Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 12.04% → 1.84%

Time: 11.9s

Precision: binary64

Cost: 20424

Math

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

↓

\[\begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{x}}{z}\\ \end{array} \]

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 (<= t_0 -1e+168)
     (/ (/ y z) x)
     (if (<= t_0 2e+99) (/ t_0 z) (* y (/ (/ (cosh x) x) 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+168) {
		tmp = (y / z) / x;
	} else if (t_0 <= 2e+99) {
		tmp = t_0 / z;
	} else {
		tmp = y * ((cosh(x) / x) / 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+168)) then
        tmp = (y / z) / x
    else if (t_0 <= 2d+99) then
        tmp = t_0 / z
    else
        tmp = y * ((cosh(x) / x) / 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+168) {
		tmp = (y / z) / x;
	} else if (t_0 <= 2e+99) {
		tmp = t_0 / z;
	} else {
		tmp = y * ((Math.cosh(x) / x) / 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+168:
		tmp = (y / z) / x
	elif t_0 <= 2e+99:
		tmp = t_0 / z
	else:
		tmp = y * ((math.cosh(x) / x) / 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+168)
		tmp = Float64(Float64(y / z) / x);
	elseif (t_0 <= 2e+99)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(y * Float64(Float64(cosh(x) / x) / 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+168)
		tmp = (y / z) / x;
	elseif (t_0 <= 2e+99)
		tmp = t_0 / z;
	else
		tmp = y * ((cosh(x) / x) / 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[LessEqual[t$95$0, -1e+168], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 2e+99], N[(t$95$0 / z), $MachinePrecision], N[(y * N[(N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	12.04%
Target	0.71%
Herbie	1.84%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < -9.9999999999999993e167

   1. Initial program 37.55

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

   2. Simplified 2.49

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

      [Start] 37.55	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-*r/ [=>] 37.55	\[ \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]
      associate-/l/ [=>] 2.49	\[ \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}} \]

   3. Taylor expanded in x around 0 3.51

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

   4. Simplified 3.19

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

      [Start] 3.51	\[ \frac{y}{z \cdot x} \]
      associate-/r* [=>] 3.19	\[ \color{blue}{\frac{\frac{y}{z}}{x}} \]

   if -9.9999999999999993e167 < (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.9999999999999999e99

   1. Initial program 0.4

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

   if 1.9999999999999999e99 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 27.04

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

   2. Simplified 5.05

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

      [Start] 27.04	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-*l/ [<=] 27.2	\[ \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]
      times-frac [<=] 4.28	\[ \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}} \]
      associate-*l/ [<=] 5.05	\[ \color{blue}{\frac{\cosh x}{z \cdot x} \cdot y} \]
      *-commutative [=>] 5.05	\[ \frac{\cosh x}{\color{blue}{x \cdot z}} \cdot y \]

   3. Applied egg-rr 69.24

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

   4. Simplified 5.07

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

      [Start] 69.24	\[ \left(e^{\mathsf{log1p}\left(\frac{\cosh x}{x \cdot z}\right)} - 1\right) \cdot y \]
      expm1-def [=>] 33.47	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\cosh x}{x \cdot z}\right)\right)} \cdot y \]
      expm1-log1p [=>] 5.05	\[ \color{blue}{\frac{\cosh x}{x \cdot z}} \cdot y \]
      associate-/r* [=>] 5.07	\[ \color{blue}{\frac{\frac{\cosh x}{x}}{z}} \cdot y \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.84

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq -1 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{x}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.89%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -14000000 \lor \neg \left(z \leq 6.5 \cdot 10^{+170}\right):\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 2
Error	2.93%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;z \leq -40000000000:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+174}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \end{array} \]

Alternative 3
Error	2.82%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;z \leq -500000000:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+170}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{x}}{z}\\ \end{array} \]

Alternative 4
Error	2.33%
Cost	7112

\[\begin{array}{l} t_0 := \frac{\frac{y}{z}}{x}\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{-97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{-68}:\\ \;\;\;\;\frac{\cosh x}{\frac{z}{\frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot t_0\\ \end{array} \]

Alternative 5
Error	2.58%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-80}:\\ \;\;\;\;\frac{\cosh x}{\frac{z}{\frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array} \]

Alternative 6
Error	2.8%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{y}{x} + y \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \]

Alternative 7
Error	2.76%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)\\ \end{array} \]

Alternative 8
Error	2.56%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{y}{x} + y \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)\\ \end{array} \]

Alternative 9
Error	3.76%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-35} \lor \neg \left(z \leq 4.4 \cdot 10^{-148}\right):\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 10
Error	3.2%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 11
Error	13.07%
Cost	320

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

Reproduce

herbie shell --seed 2023090 
(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))