Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.9% → 97.4%

Time: 7.1s

Alternatives: 12

Speedup: 1.0×

• 64.1% 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: 84.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-139}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* (cosh x) (/ y x)) z)))
   (if (<= t_0 2e-139) t_0 (/ (/ (* (cosh x) y) z) x))))

C

double code(double x, double y, double z) {
	double t_0 = (cosh(x) * (y / x)) / z;
	double tmp;
	if (t_0 <= 2e-139) {
		tmp = t_0;
	} else {
		tmp = ((cosh(x) * 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) :: t_0
    real(8) :: tmp
    t_0 = (cosh(x) * (y / x)) / z
    if (t_0 <= 2d-139) then
        tmp = t_0
    else
        tmp = ((cosh(x) * y) / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (Math.cosh(x) * (y / x)) / z;
	double tmp;
	if (t_0 <= 2e-139) {
		tmp = t_0;
	} else {
		tmp = ((Math.cosh(x) * y) / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (math.cosh(x) * (y / x)) / z
	tmp = 0
	if t_0 <= 2e-139:
		tmp = t_0
	else:
		tmp = ((math.cosh(x) * y) / z) / x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(cosh(x) * Float64(y / x)) / z)
	tmp = 0.0
	if (t_0 <= 2e-139)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(cosh(x) * y) / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (cosh(x) * (y / x)) / z;
	tmp = 0.0;
	if (t_0 <= 2e-139)
		tmp = t_0;
	else
		tmp = ((cosh(x) * y) / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-139], t$95$0, N[(N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 2.00000000000000006e-139

   1. Initial program 97.4%

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

   if 2.00000000000000006e-139 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 79.4%

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

      1. associate-*l/ 79.4%

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

   3. Simplified 79.4%

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

      1. associate-*r/ 99.8%

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

      2. associate-*l/ 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 2 \cdot 10^{-139}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \end{array} \]

Alternative 2: 88.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.5%

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

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

   1. Initial program 0.0%

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

      1. associate-/l* 0.0%

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

      2. associate-/r/ 57.1%

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

      3. associate-*l/ 57.1%

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

      4. *-commutative 57.1%

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

   3. Simplified 57.1%

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

4. Final simplification 92.3%

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

Alternative 3: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{-282} \lor \neg \left(x \leq 1.2 \cdot 10^{-91}\right):\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x 9.2e-282) (not (<= x 1.2e-91)))
   (* (/ y x) (/ (cosh x) z))
   (/ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= 9.2e-282) || !(x <= 1.2e-91)) {
		tmp = (y / x) * (cosh(x) / z);
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= 9.2d-282) .or. (.not. (x <= 1.2d-91))) then
        tmp = (y / x) * (cosh(x) / z)
    else
        tmp = y / (x * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= 9.2e-282) || !(x <= 1.2e-91)) {
		tmp = (y / x) * (Math.cosh(x) / z);
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= 9.2e-282) or not (x <= 1.2e-91):
		tmp = (y / x) * (math.cosh(x) / z)
	else:
		tmp = y / (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= 9.2e-282) || !(x <= 1.2e-91))
		tmp = Float64(Float64(y / x) * Float64(cosh(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 ((x <= 9.2e-282) || ~((x <= 1.2e-91)))
		tmp = (y / x) * (cosh(x) / z);
	else
		tmp = y / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, 9.2e-282], N[Not[LessEqual[x, 1.2e-91]], $MachinePrecision]], N[(N[(y / x), $MachinePrecision] * N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.1999999999999996e-282 or 1.20000000000000005e-91 < x

