Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.0% → 99.8%

Time: 12.7s

Precision: binary64

Cost: 20681

• 64.4% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+295} \lor \neg \left(t_0 \leq 2 \cdot 10^{+290}\right):\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (or (<= t_0 -1e+295) (not (<= t_0 2e+290)))
     (/ (* y (/ (cosh x) z)) x)
     t_0)))

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+295) || !(t_0 <= 2e+290)) {
		tmp = (y * (cosh(x) / z)) / x;
	} 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
    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+295)) .or. (.not. (t_0 <= 2d+290))) then
        tmp = (y * (cosh(x) / z)) / x
    else
        tmp = t_0
    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+295) || !(t_0 <= 2e+290)) {
		tmp = (y * (Math.cosh(x) / z)) / x;
	} else {
		tmp = t_0;
	}
	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+295) or not (t_0 <= 2e+290):
		tmp = (y * (math.cosh(x) / z)) / x
	else:
		tmp = t_0
	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+295) || !(t_0 <= 2e+290))
		tmp = Float64(Float64(y * Float64(cosh(x) / z)) / x);
	else
		tmp = t_0;
	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+295) || ~((t_0 <= 2e+290)))
		tmp = (y * (cosh(x) / z)) / x;
	else
		tmp = t_0;
	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[Or[LessEqual[t$95$0, -1e+295], N[Not[LessEqual[t$95$0, 2e+290]], $MachinePrecision]], N[(N[(y * N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$0]]

Target

Original	84.0%
Target	96.9%
Herbie	99.8%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < -9.9999999999999998e294 or 2.00000000000000012e290 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 78.1%

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

   2. Simplified 78.1%

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

      [Start] 78.1	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      *-commutative [=>] 78.1	\[ \frac{\color{blue}{\frac{y}{x} \cdot \cosh x}}{z} \]
      associate-*r/ [<=] 78.1	\[ \color{blue}{\frac{y}{x} \cdot \frac{\cosh x}{z}} \]

   3. Applied egg-rr 99.9%

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

      [Start] 78.1	\[ \frac{y}{x} \cdot \frac{\cosh x}{z} \]
      associate-*l/ [=>] 99.9	\[ \color{blue}{\frac{y \cdot \frac{\cosh x}{z}}{x}} \]

   if -9.9999999999999998e294 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 2.00000000000000012e290

   1. Initial program 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq -1 \cdot 10^{+295} \lor \neg \left(\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 2 \cdot 10^{+290}\right):\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.7%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+158} \lor \neg \left(y \leq 6.8 \cdot 10^{+198}\right):\\ \;\;\;\;\frac{y}{x \cdot z} + \frac{0.5}{\frac{z}{x \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \end{array} \]

Alternative 2
Accuracy	87.9%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{-9} \lor \neg \left(z \leq 0.0046\right):\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 3
Accuracy	84.1%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-16}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 4
Accuracy	96.2%
Cost	6848

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

Alternative 5
Accuracy	65.6%
Cost	1100

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{if}\;x \leq -63000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-262}:\\ \;\;\;\;\frac{1}{z \cdot \frac{x}{y}}\\ \mathbf{elif}\;x \leq 10^{+34}:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{1}{x} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	64.5%
Cost	976

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;x \leq -63000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-278}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-200}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Accuracy	65.3%
Cost	976

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{if}\;x \leq -63000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-278}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-209}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Accuracy	65.2%
Cost	976

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{if}\;x \leq -63000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-278}:\\ \;\;\;\;\frac{1}{z \cdot \frac{x}{y}}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-198}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Accuracy	66.3%
Cost	964

\[\begin{array}{l} \mathbf{if}\;z \leq 1.35 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{y}{x} + x \cdot \left(y \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + \frac{0.5}{\frac{z}{x \cdot y}}\\ \end{array} \]

Alternative 10
Accuracy	65.1%
Cost	704

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

Alternative 11
Accuracy	50.0%
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq 4.6 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 12
Accuracy	54.8%
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-60}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 13
Accuracy	49.4%
Cost	320

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

Reproduce

herbie shell --seed 2023163 
(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))