Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Accuracy: 88.0% → 91.7%

Time: 13.7s

Precision: binary64

Cost: 7113

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-12} \lor \neg \left(z \leq 3.7 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3e-12) (not (<= z 3.7e-14)))
   (/ (/ x (/ y (sin y))) z)
   (/ x (* y (/ z (sin 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 ((z <= -3e-12) || !(z <= 3.7e-14)) {
		tmp = (x / (y / sin(y))) / z;
	} else {
		tmp = x / (y * (z / sin(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 ((z <= (-3d-12)) .or. (.not. (z <= 3.7d-14))) then
        tmp = (x / (y / sin(y))) / z
    else
        tmp = x / (y * (z / sin(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 ((z <= -3e-12) || !(z <= 3.7e-14)) {
		tmp = (x / (y / Math.sin(y))) / z;
	} else {
		tmp = x / (y * (z / Math.sin(y)));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if (z <= -3e-12) or not (z <= 3.7e-14):
		tmp = (x / (y / math.sin(y))) / z
	else:
		tmp = x / (y * (z / math.sin(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 ((z <= -3e-12) || !(z <= 3.7e-14))
		tmp = Float64(Float64(x / Float64(y / sin(y))) / z);
	else
		tmp = Float64(x / Float64(y * Float64(z / sin(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 ((z <= -3e-12) || ~((z <= 3.7e-14)))
		tmp = (x / (y / sin(y))) / z;
	else
		tmp = x / (y * (z / sin(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[z, -3e-12], N[Not[LessEqual[z, 3.7e-14]], $MachinePrecision]], N[(N[(x / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x / N[(y * N[(z / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	88.0%
Target	91.5%
Herbie	91.7%

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

Derivation

1. Split input into 2 regimes

2. if z < -3.0000000000000001e-12 or 3.70000000000000001e-14 < z

   1. Initial program 99.8%

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

   2. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{\frac{x}{\frac{y}{\sin y}}}}{z} \]
      Step-by-step derivation

      [Start] 99.8	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      clear-num [=>] 99.8	\[ \frac{x \cdot \color{blue}{\frac{1}{\frac{y}{\sin y}}}}{z} \]
      un-div-inv [=>] 99.9	\[ \frac{\color{blue}{\frac{x}{\frac{y}{\sin y}}}}{z} \]

   if -3.0000000000000001e-12 < z < 3.70000000000000001e-14

   1. Initial program 73.7%

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

   2. Simplified 80.7%

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\sin y} \cdot y}} \]
      Step-by-step derivation

      [Start] 73.7	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      associate-/l* [=>] 80.7	\[ \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
      associate-/r/ [=>] 80.7	\[ \frac{x}{\color{blue}{\frac{z}{\sin y} \cdot y}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-12} \lor \neg \left(z \leq 3.7 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	87.7%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-8} \lor \neg \left(y \leq 1.65 \cdot 10^{-28}\right):\\ \;\;\;\;x \cdot \frac{\sin y}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 2
Accuracy	87.7%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-8} \lor \neg \left(y \leq 2 \cdot 10^{-8}\right):\\ \;\;\;\;\sin y \cdot \frac{x}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 3
Accuracy	89.0%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+126} \lor \neg \left(z \leq 6.2 \cdot 10^{-19}\right):\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \end{array} \]

Alternative 4
Accuracy	91.0%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-167} \lor \neg \left(z \leq 9 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

Alternative 5
Accuracy	87.8%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-8}:\\ \;\;\;\;\sin y \cdot \frac{x}{z \cdot y}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-19}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 6
Accuracy	59.5%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -3000000:\\ \;\;\;\;6 \cdot \frac{x}{y \cdot \left(z \cdot y\right)}\\ \mathbf{elif}\;y \leq 6.3:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(\left(z \cdot y\right) \cdot 0.16666666666666666\right)}\\ \end{array} \]

Alternative 7
Accuracy	59.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -2.5 \lor \neg \left(y \leq 2.45\right):\\ \;\;\;\;6 \cdot \frac{x}{y \cdot \left(z \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 8
Accuracy	59.3%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -2.5:\\ \;\;\;\;6 \cdot \frac{x}{y \cdot \left(z \cdot y\right)}\\ \mathbf{elif}\;y \leq 2.45:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(\left(z \cdot y\right) \cdot 0.16666666666666666\right)}\\ \end{array} \]

Alternative 9
Accuracy	58.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -15000000000000 \lor \neg \left(y \leq 9.5\right):\\ \;\;\;\;\left(\frac{x}{z} + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 10
Accuracy	51.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-264}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 11
Accuracy	58.9%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -15000000000000:\\ \;\;\;\;\left(\frac{x}{z} + 1\right) + -1\\ \mathbf{elif}\;y \leq 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 12
Accuracy	59.5%
Cost	704

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

Alternative 13
Accuracy	51.0%
Cost	320

\[\frac{1}{\frac{z}{x}} \]

Alternative 14
Accuracy	51.1%
Cost	192

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

Reproduce

herbie shell --seed 2023157 
(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))