Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Accuracy: 95.9% → 98.0%

Time: 10.9s

Precision: binary64

Cost: 20680

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \frac{x \cdot t_0}{z}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+114}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{-296}:\\ \;\;\;\;\frac{\sin y}{\frac{y}{\frac{x}{z}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (/ (* x t_0) z)))
   (if (<= t_1 -5e+114)
     (/ x (/ z t_0))
     (if (<= t_1 4e-296) (/ (sin y) (/ y (/ x z))) t_1))))

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 t_1 = (x * t_0) / z;
	double tmp;
	if (t_1 <= -5e+114) {
		tmp = x / (z / t_0);
	} else if (t_1 <= 4e-296) {
		tmp = sin(y) / (y / (x / z));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = sin(y) / y
    t_1 = (x * t_0) / z
    if (t_1 <= (-5d+114)) then
        tmp = x / (z / t_0)
    else if (t_1 <= 4d-296) then
        tmp = sin(y) / (y / (x / z))
    else
        tmp = t_1
    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 t_1 = (x * t_0) / z;
	double tmp;
	if (t_1 <= -5e+114) {
		tmp = x / (z / t_0);
	} else if (t_1 <= 4e-296) {
		tmp = Math.sin(y) / (y / (x / z));
	} else {
		tmp = t_1;
	}
	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
	t_1 = (x * t_0) / z
	tmp = 0
	if t_1 <= -5e+114:
		tmp = x / (z / t_0)
	elif t_1 <= 4e-296:
		tmp = math.sin(y) / (y / (x / z))
	else:
		tmp = t_1
	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)
	t_1 = Float64(Float64(x * t_0) / z)
	tmp = 0.0
	if (t_1 <= -5e+114)
		tmp = Float64(x / Float64(z / t_0));
	elseif (t_1 <= 4e-296)
		tmp = Float64(sin(y) / Float64(y / Float64(x / z)));
	else
		tmp = t_1;
	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;
	t_1 = (x * t_0) / z;
	tmp = 0.0;
	if (t_1 <= -5e+114)
		tmp = x / (z / t_0);
	elseif (t_1 <= 4e-296)
		tmp = sin(y) / (y / (x / z));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+114], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-296], N[(N[Sin[y], $MachinePrecision] / N[(y / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Target

Original	95.9%
Target	99.5%
Herbie	98.0%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -5.0000000000000001e114

   1. Initial program 99.7%

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

   2. Simplified 99.7%

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

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

   if -5.0000000000000001e114 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 4e-296

   1. Initial program 93.2%

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

   2. Simplified 90.6%

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

      [Start] 93.2	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      associate-*l/ [<=] 99.0	\[ \color{blue}{\frac{x}{z} \cdot \frac{\sin y}{y}} \]
      times-frac [<=] 87.9	\[ \color{blue}{\frac{x \cdot \sin y}{z \cdot y}} \]
      *-commutative [=>] 87.9	\[ \frac{\color{blue}{\sin y \cdot x}}{z \cdot y} \]
      associate-*r/ [<=] 90.6	\[ \color{blue}{\sin y \cdot \frac{x}{z \cdot y}} \]
      *-commutative [=>] 90.6	\[ \sin y \cdot \frac{x}{\color{blue}{y \cdot z}} \]

   3. Applied egg-rr 96.9%

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

      [Start] 90.6	\[ \sin y \cdot \frac{x}{y \cdot z} \]
      clear-num [=>] 90.0	\[ \sin y \cdot \color{blue}{\frac{1}{\frac{y \cdot z}{x}}} \]
      un-div-inv [=>] 90.1	\[ \color{blue}{\frac{\sin y}{\frac{y \cdot z}{x}}} \]
      associate-/l* [=>] 96.9	\[ \frac{\sin y}{\color{blue}{\frac{y}{\frac{x}{z}}}} \]

   if 4e-296 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.3%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -5 \cdot 10^{+114}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{elif}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq 4 \cdot 10^{-296}:\\ \;\;\;\;\frac{\sin y}{\frac{y}{\frac{x}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	95.0%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -0.011 \lor \neg \left(y \leq 2.2 \cdot 10^{-8}\right):\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 2
Accuracy	94.1%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -0.011:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-155}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

Alternative 3
Accuracy	95.1%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+186}:\\ \;\;\;\;\frac{\sin y}{\frac{y}{\frac{x}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array} \]

Alternative 4
Accuracy	95.1%
Cost	6848

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

Alternative 5
Accuracy	64.8%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;y \leq -12.5 \lor \neg \left(y \leq 1.95\right):\\ \;\;\;\;\frac{x}{y \cdot \left(\left(y \cdot z\right) \cdot 0.16666666666666666 + \frac{z}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}\\ \end{array} \]

Alternative 6
Accuracy	64.3%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -12.5 \lor \neg \left(y \leq 100000000\right):\\ \;\;\;\;\left(\frac{x}{z} + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}\\ \end{array} \]

Alternative 7
Accuracy	63.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+69} \lor \neg \left(y \leq 5.5 \cdot 10^{-66}\right):\\ \;\;\;\;y \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 8
Accuracy	64.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -21000000000000 \lor \neg \left(y \leq 6.2 \cdot 10^{+16}\right):\\ \;\;\;\;\left(\frac{x}{z} + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 9
Accuracy	55.0%
Cost	320

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

Alternative 10
Accuracy	55.1%
Cost	192

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

Reproduce

herbie shell --seed 2023151 
(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))