Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 99.7% → 99.8%

Time: 44.7s

Alternatives: 6

Speedup: 1.0×

• could not determine a ground truth

  1. x = 9.429174205771284e-145
  2. y = 1.1730846309764473e-229
  3. z = -6.994430973655785e+179

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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*r/ 99.4%

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

   2. associate-/l/ 83.3%

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

   3. associate-/r* 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-44}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{1}{x} + x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.4e-44)
   (* y (/ (cosh x) (* x z)))
   (* (/ y z) (+ (/ 1.0 x) (* x 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4e-44) {
		tmp = y * (cosh(x) / (x * z));
	} else {
		tmp = (y / z) * ((1.0 / x) + (x * 0.5));
	}
	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 <= (-1.4d-44)) then
        tmp = y * (cosh(x) / (x * z))
    else
        tmp = (y / z) * ((1.0d0 / x) + (x * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4e-44) {
		tmp = y * (Math.cosh(x) / (x * z));
	} else {
		tmp = (y / z) * ((1.0 / x) + (x * 0.5));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.4e-44:
		tmp = y * (math.cosh(x) / (x * z))
	else:
		tmp = (y / z) * ((1.0 / x) + (x * 0.5))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.4e-44)
		tmp = Float64(y * Float64(cosh(x) / Float64(x * z)));
	else
		tmp = Float64(Float64(y / z) * Float64(Float64(1.0 / x) + Float64(x * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.4e-44)
		tmp = y * (cosh(x) / (x * z));
	else
		tmp = (y / z) * ((1.0 / x) + (x * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.4e-44], N[(y * N[(N[Cosh[x], $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4e-44

   1. Initial program 99.7%

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

      1. associate-*r/ 99.6%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l/ 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   if -1.4e-44 < x

   1. Initial program 99.4%

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

      1. associate-*r/ 99.4%

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

      2. associate-/l/ 81.2%

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

      3. associate-*l/ 79.6%

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

      4. *-commutative 79.6%

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

      5. *-commutative 79.6%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in x around 0 80.2%

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

      1. +-commutative 80.2%

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

      2. associate-/l* 80.2%

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

      3. *-commutative 80.2%

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

   6. Simplified 80.2%

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

      1. clear-num 80.2%

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

      2. un-div-inv 80.2%

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

      3. associate-/l/ 80.2%

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

   8. Applied egg-rr 80.2%

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

   9. Taylor expanded in y around 0 78.7%

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

       1. distribute-rgt-in 78.7%

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

       2. associate-/l/ 78.7%

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

       3. associate-*l/ 98.4%

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

       4. associate-*r/ 98.9%

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

       5. associate-*r/ 98.9%

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

       6. associate-*l/ 98.9%

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

       7. associate-*r/ 98.9%

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

       8. associate-*r* 98.9%

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

       9. associate-*r* 98.9%

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

       10. distribute-rgt-out 98.9%

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

   11. Simplified 98.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-44}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{1}{x} + x \cdot 0.5\right)\\ \end{array} \]

Alternative 3: 95.8% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

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) * ((1.0d0 / x) + (x * 0.5d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*r/ 99.4%

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

   2. associate-/l/ 83.3%

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

   3. associate-*l/ 81.9%

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

   4. *-commutative 81.9%

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

   5. *-commutative 81.9%

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

3. Simplified 81.9%

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

4. Taylor expanded in x around 0 78.8%

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

   1. +-commutative 78.8%

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

   2. associate-/l* 78.8%

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

   3. *-commutative 78.8%

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

6. Simplified 78.8%

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

   1. clear-num 78.8%

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

   2. un-div-inv 78.8%

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

   3. associate-/l/ 78.8%

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

8. Applied egg-rr 78.8%

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

9. Taylor expanded in y around 0 77.4%

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

    1. distribute-rgt-in 77.4%

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

    2. associate-/l/ 77.4%

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

    3. associate-*l/ 94.9%

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

    4. associate-*r/ 95.3%

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

    5. associate-*r/ 95.3%

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

    6. associate-*l/ 95.3%

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

    7. associate-*r/ 95.3%

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

    8. associate-*r* 95.3%

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

    9. associate-*r* 95.3%

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

    10. distribute-rgt-out 95.3%

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

11. Simplified 95.3%

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

12. Final simplification 95.3%

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

Alternative 4: 82.7% 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 99.4%

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

   1. associate-*r/ 99.4%

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

   2. associate-/l/ 83.3%

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

   3. associate-*l/ 81.9%

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

   4. *-commutative 81.9%

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

   5. *-commutative 81.9%

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

3. Simplified 81.9%

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

4. Taylor expanded in x around 0 78.3%

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

5. Final simplification 78.3%

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

Alternative 5: 95.3% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*r/ 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 94.4%

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

   1. un-div-inv 94.4%

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

6. Applied egg-rr 94.4%

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

7. Final simplification 94.4%

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

Alternative 6: 95.5% 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 99.4%

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

   1. associate-/l* 99.4%

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

   2. associate-/r/ 99.7%

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

3. Simplified 99.7%

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

   1. *-commutative 99.7%

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

   2. clear-num 99.7%

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

   3. div-inv 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in x around 0 78.3%

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

   1. associate-/r* 94.8%

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

8. Simplified 94.8%

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

9. Final simplification 94.8%

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

Developer target: 99.8% 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 2023278 
(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))