   1. Initial program 88.0%

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

      1. associate-*l/ 88.0%

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

   3. Simplified 88.0%

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

   if 9.1999999999999996e-282 < x < 1.20000000000000005e-91

   1. Initial program 86.3%

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

      1. associate-*l/ 86.2%

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

   3. Simplified 86.2%

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

   4. Taylor expanded in x around 0 98.1%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{-282} \lor \neg \left(x \leq 1.2 \cdot 10^{-91}\right):\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 4: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-153} \lor \neg \left(y \leq 9 \cdot 10^{-248}\right):\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{\frac{x \cdot z}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1e-153) (not (<= y 9e-248)))
   (* (/ y x) (/ (cosh x) z))
   (/ (cosh x) (/ (* x z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1e-153) || !(y <= 9e-248)) {
		tmp = (y / x) * (cosh(x) / z);
	} else {
		tmp = cosh(x) / ((x * z) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1d-153)) .or. (.not. (y <= 9d-248))) then
        tmp = (y / x) * (cosh(x) / z)
    else
        tmp = cosh(x) / ((x * z) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1e-153) || !(y <= 9e-248)) {
		tmp = (y / x) * (Math.cosh(x) / z);
	} else {
		tmp = Math.cosh(x) / ((x * z) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1e-153) or not (y <= 9e-248):
		tmp = (y / x) * (math.cosh(x) / z)
	else:
		tmp = math.cosh(x) / ((x * z) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1e-153) || !(y <= 9e-248))
		tmp = Float64(Float64(y / x) * Float64(cosh(x) / z));
	else
		tmp = Float64(cosh(x) / Float64(Float64(x * z) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1e-153) || ~((y <= 9e-248)))
		tmp = (y / x) * (cosh(x) / z);
	else
		tmp = cosh(x) / ((x * z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1e-153], N[Not[LessEqual[y, 9e-248]], $MachinePrecision]], N[(N[(y / x), $MachinePrecision] * N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] / N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.00000000000000004e-153 or 8.9999999999999992e-248 < y

   1. Initial program 92.4%

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

      1. associate-*l/ 92.3%

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

   3. Simplified 92.3%

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

   if -1.00000000000000004e-153 < y < 8.9999999999999992e-248

   1. Initial program 60.4%

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

      1. associate-/l* 56.2%

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

      2. associate-/r/ 75.0%

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

      3. associate-*l/ 80.0%

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

      4. *-commutative 80.0%

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

   3. Simplified 80.0%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-153} \lor \neg \left(y \leq 9 \cdot 10^{-248}\right):\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{\frac{x \cdot z}{y}}\\ \end{array} \]

Alternative 5: 70.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-244}:\\ \;\;\;\;\frac{-y}{\frac{z}{x \cdot -0.5 + \frac{-1}{x}}}\\ \mathbf{elif}\;y \leq 10^{+79}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ y (* x z)) (* 0.5 (/ (* x y) z)))))
   (if (<= y -2.8e-58)
     t_0
     (if (<= y 7.5e-244)
       (/ (- y) (/ z (+ (* x -0.5) (/ -1.0 x))))
       (if (<= y 1e+79) (/ (+ (/ y x) (* 0.5 (* x y))) z) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (y / (x * z)) + (0.5 * ((x * y) / z));
	double tmp;
	if (y <= -2.8e-58) {
		tmp = t_0;
	} else if (y <= 7.5e-244) {
		tmp = -y / (z / ((x * -0.5) + (-1.0 / x)));
	} else if (y <= 1e+79) {
		tmp = ((y / x) + (0.5 * (x * y))) / 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 / (x * z)) + (0.5d0 * ((x * y) / z))
    if (y <= (-2.8d-58)) then
        tmp = t_0
    else if (y <= 7.5d-244) then
        tmp = -y / (z / ((x * (-0.5d0)) + ((-1.0d0) / x)))
    else if (y <= 1d+79) then
        tmp = ((y / x) + (0.5d0 * (x * y))) / 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 / (x * z)) + (0.5 * ((x * y) / z));
	double tmp;
	if (y <= -2.8e-58) {
		tmp = t_0;
	} else if (y <= 7.5e-244) {
		tmp = -y / (z / ((x * -0.5) + (-1.0 / x)));
	} else if (y <= 1e+79) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y / (x * z)) + (0.5 * ((x * y) / z))
	tmp = 0
	if y <= -2.8e-58:
		tmp = t_0
	elif y <= 7.5e-244:
		tmp = -y / (z / ((x * -0.5) + (-1.0 / x)))
	elif y <= 1e+79:
		tmp = ((y / x) + (0.5 * (x * y))) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y / Float64(x * z)) + Float64(0.5 * Float64(Float64(x * y) / z)))
	tmp = 0.0
	if (y <= -2.8e-58)
		tmp = t_0;
	elseif (y <= 7.5e-244)
		tmp = Float64(Float64(-y) / Float64(z / Float64(Float64(x * -0.5) + Float64(-1.0 / x))));
	elseif (y <= 1e+79)
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(x * y))) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y / (x * z)) + (0.5 * ((x * y) / z));
	tmp = 0.0;
	if (y <= -2.8e-58)
		tmp = t_0;
	elseif (y <= 7.5e-244)
		tmp = -y / (z / ((x * -0.5) + (-1.0 / x)));
	elseif (y <= 1e+79)
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.8e-58], t$95$0, If[LessEqual[y, 7.5e-244], N[((-y) / N[(z / N[(N[(x * -0.5), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+79], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.8000000000000001e-58 or 9.99999999999999967e78 < y

