Linear.Quaternion:$ctan from linear-1.19.1.3

Average Accuracy: 88.2% → 99.3%

Time: 11.9s

Precision: binary64

Cost: 20680

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\cosh x \cdot \frac{y}{x}}{z}\\ t_1 := \frac{\frac{y}{z}}{x}\\ \mathbf{if}\;t_0 \leq -2000:\\ \;\;\;\;\cosh x \cdot t_1\\ \mathbf{elif}\;t_0 \leq 10^{+126}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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)) (t_1 (/ (/ y z) x)))
   (if (<= t_0 -2000.0) (* (cosh x) t_1) (if (<= t_0 1e+126) t_0 t_1))))

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 t_1 = (y / z) / x;
	double tmp;
	if (t_0 <= -2000.0) {
		tmp = cosh(x) * t_1;
	} else if (t_0 <= 1e+126) {
		tmp = t_0;
	} 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
    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) :: t_1
    real(8) :: tmp
    t_0 = (cosh(x) * (y / x)) / z
    t_1 = (y / z) / x
    if (t_0 <= (-2000.0d0)) then
        tmp = cosh(x) * t_1
    else if (t_0 <= 1d+126) then
        tmp = t_0
    else
        tmp = t_1
    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 t_1 = (y / z) / x;
	double tmp;
	if (t_0 <= -2000.0) {
		tmp = Math.cosh(x) * t_1;
	} else if (t_0 <= 1e+126) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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
	t_1 = (y / z) / x
	tmp = 0
	if t_0 <= -2000.0:
		tmp = math.cosh(x) * t_1
	elif t_0 <= 1e+126:
		tmp = t_0
	else:
		tmp = t_1
	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)
	t_1 = Float64(Float64(y / z) / x)
	tmp = 0.0
	if (t_0 <= -2000.0)
		tmp = Float64(cosh(x) * t_1);
	elseif (t_0 <= 1e+126)
		tmp = t_0;
	else
		tmp = t_1;
	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;
	t_1 = (y / z) / x;
	tmp = 0.0;
	if (t_0 <= -2000.0)
		tmp = cosh(x) * t_1;
	elseif (t_0 <= 1e+126)
		tmp = t_0;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2000.0], N[(N[Cosh[x], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 1e+126], t$95$0, t$95$1]]]]

Target

Original	88.2%
Target	99.4%
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 3 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < -2e3

   1. Initial program 79.3%

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

   2. Simplified 99.6%

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

      [Start] 79.3	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-*r/ [<=] 79.4	\[ \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]
      associate-/l/ [=>] 80.6	\[ \cosh x \cdot \color{blue}{\frac{y}{z \cdot x}} \]
      associate-/r* [=>] 99.6	\[ \cosh x \cdot \color{blue}{\frac{\frac{y}{z}}{x}} \]

   if -2e3 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 9.99999999999999925e125

   1. Initial program 99.6%

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

   if 9.99999999999999925e125 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 70.0%

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

   2. Simplified 99.2%

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

      [Start] 70.0	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-*r/ [=>] 70.0	\[ \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]
      associate-/r* [<=] 76.7	\[ \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]
      times-frac [=>] 99.2	\[ \color{blue}{\frac{\cosh x}{x} \cdot \frac{y}{z}} \]

   3. Taylor expanded in x around 0 97.7%

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

   4. Applied egg-rr 98.0%

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

      [Start] 97.7	\[ \frac{1}{x} \cdot \frac{y}{z} \]
      associate-*l/ [=>] 98.0	\[ \color{blue}{\frac{1 \cdot \frac{y}{z}}{x}} \]
      *-un-lft-identity [<=] 98.0	\[ \frac{\color{blue}{\frac{y}{z}}}{x} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq -2000:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 10^{+126}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.7%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-15} \lor \neg \left(z \leq 2.7 \cdot 10^{-73}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)\\ \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{+33} \lor \neg \left(z \leq 5.8 \cdot 10^{-57}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 3
Accuracy	98.4%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;z \leq -1.62 \cdot 10^{-13} \lor \neg \left(z \leq 8 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)\\ \end{array} \]

Alternative 4
Accuracy	98.4%
Cost	1096

\[\begin{array}{l} t_0 := \frac{y}{x \cdot z}\\ \mathbf{if}\;z \leq -2 \cdot 10^{-13}:\\ \;\;\;\;t_0 + \left(y \cdot 0.5\right) \cdot \left(x \cdot \frac{1}{z}\right)\\ \mathbf{elif}\;z \leq 8.9 \cdot 10^{-38}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + 0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \]

Alternative 5
Accuracy	98.1%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+34} \lor \neg \left(z \leq 9.2 \cdot 10^{+28}\right):\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)\\ \end{array} \]

Alternative 6
Accuracy	97.4%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot z}{y}}\\ \end{array} \]

Alternative 7
Accuracy	97.5%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-77} \lor \neg \left(y \leq 10^{-14}\right):\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 8
Accuracy	97.3%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 9
Accuracy	87.3%
Cost	320

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

Reproduce

herbie shell --seed 2023137 
(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))