Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Accuracy: 95.5% → 99.7%

Time: 12.9s

Precision: binary64

Cost: 7113

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;z \leq -6 \cdot 10^{-28} \lor \neg \left(z \leq 5 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \]

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 (or (<= z -6e-28) (not (<= z 5e+30))) (/ (* x t_0) z) (/ x (/ z t_0)))))

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 ((z <= -6e-28) || !(z <= 5e+30)) {
		tmp = (x * t_0) / z;
	} else {
		tmp = x / (z / t_0);
	}
	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 ((z <= (-6d-28)) .or. (.not. (z <= 5d+30))) then
        tmp = (x * t_0) / z
    else
        tmp = x / (z / t_0)
    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 ((z <= -6e-28) || !(z <= 5e+30)) {
		tmp = (x * t_0) / z;
	} else {
		tmp = x / (z / t_0);
	}
	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 (z <= -6e-28) or not (z <= 5e+30):
		tmp = (x * t_0) / z
	else:
		tmp = x / (z / t_0)
	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 ((z <= -6e-28) || !(z <= 5e+30))
		tmp = Float64(Float64(x * t_0) / z);
	else
		tmp = Float64(x / Float64(z / t_0));
	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 ((z <= -6e-28) || ~((z <= 5e+30)))
		tmp = (x * t_0) / z;
	else
		tmp = x / (z / t_0);
	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[Or[LessEqual[z, -6e-28], N[Not[LessEqual[z, 5e+30]], $MachinePrecision]], N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Target

Original	95.5%
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 < -6.00000000000000005e-28 or 4.9999999999999998e30 < z

   1. Initial program 99.8%

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

   if -6.00000000000000005e-28 < z < 4.9999999999999998e30

   1. Initial program 90.6%

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

   2. Simplified 99.6%

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

      [Start] 90.6	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      associate-/l* [=>] 99.6	\[ \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-28} \lor \neg \left(z \leq 5 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	95.6%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -0.00031 \lor \neg \left(y \leq 10^{-7}\right):\\ \;\;\;\;x \cdot \frac{\sin y}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \end{array} \]

Alternative 2
Accuracy	95.7%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -0.000215:\\ \;\;\;\;x \cdot \frac{\sin y}{z \cdot y}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 3
Accuracy	95.7%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4
Accuracy	95.6%
Cost	6848

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

Alternative 5
Accuracy	65.3%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;y \leq -6.2 \lor \neg \left(y \leq 8200000000000\right):\\ \;\;\;\;\frac{x}{y \cdot \left(\left(z \cdot y\right) \cdot 0.16666666666666666 + \frac{z}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \end{array} \]

Alternative 6
Accuracy	65.1%
Cost	1032

\[\begin{array}{l} \mathbf{if}\;y \leq -6.2:\\ \;\;\;\;\left(1 + \frac{x}{z}\right) + -1\\ \mathbf{elif}\;y \leq 3.5:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \frac{\frac{-x}{z}}{y \cdot y}\right)\\ \end{array} \]

Alternative 7
Accuracy	64.9%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -6.2:\\ \;\;\;\;\left(1 + \frac{x}{z}\right) + -1\\ \mathbf{elif}\;y \leq 3.5:\\ \;\;\;\;\frac{x}{\frac{z}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{z \cdot y}{x}}\\ \end{array} \]

Alternative 8
Accuracy	64.9%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -6.2:\\ \;\;\;\;\left(1 + \frac{x}{z}\right) + -1\\ \mathbf{elif}\;y \leq 3.5:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{z \cdot y}{x}}\\ \end{array} \]

Alternative 9
Accuracy	64.7%
Cost	776

\[\begin{array}{l} \mathbf{if}\;y \leq -7500000:\\ \;\;\;\;\left(1 + \frac{x}{z}\right) + -1\\ \mathbf{elif}\;y \leq 3.1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{z \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 10
Accuracy	64.7%
Cost	776

\[\begin{array}{l} \mathbf{if}\;y \leq -7500000:\\ \;\;\;\;\left(1 + \frac{x}{z}\right) + -1\\ \mathbf{elif}\;y \leq 3.1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{z}{\frac{x}{y}}}\\ \end{array} \]

Alternative 11
Accuracy	64.7%
Cost	776

\[\begin{array}{l} \mathbf{if}\;y \leq -7500000:\\ \;\;\;\;\left(1 + \frac{x}{z}\right) + -1\\ \mathbf{elif}\;y \leq 3.1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{z \cdot y}{x}}\\ \end{array} \]

Alternative 12
Accuracy	64.3%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-8} \lor \neg \left(y \leq 10^{-8}\right):\\ \;\;\;\;y \cdot \frac{x}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 13
Accuracy	64.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -7400000 \lor \neg \left(y \leq 5.1 \cdot 10^{+35}\right):\\ \;\;\;\;\left(1 + \frac{x}{z}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 14
Accuracy	55.6%
Cost	320

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

Alternative 15
Accuracy	55.9%
Cost	192

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

Reproduce

herbie shell --seed 2023129 
(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))