   1. Initial program 92.0%

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

      1. associate-*l/ 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in x around 0 84.4%

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

   if -2.8000000000000001e-58 < y < 7.5000000000000003e-244

   1. Initial program 69.3%

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

   2. Taylor expanded in x around 0 52.3%

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

   3. Taylor expanded in y around -inf 52.2%

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

      1. mul-1-neg 52.2%

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

      2. associate-/l* 64.8%

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

      3. *-commutative 64.8%

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

   5. Simplified 64.8%

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

   if 7.5000000000000003e-244 < y < 9.99999999999999967e78

   1. Initial program 95.4%

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

   2. Taylor expanded in x around 0 71.3%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-244}:\\ \;\;\;\;\frac{-y}{\frac{z}{x \cdot -0.5 + \frac{-1}{x}}}\\ \mathbf{elif}\;y \leq 10^{+79}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \]

Alternative 6: 70.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x \cdot z}\\ t_1 := t_0 + 0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{+92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-250}:\\ \;\;\;\;t_0 + 0.5 \cdot \frac{\frac{x}{z}}{\frac{1}{y}}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (* x z))) (t_1 (+ t_0 (* 0.5 (/ (* x y) z)))))
   (if (<= y -1.6e+92)
     t_1
     (if (<= y 6.8e-250)
       (+ t_0 (* 0.5 (/ (/ x z) (/ 1.0 y))))
       (if (<= y 6e+78) (/ (+ (/ y x) (* 0.5 (* x y))) z) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = y / (x * z);
	double t_1 = t_0 + (0.5 * ((x * y) / z));
	double tmp;
	if (y <= -1.6e+92) {
		tmp = t_1;
	} else if (y <= 6.8e-250) {
		tmp = t_0 + (0.5 * ((x / z) / (1.0 / y)));
	} else if (y <= 6e+78) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = y / (x * z)
    t_1 = t_0 + (0.5d0 * ((x * y) / z))
    if (y <= (-1.6d+92)) then
        tmp = t_1
    else if (y <= 6.8d-250) then
        tmp = t_0 + (0.5d0 * ((x / z) / (1.0d0 / y)))
    else if (y <= 6d+78) then
        tmp = ((y / x) + (0.5d0 * (x * y))) / z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / (x * z);
	double t_1 = t_0 + (0.5 * ((x * y) / z));
	double tmp;
	if (y <= -1.6e+92) {
		tmp = t_1;
	} else if (y <= 6.8e-250) {
		tmp = t_0 + (0.5 * ((x / z) / (1.0 / y)));
	} else if (y <= 6e+78) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / (x * z)
	t_1 = t_0 + (0.5 * ((x * y) / z))
	tmp = 0
	if y <= -1.6e+92:
		tmp = t_1
	elif y <= 6.8e-250:
		tmp = t_0 + (0.5 * ((x / z) / (1.0 / y)))
	elif y <= 6e+78:
		tmp = ((y / x) + (0.5 * (x * y))) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / Float64(x * z))
	t_1 = Float64(t_0 + Float64(0.5 * Float64(Float64(x * y) / z)))
	tmp = 0.0
	if (y <= -1.6e+92)
		tmp = t_1;
	elseif (y <= 6.8e-250)
		tmp = Float64(t_0 + Float64(0.5 * Float64(Float64(x / z) / Float64(1.0 / y))));
	elseif (y <= 6e+78)
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(x * y))) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / (x * z);
	t_1 = t_0 + (0.5 * ((x * y) / z));
	tmp = 0.0;
	if (y <= -1.6e+92)
		tmp = t_1;
	elseif (y <= 6.8e-250)
		tmp = t_0 + (0.5 * ((x / z) / (1.0 / y)));
	elseif (y <= 6e+78)
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(0.5 * N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.6e+92], t$95$1, If[LessEqual[y, 6.8e-250], N[(t$95$0 + N[(0.5 * N[(N[(x / z), $MachinePrecision] / N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e+78], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.60000000000000013e92 or 5.99999999999999964e78 < y

