Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.1% → 99.7%

Time: 15.1s

Precision: binary64

Cost: 7113

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-34} \lor \neg \left(z \leq 65\right):\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5e-34) (not (<= z 65.0)))
   (/ (* x (/ (sin y) 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 <= -5e-34) || !(z <= 65.0)) {
		tmp = (x * (sin(y) / 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 <= (-5d-34)) .or. (.not. (z <= 65.0d0))) then
        tmp = (x * (sin(y) / 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 <= -5e-34) || !(z <= 65.0)) {
		tmp = (x * (Math.sin(y) / 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 <= -5e-34) or not (z <= 65.0):
		tmp = (x * (math.sin(y) / 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 <= -5e-34) || !(z <= 65.0))
		tmp = Float64(Float64(x * Float64(sin(y) / 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 <= -5e-34) || ~((z <= 65.0)))
		tmp = (x * (sin(y) / 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, -5e-34], N[Not[LessEqual[z, 65.0]], $MachinePrecision]], N[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x / N[(y * N[(z / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	96.1%
Target	99.6%
Herbie	99.7%

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

Derivation

1. Split input into 2 regimes

2. if z < -5.0000000000000003e-34 or 65 < z

   1. Initial program 99.9%

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

   if -5.0000000000000003e-34 < z < 65

   1. Initial program 91.8%

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

   2. Simplified 99.7%

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

      [Start] 91.8%	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      associate-/l* [=>] 99.7%	\[ \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
      associate-/r/ [=>] 99.7%	\[ \frac{x}{\color{blue}{\frac{z}{\sin y} \cdot y}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-34} \lor \neg \left(z \leq 65\right):\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-34} \lor \neg \left(z \leq 65\right):\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

Alternative 2
Accuracy	95.8%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -0.092 \lor \neg \left(y \leq 1.52 \cdot 10^{-10}\right):\\ \;\;\;\;\sin y \cdot \frac{x}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 3
Accuracy	96.0%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -0.092 \lor \neg \left(y \leq 1.52 \cdot 10^{-10}\right):\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 4
Accuracy	96.1%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -0.092 \lor \neg \left(y \leq 4 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \end{array} \]

Alternative 5
Accuracy	97.6%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+114} \lor \neg \left(z \leq 6.2 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{\sin y}{y \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

Alternative 6
Accuracy	97.9%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+110} \lor \neg \left(z \leq 1.4 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{\sin y}{\frac{y}{\frac{x}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

Alternative 7
Accuracy	66.5%
Cost	968

\[\begin{array}{l} t_0 := z \cdot \left(y \cdot 0.16666666666666666\right)\\ \mathbf{if}\;y \leq -2.45 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{t_0}\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y \cdot t_0}{x}}\\ \end{array} \]

Alternative 8
Accuracy	65.7%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+99}:\\ \;\;\;\;\left(\frac{x}{z} \cdot \frac{6}{y}\right) \cdot \frac{1}{y}\\ \mathbf{elif}\;y \leq 19:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(z \cdot \left(y \cdot 0.16666666666666666\right)\right)}{x}}\\ \end{array} \]

Alternative 9
Accuracy	65.9%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+118} \lor \neg \left(y \leq 2.5\right):\\ \;\;\;\;6 \cdot \frac{x}{y \cdot \left(z \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 10
Accuracy	66.7%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -390000000000 \lor \neg \left(y \leq 2.5\right):\\ \;\;\;\;\frac{6}{y \cdot \left(z \cdot \frac{y}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 11
Accuracy	66.4%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -550000000000:\\ \;\;\;\;\frac{6}{y \cdot \left(z \cdot \frac{y}{x}\right)}\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y \cdot \left(z \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]

Alternative 12
Accuracy	66.5%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -390000000000:\\ \;\;\;\;\left(\frac{x}{z} \cdot \frac{6}{y}\right) \cdot \frac{1}{y}\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y \cdot \left(z \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]

Alternative 13
Accuracy	66.5%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -390000000000:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{z \cdot \left(y \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y \cdot \left(z \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]

Alternative 14
Accuracy	60.6%
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq 10000000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot y}{y}}\\ \end{array} \]

Alternative 15
Accuracy	58.5%
Cost	192

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

Reproduce

herbie shell --seed 2023263 
(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))