Linear.Quaternion:$ctan from linear-1.19.1.3

Average Accuracy: 87.5% → 99.3%

Time: 12.7s

Precision: binary64

Cost: 20425

Math

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

↓

\[\begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+260} \lor \neg \left(t_0 \leq 5 \cdot 10^{+259}\right):\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{z}\\ \end{array} \]

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))))
   (if (or (<= t_0 -1e+260) (not (<= t_0 5e+259))) (/ (/ y z) x) (/ t_0 z))))

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);
	double tmp;
	if ((t_0 <= -1e+260) || !(t_0 <= 5e+259)) {
		tmp = (y / z) / x;
	} else {
		tmp = t_0 / 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
    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)
    if ((t_0 <= (-1d+260)) .or. (.not. (t_0 <= 5d+259))) then
        tmp = (y / z) / x
    else
        tmp = t_0 / z
    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);
	double tmp;
	if ((t_0 <= -1e+260) || !(t_0 <= 5e+259)) {
		tmp = (y / z) / x;
	} else {
		tmp = t_0 / z;
	}
	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)
	tmp = 0
	if (t_0 <= -1e+260) or not (t_0 <= 5e+259):
		tmp = (y / z) / x
	else:
		tmp = t_0 / z
	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(cosh(x) * Float64(y / x))
	tmp = 0.0
	if ((t_0 <= -1e+260) || !(t_0 <= 5e+259))
		tmp = Float64(Float64(y / z) / x);
	else
		tmp = Float64(t_0 / z);
	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);
	tmp = 0.0;
	if ((t_0 <= -1e+260) || ~((t_0 <= 5e+259)))
		tmp = (y / z) / x;
	else
		tmp = t_0 / z;
	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[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e+260], N[Not[LessEqual[t$95$0, 5e+259]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], N[(t$95$0 / z), $MachinePrecision]]]

Target

Original	87.5%
Target	99.2%
Herbie	99.3%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < -1.00000000000000007e260 or 5.00000000000000033e259 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 33.4%

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

   2. Simplified 98.7%

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

      [Start] 33.4	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-*r/ [=>] 33.4	\[ \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]
      associate-/r* [<=] 98.5	\[ \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]
      times-frac [=>] 98.7	\[ \color{blue}{\frac{\cosh x}{x} \cdot \frac{y}{z}} \]

   3. Taylor expanded in x around 0 98.0%

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

   4. Applied egg-rr 98.1%

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

      [Start] 98.0	\[ \frac{1}{x} \cdot \frac{y}{z} \]
      associate-*l/ [=>] 98.1	\[ \color{blue}{\frac{1 \cdot \frac{y}{z}}{x}} \]
      *-un-lft-identity [<=] 98.1	\[ \frac{\color{blue}{\frac{y}{z}}}{x} \]

   if -1.00000000000000007e260 < (*.f64 (cosh.f64 x) (/.f64 y x)) < 5.00000000000000033e259

   1. Initial program 99.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq -1 \cdot 10^{+260} \lor \neg \left(\cosh x \cdot \frac{y}{x} \leq 5 \cdot 10^{+259}\right):\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.7%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -2200000000000 \lor \neg \left(z \leq 3.8 \cdot 10^{-53}\right):\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \]

Alternative 2
Accuracy	98.8%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -2200000000000 \lor \neg \left(z \leq 3.8 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \]

Alternative 3
Accuracy	98.7%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;z \leq -2200000000000:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \end{array} \]

Alternative 4
Accuracy	98.4%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\cosh x}{x}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 5
Accuracy	97.6%
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -2200000000000:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \end{array} \]

Alternative 6
Accuracy	98.0%
Cost	968

\[\begin{array}{l} t_0 := x \cdot 0.5 + \frac{1}{x}\\ \mathbf{if}\;y \leq -8.8 \cdot 10^{+20}:\\ \;\;\;\;y \cdot \frac{t_0}{z}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-16}:\\ \;\;\;\;\frac{y \cdot t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 7
Accuracy	98.1%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+20}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 8
Accuracy	97.7%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+20} \lor \neg \left(y \leq 4 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 9
Accuracy	97.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+19} \lor \neg \left(y \leq 2 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 10
Accuracy	87.2%
Cost	320

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

Reproduce

herbie shell --seed 2023135 
(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))