 41.7%

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

   8. Simplified 41.7%

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

      1. associate-/r/ 41.7%

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

   10. Applied egg-rr 41.7%

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

4. Final simplification 61.8%

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

Alternative 12: 50.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 5e-20) (/ (/ y x) z) (/ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 5e-20) {
		tmp = (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 (z <= 5d-20) then
        tmp = (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 (z <= 5e-20) {
		tmp = (y / x) / z;
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 5e-20:
		tmp = (y / x) / z
	else:
		tmp = y / (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 5e-20)
		tmp = Float64(Float64(y / x) / z);
	else
		tmp = Float64(y / Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 5e-20)
		tmp = (y / x) / z;
	else
		tmp = y / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 5e-20], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.9999999999999999e-20

   1. Initial program 82.8%

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

   3. Taylor expanded in x around 0 50.9%

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

   if 4.9999999999999999e-20 < z

   1. Initial program 78.1%

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

      1. associate-*l/ 78.1%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in x around 0 43.6%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.1% 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 81.5%

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

   1. associate-*l/ 81.5%

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

3. Simplified 81.5%

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

5. Taylor expanded in x around 0 49.2%

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

6. Final simplification 49.2%

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

Alternative 14: 53.2% accurate, 21.4× speedup

Math

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

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(Float64(y / z) / x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.5%

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

   1. associate-*l/ 81.5%

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

3. Simplified 81.5%

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

   1. associate-*r/ 98.3%

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

6. Applied egg-rr 98.3%

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

7. Taylor expanded in x around 0 56.0%

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

8. Final simplification 56.0%

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

Developer target: 96.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024041 
(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))