Linear.Quaternion:$ctan from linear-1.19.1.3

Average Accuracy: 87.8% → 99.0%

Time: 12.9s

Precision: binary64

Cost: 20424

Math

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

↓

\[\begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+213}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+159}:\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{1}{x}}}\\ \end{array} \]

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 (<= t_0 -1e+213)
     (* y (/ (+ (* x 0.5) (/ 1.0 x)) z))
     (if (<= t_0 5e+159) (/ t_0 z) (/ y (/ z (/ 1.0 x)))))))

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+213) {
		tmp = y * (((x * 0.5) + (1.0 / x)) / z);
	} else if (t_0 <= 5e+159) {
		tmp = t_0 / z;
	} else {
		tmp = y / (z / (1.0 / 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
    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+213)) then
        tmp = y * (((x * 0.5d0) + (1.0d0 / x)) / z)
    else if (t_0 <= 5d+159) then
        tmp = t_0 / z
    else
        tmp = y / (z / (1.0d0 / x))
    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+213) {
		tmp = y * (((x * 0.5) + (1.0 / x)) / z);
	} else if (t_0 <= 5e+159) {
		tmp = t_0 / z;
	} else {
		tmp = y / (z / (1.0 / x));
	}
	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+213:
		tmp = y * (((x * 0.5) + (1.0 / x)) / z)
	elif t_0 <= 5e+159:
		tmp = t_0 / z
	else:
		tmp = y / (z / (1.0 / x))
	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+213)
		tmp = Float64(y * Float64(Float64(Float64(x * 0.5) + Float64(1.0 / x)) / z));
	elseif (t_0 <= 5e+159)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(y / Float64(z / Float64(1.0 / x)));
	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+213)
		tmp = y * (((x * 0.5) + (1.0 / x)) / z);
	elseif (t_0 <= 5e+159)
		tmp = t_0 / z;
	else
		tmp = y / (z / (1.0 / x));
	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[LessEqual[t$95$0, -1e+213], N[(y * N[(N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+159], N[(t$95$0 / z), $MachinePrecision], N[(y / N[(z / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	87.8%
Target	99.3%
Herbie	99.0%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < -9.99999999999999984e212

   1. Initial program 52.3%

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

   2. Simplified 98.4%

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

      [Start] 52.3	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-*l/ [<=] 52.4	\[ \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]
      times-frac [<=] 98.4	\[ \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}} \]
      associate-*l/ [<=] 98.4	\[ \color{blue}{\frac{\cosh x}{z \cdot x} \cdot y} \]
      *-commutative [=>] 98.4	\[ \frac{\cosh x}{\color{blue}{x \cdot z}} \cdot y \]

   3. Applied egg-rr 27.7%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\cosh x}{x \cdot z}\right)} - 1\right)} \cdot y \]

   4. Simplified 98.3%

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

      [Start] 27.7	\[ \left(e^{\mathsf{log1p}\left(\frac{\cosh x}{x \cdot z}\right)} - 1\right) \cdot y \]
      expm1-def [=>] 70.1	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\cosh x}{x \cdot z}\right)\right)} \cdot y \]
      expm1-log1p [=>] 98.4	\[ \color{blue}{\frac{\cosh x}{x \cdot z}} \cdot y \]
      associate-/r* [=>] 98.3	\[ \color{blue}{\frac{\frac{\cosh x}{x}}{z}} \cdot y \]

   5. Taylor expanded in x around 0 97.8%

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

   if -9.99999999999999984e212 < (*.f64 (cosh.f64 x) (/.f64 y x)) < 5.00000000000000003e159

   1. Initial program 99.6%

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

   if 5.00000000000000003e159 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 63.0%

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

   2. Simplified 98.0%

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

      [Start] 63.0	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-*r/ [=>] 63.0	\[ \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]
      associate-/r* [<=] 98.2	\[ \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]
      times-frac [=>] 98.0	\[ \color{blue}{\frac{\cosh x}{x} \cdot \frac{y}{z}} \]

   3. Taylor expanded in x around 0 97.0%

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

   4. Applied egg-rr 97.1%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq -1 \cdot 10^{+213}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \leq 5 \cdot 10^{+159}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{1}{x}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.4%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -200:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-48}:\\ \;\;\;\;\frac{\frac{y}{x} + y \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \end{array} \]

Alternative 2
Accuracy	98.8%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-18}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-49}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \end{array} \]

Alternative 3
Accuracy	98.7%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;z \leq -50000000:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{x}}{z}\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-49}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \end{array} \]

Alternative 4
Accuracy	98.1%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;y \leq -190:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{y}{x} + y \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{x \cdot 0.5}{z} + \frac{1}{x \cdot z}\right)\\ \end{array} \]

Alternative 5
Accuracy	98.1%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;y \leq -200:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{y}{x} + y \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + 0.5 \cdot \left(x \cdot x\right)}{x \cdot \frac{z}{y}}\\ \end{array} \]

Alternative 6
Accuracy	97.6%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -122 \lor \neg \left(y \leq 1.5 \cdot 10^{-66}\right):\\ \;\;\;\;\left(x \cdot 0.5 + \frac{1}{x}\right) \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 7
Accuracy	97.1%
Cost	969

\[\begin{array}{l} t_0 := x \cdot 0.5 + \frac{1}{x}\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{+100} \lor \neg \left(z \leq 2.9 \cdot 10^{-49}\right):\\ \;\;\;\;y \cdot \frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{y}{z}\\ \end{array} \]

Alternative 8
Accuracy	98.1%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -800 \lor \neg \left(y \leq 4 \cdot 10^{+21}\right):\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + y \cdot \left(x \cdot 0.5\right)}{z}\\ \end{array} \]

Alternative 9
Accuracy	97.5%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -160:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{1}{x}}}\\ \end{array} \]

Alternative 10
Accuracy	97.3%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+42} \lor \neg \left(z \leq 5 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 11
Accuracy	97.6%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-18} \lor \neg \left(z \leq 1000\right):\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 12
Accuracy	87.3%
Cost	320

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

Reproduce

herbie shell --seed 2023141 
(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))