Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.7% → 95.4%

Time: 7.8s

Alternatives: 7

Speedup: 1.0×

• 62.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

   1. add-log-exp 57.8%

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

   2. *-un-lft-identity 57.8%

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

   3. log-prod 57.8%

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

   4. metadata-eval 57.8%

      \[\leadsto \frac{\color{blue}{0} + \log \left(e^{\cosh x \cdot \frac{y}{x}}\right)}{z} \]

   5. add-log-exp 84.2%

      \[\leadsto \frac{0 + \color{blue}{\cosh x \cdot \frac{y}{x}}}{z} \]

   6. *-commutative 84.2%

      \[\leadsto \frac{0 + \color{blue}{\frac{y}{x} \cdot \cosh x}}{z} \]

4. Applied egg-rr 84.2%

   \[\leadsto \frac{\color{blue}{0 + \frac{y}{x} \cdot \cosh x}}{z} \]
5. Step-by-step derivation

   1. +-lft-identity 84.2%

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

   2. associate-*l/ 95.9%

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

6. Simplified 95.9%

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

Alternative 2: 91.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x}\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ y x))))
   (if (<= t_0 INFINITY) (/ t_0 z) (* y (/ (cosh x) (* x z))))))

C

double code(double x, double y, double z) {
	double t_0 = cosh(x) * (y / x);
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 / z;
	} else {
		tmp = y * (cosh(x) / (x * z));
	}
	return tmp;
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < +inf.0

   1. Initial program 95.3%

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

   if +inf.0 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/l* 0.0%

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

   3. Simplified 0.0%

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

      1. add-log-exp 0.0%

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

      2. *-un-lft-identity 0.0%

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

      3. log-prod 0.0%

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

      4. metadata-eval 0.0%

         \[\leadsto \color{blue}{0} + \log \left(e^{\frac{y}{x} \cdot \frac{\cosh x}{z}}\right) \]

      5. add-log-exp 0.0%

         \[\leadsto 0 + \color{blue}{\frac{y}{x} \cdot \frac{\cosh x}{z}} \]

      6. clear-num 0.0%

         \[\leadsto 0 + \color{blue}{\frac{1}{\frac{x}{y}}} \cdot \frac{\cosh x}{z} \]

      7. frac-times 0.0%

         \[\leadsto 0 + \color{blue}{\frac{1 \cdot \cosh x}{\frac{x}{y} \cdot z}} \]

      8. *-un-lft-identity 0.0%

         \[\leadsto 0 + \frac{\color{blue}{\cosh x}}{\frac{x}{y} \cdot z} \]

   6. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{0 + \frac{\cosh x}{\frac{x}{y} \cdot z}} \]
   7. Step-by-step derivation

      1. +-lft-identity 0.0%

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

      2. associate-/l/ 0.0%

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

      3. associate-/r/ 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-/l/ 70.0%

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

   8. Simplified 70.0%

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

Alternative 3: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.6 \cdot 10^{-113}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9.6e-113) (* y (/ (cosh x) (* x z))) (* (/ y x) (/ (cosh x) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.6e-113) {
		tmp = y * (cosh(x) / (x * z));
	} else {
		tmp = (y / x) * (cosh(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 <= 9.6d-113) then
        tmp = y * (cosh(x) / (x * z))
    else
        tmp = (y / x) * (cosh(x) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.6e-113) {
		tmp = y * (Math.cosh(x) / (x * z));
	} else {
		tmp = (y / x) * (Math.cosh(x) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 9.6e-113:
		tmp = y * (math.cosh(x) / (x * z))
	else:
		tmp = (y / x) * (math.cosh(x) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9.6e-113)
		tmp = Float64(y * Float64(cosh(x) / Float64(x * z)));
	else
		tmp = Float64(Float64(y / x) * Float64(cosh(x) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 9.6e-113)
		tmp = y * (cosh(x) / (x * z));
	else
		tmp = (y / x) * (cosh(x) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9.6e-113], N[(y * N[(N[Cosh[x], $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / x), $MachinePrecision] * N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.60000000000000049e-113

   1. Initial program 78.9%

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

      1. *-commutative 78.9%

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

      2. associate-/l* 78.9%

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

   3. Simplified 78.9%

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

      1. add-log-exp 46.7%

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

      2. *-un-lft-identity 46.7%

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

      3. log-prod 46.7%

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

      4. metadata-eval 46.7%

         \[\leadsto \color{blue}{0} + \log \left(e^{\frac{y}{x} \cdot \frac{\cosh x}{z}}\right) \]

      5. add-log-exp 78.9%

         \[\leadsto 0 + \color{blue}{\frac{y}{x} \cdot \frac{\cosh x}{z}} \]

      6. clear-num 78.2%

         \[\leadsto 0 + \color{blue}{\frac{1}{\frac{x}{y}}} \cdot \frac{\cosh x}{z} \]

