Linear.Quaternion:$ctan from linear-1.19.1.3

Average Accuracy: 87.6% → 98.8%

Time: 12.0s

Precision: binary64

Cost: 20680

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+109}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;t_0 \leq 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{z}{y}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* (cosh x) (/ y x)) z)))
   (if (<= t_0 -1e+109)
     (/ (/ y z) x)
     (if (<= t_0 1e+23) t_0 (/ (/ 1.0 x) (/ z y))))))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = (cosh(x) * (y / x)) / z;
	double tmp;
	if (t_0 <= -1e+109) {
		tmp = (y / z) / x;
	} else if (t_0 <= 1e+23) {
		tmp = t_0;
	} else {
		tmp = (1.0 / 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
    code = (cosh(x) * (y / x)) / z
end function

↓

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 <= (-1d+109)) then
        tmp = (y / z) / x
    else if (t_0 <= 1d+23) then
        tmp = t_0
    else
        tmp = (1.0d0 / x) / (z / y)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z) {
	double t_0 = (Math.cosh(x) * (y / x)) / z;
	double tmp;
	if (t_0 <= -1e+109) {
		tmp = (y / z) / x;
	} else if (t_0 <= 1e+23) {
		tmp = t_0;
	} else {
		tmp = (1.0 / x) / (z / y);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (math.cosh(x) * (y / x)) / z
	tmp = 0
	if t_0 <= -1e+109:
		tmp = (y / z) / x
	elif t_0 <= 1e+23:
		tmp = t_0
	else:
		tmp = (1.0 / x) / (z / y)
	return tmp

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(Float64(cosh(x) * Float64(y / x)) / z)
	tmp = 0.0
	if (t_0 <= -1e+109)
		tmp = Float64(Float64(y / z) / x);
	elseif (t_0 <= 1e+23)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / x) / Float64(z / y));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	t_0 = (cosh(x) * (y / x)) / z;
	tmp = 0.0;
	if (t_0 <= -1e+109)
		tmp = (y / z) / x;
	elseif (t_0 <= 1e+23)
		tmp = t_0;
	else
		tmp = (1.0 / x) / (z / y);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

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, -1e+109], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1e+23], t$95$0, N[(N[(1.0 / x), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	87.6%
Target	99.3%
Herbie	98.8%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < -9.99999999999999982e108

   1. Initial program 69.6%

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

   2. Simplified 99.3%

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

      [Start] 69.6	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-*r/ [=>] 69.6	\[ \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]
      associate-/r* [<=] 78.2	\[ \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]
      times-frac [=>] 99.3	\[ \color{blue}{\frac{\cosh x}{x} \cdot \frac{y}{z}} \]

   3. Taylor expanded in x around 0 97.9%

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

   4. Applied egg-rr 98.1%

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

   if -9.99999999999999982e108 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 9.9999999999999992e22

   1. Initial program 99.6%

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

   if 9.9999999999999992e22 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 78.4%

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

   2. Simplified 99.4%

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

      [Start] 78.4	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-*r/ [=>] 78.4	\[ \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]
      associate-/r* [<=] 81.3	\[ \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]
      times-frac [=>] 99.4	\[ \color{blue}{\frac{\cosh x}{x} \cdot \frac{y}{z}} \]

   3. Taylor expanded in x around 0 97.8%

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

   4. Applied egg-rr 97.8%

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{\frac{z}{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq -1 \cdot 10^{+109}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 10^{+23}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{z}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	96.9%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+186} \lor \neg \left(z \leq 1.2 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 2
Accuracy	97.3%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+186} \lor \neg \left(z \leq 2 \cdot 10^{-31}\right):\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 3
Accuracy	97.1%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+186} \lor \neg \left(z \leq 1.8 \cdot 10^{-64}\right):\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 4
Accuracy	96.5%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+156} \lor \neg \left(z \leq 1.42 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{1}{x} + x \cdot 0.5\right)\\ \end{array} \]

Alternative 5
Accuracy	97.7%
Cost	968

\[\begin{array}{l} t_0 := \frac{1}{x} + x \cdot 0.5\\ \mathbf{if}\;y \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{z} \cdot t_0\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-63}:\\ \;\;\;\;\frac{y \cdot t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 6
Accuracy	97.8%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{1}{x} + x \cdot 0.5\right)\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 7
Accuracy	97.8%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -5000:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 8
Accuracy	97.4%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+40} \lor \neg \left(z \leq 5 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 9
Accuracy	97.4%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -35000000000000 \lor \neg \left(y \leq 3 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 10
Accuracy	87.6%
Cost	320

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

Reproduce

herbie shell --seed 2023129 
(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))