Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.3% → 97.4%

Time: 12.2s

Precision: binary64

Cost: 6980

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;x \leq 3.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{t_0}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= x 3.5e-77) (/ t_0 (/ z x)) (/ (* x t_0) z))))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (x <= 3.5e-77) {
		tmp = t_0 / (z / x);
	} else {
		tmp = (x * t_0) / 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 = (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) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (x <= 3.5d-77) then
        tmp = t_0 / (z / x)
    else
        tmp = (x * t_0) / z
    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 t_0 = Math.sin(y) / y;
	double tmp;
	if (x <= 3.5e-77) {
		tmp = t_0 / (z / x);
	} else {
		tmp = (x * t_0) / z;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if x <= 3.5e-77:
		tmp = t_0 / (z / x)
	else:
		tmp = (x * t_0) / z
	return tmp

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (x <= 3.5e-77)
		tmp = Float64(t_0 / Float64(z / x));
	else
		tmp = Float64(Float64(x * t_0) / z);
	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)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (x <= 3.5e-77)
		tmp = t_0 / (z / x);
	else
		tmp = (x * t_0) / z;
	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_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, 3.5e-77], N[(t$95$0 / N[(z / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision]]]

Target

Original	96.3%
Target	99.6%
Herbie	97.4%

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

Derivation

1. Split input into 2 regimes

2. if x < 3.50000000000000013e-77

   1. Initial program 94.7%

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

   2. Simplified 87.9%

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

      [Start] 94.7%	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      associate-*l/ [<=] 98.0%	\[ \color{blue}{\frac{x}{z} \cdot \frac{\sin y}{y}} \]
      times-frac [<=] 80.0%	\[ \color{blue}{\frac{x \cdot \sin y}{z \cdot y}} \]
      *-commutative [=>] 80.0%	\[ \frac{\color{blue}{\sin y \cdot x}}{z \cdot y} \]
      associate-*r/ [<=] 87.9%	\[ \color{blue}{\sin y \cdot \frac{x}{z \cdot y}} \]
      *-commutative [=>] 87.9%	\[ \sin y \cdot \frac{x}{\color{blue}{y \cdot z}} \]

   3. Applied egg-rr 97.9%

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

      [Start] 87.9%	\[ \sin y \cdot \frac{x}{y \cdot z} \]
      associate-*r/ [=>] 80.0%	\[ \color{blue}{\frac{\sin y \cdot x}{y \cdot z}} \]
      associate-/r* [=>] 79.6%	\[ \color{blue}{\frac{\frac{\sin y \cdot x}{y}}{z}} \]
      associate-*l/ [<=] 94.7%	\[ \frac{\color{blue}{\frac{\sin y}{y} \cdot x}}{z} \]
      associate-/l* [=>] 97.9%	\[ \color{blue}{\frac{\frac{\sin y}{y}}{\frac{z}{x}}} \]

   if 3.50000000000000013e-77 < x

   1. Initial program 99.8%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{\sin y}{y}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.4%
Cost	6980

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;x \leq 3.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{t_0}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \end{array} \]

Alternative 2
Accuracy	97.3%
Cost	13764

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \frac{x \cdot t_0}{z}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-279}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \]

Alternative 3
Accuracy	97.3%
Cost	13764

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;\frac{x \cdot t_0}{z} \leq -2 \cdot 10^{-279}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \]

Alternative 4
Accuracy	95.7%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -0.85 \lor \neg \left(y \leq 7.5 \cdot 10^{-37}\right):\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(y \cdot y\right) + 1\right) \cdot \frac{x}{z}\\ \end{array} \]

Alternative 5
Accuracy	95.6%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -0.85:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-72}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(y \cdot y\right) + 1\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

Alternative 6
Accuracy	95.8%
Cost	6848

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

Alternative 7
Accuracy	66.3%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3:\\ \;\;\;\;\frac{\frac{x}{z}}{y \cdot \left(y \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+16}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(y \cdot y\right) + 1\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6 \cdot \frac{x}{y \cdot y}}{z}\\ \end{array} \]

Alternative 8
Accuracy	66.5%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3:\\ \;\;\;\;\frac{x \cdot \frac{\frac{1}{z}}{y \cdot 0.16666666666666666}}{y}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+16}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(y \cdot y\right) + 1\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6 \cdot \frac{x}{y \cdot y}}{z}\\ \end{array} \]

Alternative 9
Accuracy	65.7%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+108} \lor \neg \left(y \leq 2.5\right):\\ \;\;\;\;\frac{6}{z} \cdot \frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 10
Accuracy	65.7%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+107}:\\ \;\;\;\;\frac{\frac{6}{y}}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{z} \cdot \frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 11
Accuracy	65.6%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+107}:\\ \;\;\;\;\frac{\frac{6}{y}}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+57}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0.16666666666666666 \cdot \left(z \cdot \left(y \cdot y\right)\right)}\\ \end{array} \]

Alternative 12
Accuracy	65.7%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+107}:\\ \;\;\;\;\frac{\frac{6}{y}}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6 \cdot \frac{x}{y \cdot y}}{z}\\ \end{array} \]

Alternative 13
Accuracy	66.2%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4:\\ \;\;\;\;\frac{\frac{x}{z}}{y \cdot \left(y \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6 \cdot \frac{x}{y \cdot y}}{z}\\ \end{array} \]

Alternative 14
Accuracy	66.2%
Cost	704

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

Alternative 15
Accuracy	58.4%
Cost	192

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

Reproduce

herbie shell --seed 2023272 
(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))