      7. frac-times 73.9%

         \[\leadsto 0 + \color{blue}{\frac{1 \cdot \cosh x}{\frac{x}{y} \cdot z}} \]

      8. *-un-lft-identity 73.9%

         \[\leadsto 0 + \frac{\color{blue}{\cosh x}}{\frac{x}{y} \cdot z} \]

   6. Applied egg-rr 73.9%

      \[\leadsto \color{blue}{0 + \frac{\cosh x}{\frac{x}{y} \cdot z}} \]
   7. Step-by-step derivation

      1. +-lft-identity 73.9%

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

      2. associate-/l/ 78.2%

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

      3. associate-/r/ 95.2%

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

      4. *-commutative 95.2%

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

      5. associate-/l/ 85.2%

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

   8. Simplified 85.2%

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

   if 9.60000000000000049e-113 < y

   1. Initial program 93.1%

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

      1. *-commutative 93.1%

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

      2. associate-/l* 93.1%

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

   3. Simplified 93.1%

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

Alternative 4: 83.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

   1. *-commutative 84.2%

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

   2. associate-/l* 84.1%

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

3. Simplified 84.1%

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

   1. add-log-exp 57.2%

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

   2. *-un-lft-identity 57.2%

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

   3. log-prod 57.2%

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

   4. metadata-eval 57.2%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{y}{x} \cdot \frac{\cosh x}{z}}\right) \]

   5. add-log-exp 84.1%

      \[\leadsto 0 + \color{blue}{\frac{y}{x} \cdot \frac{\cosh x}{z}} \]

   6. clear-num 83.7%

      \[\leadsto 0 + \color{blue}{\frac{1}{\frac{x}{y}}} \cdot \frac{\cosh x}{z} \]

   7. frac-times 79.6%

      \[\leadsto 0 + \color{blue}{\frac{1 \cdot \cosh x}{\frac{x}{y} \cdot z}} \]

   8. *-un-lft-identity 79.6%

      \[\leadsto 0 + \frac{\color{blue}{\cosh x}}{\frac{x}{y} \cdot z} \]

6. Applied egg-rr 79.6%

   \[\leadsto \color{blue}{0 + \frac{\cosh x}{\frac{x}{y} \cdot z}} \]
7. Step-by-step derivation

   1. +-lft-identity 79.6%

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

   2. associate-/l/ 83.9%

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

   3. associate-/r/ 96.6%

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

   4. *-commutative 96.6%

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

   5. associate-/l/ 87.2%

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

8. Simplified 87.2%

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

Alternative 5: 52.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+59}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9e+59) (/ (/ y x) z) (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9e+59) {
		tmp = (y / x) / z;
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 9d+59) then
        tmp = (y / x) / z
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 9e+59) {
		tmp = (y / x) / z;
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 9e+59:
		tmp = (y / x) / z
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9e+59)
		tmp = Float64(Float64(y / x) / z);
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 9e+59)
		tmp = (y / x) / z;
	else
		tmp = (y / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9e+59], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.99999999999999919e59

   1. Initial program 82.4%

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

      1. *-commutative 82.4%

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

      2. associate-/l* 82.4%

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

   3. Simplified 82.4%

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

   5. Taylor expanded in x around 0 51.2%

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

      1. associate-/r* 53.0%

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

   7. Simplified 53.0%

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

   if 8.99999999999999919e59 < y

   1. Initial program 90.3%

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

      1. *-commutative 90.3%

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

      2. associate-/l* 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in x around 0 45.8%

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

      1. associate-/l/ 60.7%

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

   7. Simplified 60.7%

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

Alternative 6: 50.8% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.25e+48) (/ (/ y x) z) (/ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.25e+48) {
		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 (y <= 2.25d+48) 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 (y <= 2.25e+48) {
		tmp = (y / x) / z;
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.25e+48:
		tmp = (y / x) / z
	else:
		tmp = y / (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.25e+48)
		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 (y <= 2.25e+48)
		tmp = (y / x) / z;
	else
		tmp = y / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.25e+48], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.24999999999999998e48

   1. Initial program 82.3%

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

      1. *-commutative 82.3%

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

      2. associate-/l* 82.3%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in x around 0 51.4%

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

      1. associate-/r* 53.3%

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

   7. Simplified 53.3%

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

   if 2.24999999999999998e48 < y

   1. Initial program 90.4%

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

      1. *-commutative 90.4%

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

      2. associate-/l* 90.5%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in x around 0 45.0%

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

Alternative 7: 49.8% 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 84.2%

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

   1. *-commutative 84.2%

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

   2. associate-/l* 84.1%

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

3. Simplified 84.1%

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

5. Taylor expanded in x around 0 50.0%

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

Developer target: 97.0% 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 2024111 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :alt
  (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))