   1. Initial program 90.0%

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

      1. associate-*l/ 89.9%

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

   3. Simplified 89.9%

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

   4. Taylor expanded in x around 0 89.7%

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

   if -1.60000000000000013e92 < y < 6.79999999999999987e-250

   1. Initial program 78.9%

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

      1. associate-*l/ 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in x around 0 53.6%

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

      1. associate-/l* 53.6%

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

      2. clear-num 53.6%

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

      3. associate-/r/ 53.6%

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

      4. clear-num 53.6%

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

   6. Applied egg-rr 53.6%

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

      1. clear-num 53.6%

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

      2. associate-/r/ 53.6%

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

      3. div-inv 53.6%

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

      4. associate-*l/ 64.6%

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

      5. associate-/r* 64.6%

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

      6. clear-num 64.6%

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

   8. Applied egg-rr 64.6%

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

   if 6.79999999999999987e-250 < y < 5.99999999999999964e78

   1. Initial program 95.4%

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

   2. Taylor expanded in x around 0 71.3%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-250}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{\frac{x}{z}}{\frac{1}{y}}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \]

Alternative 7: 66.2% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-283}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.85e-283)
   (/ (+ (/ y x) (* 0.5 (* x y))) z)
   (if (<= x 1.4) (/ y (* x z)) (* y (* x (/ 0.5 z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.85e-283) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else if (x <= 1.4) {
		tmp = y / (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 <= 1.85d-283) then
        tmp = ((y / x) + (0.5d0 * (x * y))) / z
    else if (x <= 1.4d0) then
        tmp = y / (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 <= 1.85e-283) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else if (x <= 1.4) {
		tmp = y / (x * z);
	} else {
		tmp = y * (x * (0.5 / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1.85e-283:
		tmp = ((y / x) + (0.5 * (x * y))) / z
	elif x <= 1.4:
		tmp = y / (x * z)
	else:
		tmp = y * (x * (0.5 / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.85e-283)
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(x * y))) / z);
	elseif (x <= 1.4)
		tmp = Float64(y / Float64(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 <= 1.85e-283)
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	elseif (x <= 1.4)
		tmp = y / (x * z);
	else
		tmp = y * (x * (0.5 / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.85e-283], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[x, 1.4], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.85e-283

   1. Initial program 86.9%

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

   2. Taylor expanded in x around 0 69.9%

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

   if 1.85e-283 < x < 1.3999999999999999

   1. Initial program 89.1%

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

      1. associate-*l/ 89.0%

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

   3. Simplified 89.0%

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

   4. Taylor expanded in x around 0 98.2%

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

   if 1.3999999999999999 < x

   1. Initial program 87.5%

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

   2. Taylor expanded in x around 0 47.6%

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

   3. Taylor expanded in x around inf 47.6%

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

      1. associate-*r/ 47.6%

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

      2. associate-/l* 47.6%

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

      3. *-commutative 47.6%

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

   5. Simplified 47.6%

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

      1. associate-/r/ 47.6%

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

      2. *-commutative 47.6%

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

      3. associate-*r* 50.5%

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

   7. Applied egg-rr 50.5%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-283}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \end{array} \]

Alternative 8: 66.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{-y}{\frac{z}{x \cdot -0.5 + \frac{-1}{x}}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

2. Taylor expanded in x around 0 69.6%

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

3. Taylor expanded in y around -inf 69.5%

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

   1. mul-1-neg 69.5%

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

   2. associate-/l* 71.6%

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

   3. *-commutative 71.6%

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

5. Simplified 71.6%

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

6. Final simplification 71.6%

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

Alternative 9: 62.1% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -750 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -750.0) (not (<= x 1.4))) (* 0.5 (* x (/ y z))) (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -750.0) || !(x <= 1.4)) {
		tmp = 0.5 * (x * (y / z));
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-750.0d0)) .or. (.not. (x <= 1.4d0))) then
        tmp = 0.5d0 * (x * (y / z))
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -750.0) || !(x <= 1.4)) {
		tmp = 0.5 * (x * (y / z));
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -750.0) or not (x <= 1.4):
		tmp = 0.5 * (x * (y / z))
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -750.0) || !(x <= 1.4))
		tmp = Float64(0.5 * Float64(x * Float64(y / z)));
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -750.0) || ~((x <= 1.4)))
		tmp = 0.5 * (x * (y / z));
	else
		tmp = (y / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -750.0], N[Not[LessEqual[x, 1.4]], $MachinePrecision]], N[(0.5 * N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -750 or 1.3999999999999999 < x

