Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Accuracy: 96.5% → 99.2%

Time: 11.6s

Precision: binary64

Cost: 7113

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+73} \lor \neg \left(x \leq 5 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5e+73) (not (<= x 5e-121)))
   (/ (/ x (/ y (sin y))) z)
   (/ x (/ z (/ (sin y) y)))))

C

double code(double x, double y, double z) {
	return (x * (sin(y) / y)) / z;
}

↓

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5e+73) || !(x <= 5e-121)) {
		tmp = (x / (y / sin(y))) / z;
	} else {
		tmp = x / (z / (sin(y) / y));
	}
	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 = (x * (sin(y) / y)) / 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) :: tmp
    if ((x <= (-5d+73)) .or. (.not. (x <= 5d-121))) then
        tmp = (x / (y / sin(y))) / z
    else
        tmp = x / (z / (sin(y) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return (x * (Math.sin(y) / y)) / z;
}

↓

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5e+73) || !(x <= 5e-121)) {
		tmp = (x / (y / Math.sin(y))) / z;
	} else {
		tmp = x / (z / (Math.sin(y) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	return (x * (math.sin(y) / y)) / z

↓

def code(x, y, z):
	tmp = 0
	if (x <= -5e+73) or not (x <= 5e-121):
		tmp = (x / (y / math.sin(y))) / z
	else:
		tmp = x / (z / (math.sin(y) / y))
	return tmp

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(sin(y) / y)) / z)
end

↓

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5e+73) || !(x <= 5e-121))
		tmp = Float64(Float64(x / Float64(y / sin(y))) / z);
	else
		tmp = Float64(x / Float64(z / Float64(sin(y) / y)));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (sin(y) / y)) / z;
end

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -5e+73) || ~((x <= 5e-121)))
		tmp = (x / (y / sin(y))) / z;
	else
		tmp = x / (z / (sin(y) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

↓

code[x_, y_, z_] := If[Or[LessEqual[x, -5e+73], N[Not[LessEqual[x, 5e-121]], $MachinePrecision]], N[(N[(x / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x / N[(z / N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	96.5%
Target	99.5%
Herbie	99.2%

\[\begin{array}{l} \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if x < -4.99999999999999976e73 or 4.99999999999999989e-121 < x

   1. Initial program 99.0%

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

   2. Applied egg-rr 98.9%

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

      [Start] 99.0	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      clear-num [=>] 98.8	\[ \frac{x \cdot \color{blue}{\frac{1}{\frac{y}{\sin y}}}}{z} \]
      un-div-inv [=>] 98.9	\[ \frac{\color{blue}{\frac{x}{\frac{y}{\sin y}}}}{z} \]

   if -4.99999999999999976e73 < x < 4.99999999999999989e-121

   1. Initial program 94.1%

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

   2. Simplified 99.5%

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

      [Start] 94.1	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      associate-/l* [=>] 99.5	\[ \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+73} \lor \neg \left(x \leq 5 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	95.3%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-8} \lor \neg \left(y \leq 1.4 \cdot 10^{-14}\right):\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 2
Accuracy	99.2%
Cost	7113

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;z \leq -2 \cdot 10^{+54} \lor \neg \left(z \leq 6.4 \cdot 10^{-156}\right):\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \]

Alternative 3
Accuracy	95.9%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \]

Alternative 4
Accuracy	96.0%
Cost	7112

\[\begin{array}{l} t_0 := \frac{\sin y}{z}\\ \mathbf{if}\;y \leq -5 \cdot 10^{-8}:\\ \;\;\;\;t_0 \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-27}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t_0}{y}\\ \end{array} \]

Alternative 5
Accuracy	95.3%
Cost	6848

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

Alternative 6
Accuracy	64.4%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -25:\\ \;\;\;\;\frac{x}{0.16666666666666666 \cdot \left(z \cdot \left(y \cdot y\right)\right)}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + \left(y \cdot y\right) \cdot -0.16666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{6 \cdot \frac{\frac{x}{y}}{y}}{z}\\ \end{array} \]

Alternative 7
Accuracy	64.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45 \lor \neg \left(y \leq 4.2 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{x}{0.16666666666666666 \cdot \left(z \cdot \left(y \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 8
Accuracy	64.3%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45:\\ \;\;\;\;\frac{x}{0.16666666666666666 \cdot \left(z \cdot \left(y \cdot y\right)\right)}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6 \cdot \frac{\frac{x}{y}}{y}}{z}\\ \end{array} \]

Alternative 9
Accuracy	64.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+57} \lor \neg \left(y \leq 4.2 \cdot 10^{-14}\right):\\ \;\;\;\;\left(\frac{x}{z} + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 10
Accuracy	57.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -7.3 \cdot 10^{-236}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 11
Accuracy	65.1%
Cost	704

\[\frac{\frac{x}{1 + \left(y \cdot y\right) \cdot 0.16666666666666666}}{z} \]

Alternative 12
Accuracy	55.8%
Cost	192

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

Reproduce

herbie shell --seed 2023146 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< z -4.2173720203427147e-29) (/ (* x (/ 1.0 (/ y (sin y)))) z) (if (< z 4.446702369113811e+64) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1.0 (/ y (sin y)))) z)))

  (/ (* x (/ (sin y) y)) z))