   1. Initial program 82.4%

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

   2. Taylor expanded in x around 0 43.5%

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

   3. Taylor expanded in x around inf 43.5%

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

      1. associate-*r/ 43.5%

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

      2. associate-/l* 43.5%

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

      3. *-commutative 43.5%

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

   5. Simplified 43.5%

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

   6. Taylor expanded in z around 0 43.5%

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

      1. associate-*r/ 37.1%

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

   8. Simplified 37.1%

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

   if -750 < x < 1.3999999999999999

   1. Initial program 92.3%

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

      1. associate-*l/ 92.1%

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

   3. Simplified 92.1%

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

      1. associate-*r/ 94.3%

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

      2. associate-*l/ 94.5%

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

   5. Applied egg-rr 94.5%

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

   6. Taylor expanded in x around 0 94.5%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -750 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 10: 65.9% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -750 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -750.0) (not (<= x 1.4))) (* y (* x (/ 0.5 z))) (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -750.0) || !(x <= 1.4)) {
		tmp = y * (x * (0.5 / z));
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-750.0d0)) .or. (.not. (x <= 1.4d0))) then
        tmp = y * (x * (0.5d0 / z))
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -750.0) || !(x <= 1.4)) {
		tmp = y * (x * (0.5 / z));
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -750.0) or not (x <= 1.4):
		tmp = y * (x * (0.5 / z))
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -750.0) || !(x <= 1.4))
		tmp = Float64(y * Float64(x * Float64(0.5 / z)));
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -750.0) || ~((x <= 1.4)))
		tmp = y * (x * (0.5 / z));
	else
		tmp = (y / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -750.0], N[Not[LessEqual[x, 1.4]], $MachinePrecision]], N[(y * N[(x * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -750 or 1.3999999999999999 < x

   1. Initial program 82.4%

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

   2. Taylor expanded in x around 0 43.5%

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

   3. Taylor expanded in x around inf 43.5%

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

      1. associate-*r/ 43.5%

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

      2. associate-/l* 43.5%

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

      3. *-commutative 43.5%

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

   5. Simplified 43.5%

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

      1. associate-/r/ 43.5%

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

      2. *-commutative 43.5%

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

      3. associate-*r* 45.8%

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

   7. Applied egg-rr 45.8%

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

   if -750 < x < 1.3999999999999999

   1. Initial program 92.3%

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

      1. associate-*l/ 92.1%

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

   3. Simplified 92.1%

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

      1. associate-*r/ 94.3%

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

      2. associate-*l/ 94.5%

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

   5. Applied egg-rr 94.5%

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

   6. Taylor expanded in x around 0 94.5%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -750 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 11: 54.9% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-31}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3e-31) (/ y (* x z)) (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3e-31) {
		tmp = y / (x * z);
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-3d-31)) then
        tmp = y / (x * z)
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3e-31) {
		tmp = y / (x * z);
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3e-31:
		tmp = y / (x * z)
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3e-31)
		tmp = Float64(y / Float64(x * z));
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3e-31)
		tmp = y / (x * z);
	else
		tmp = (y / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3e-31], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.99999999999999981e-31

   1. Initial program 87.9%

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

      1. associate-*l/ 87.8%

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

   3. Simplified 87.8%

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

   4. Taylor expanded in x around 0 57.1%

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

   if -2.99999999999999981e-31 < z

   1. Initial program 87.6%

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

      1. associate-*l/ 87.5%

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

   3. Simplified 87.5%

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

      1. associate-*r/ 98.9%

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

      2. associate-*l/ 98.9%

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

   5. Applied egg-rr 98.9%

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

   6. Taylor expanded in x around 0 62.4%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-31}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 12: 49.7% 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 87.7%

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

   1. associate-*l/ 87.6%

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

3. Simplified 87.6%

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

4. Taylor expanded in x around 0 52.8%

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

5. Final simplification 52.8%

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

Developer target: 97.0% 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 2023309 
(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))