Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 4.63% → 0.55%

Time: 11.1s

Precision: binary64

Cost: 7113

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \lor \neg \left(z \leq 1.45 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.0) (not (<= z 1.45e-83)))
   (* (/ (sin y) y) (/ x 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 <= -2.0) || !(z <= 1.45e-83)) {
		tmp = (sin(y) / y) * (x / 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 <= (-2.0d0)) .or. (.not. (z <= 1.45d-83))) then
        tmp = (sin(y) / y) * (x / 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 <= -2.0) || !(z <= 1.45e-83)) {
		tmp = (Math.sin(y) / y) * (x / 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 <= -2.0) or not (z <= 1.45e-83):
		tmp = (math.sin(y) / y) * (x / 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 <= -2.0) || !(z <= 1.45e-83))
		tmp = Float64(Float64(sin(y) / y) * Float64(x / 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 <= -2.0) || ~((z <= 1.45e-83)))
		tmp = (sin(y) / y) * (x / 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, -2.0], N[Not[LessEqual[z, 1.45e-83]], $MachinePrecision]], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(z / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	4.63%
Target	0.4%
Herbie	0.55%

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

Derivation

1. Split input into 2 regimes

2. if z < -2 or 1.45e-83 < z

   1. Initial program 0.59

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

   2. Simplified 7.33

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

      [Start] 0.59	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      associate-/l* [=>] 7.33	\[ \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

   3. Applied egg-rr 0.64

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

   if -2 < z < 1.45e-83

   1. Initial program 11.03

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

   2. Simplified 0.41

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

      [Start] 11.03	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      associate-/l* [=>] 0.36	\[ \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
      associate-/r/ [=>] 0.41	\[ \frac{x}{\color{blue}{\frac{z}{\sin y} \cdot y}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.55

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \lor \neg \left(z \leq 1.45 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.71%
Cost	13636

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := x \cdot t_0\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{t_0}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{z}\\ \end{array} \]

Alternative 2
Error	2.68%
Cost	13636

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := x \cdot t_0\\ \mathbf{if}\;t_1 \leq 10^{-158}:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{z}\\ \end{array} \]

Alternative 3
Error	2.69%
Cost	13636

\[\begin{array}{l} t_0 := x \cdot \frac{\sin y}{y}\\ \mathbf{if}\;t_0 \leq 10^{-158}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{z}\\ \end{array} \]

Alternative 4
Error	4.63%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -0.024 \lor \neg \left(y \leq 0.00022\right):\\ \;\;\;\;x \cdot \frac{\sin y}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}\\ \end{array} \]

Alternative 5
Error	4.75%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+83}:\\ \;\;\;\;x \cdot \frac{\sin y}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \end{array} \]

Alternative 6
Error	4.73%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+108}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \end{array} \]

Alternative 7
Error	2.65%
Cost	6980

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;z \leq 10^{-34}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \end{array} \]

Alternative 8
Error	35.82%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4:\\ \;\;\;\;\frac{6 \cdot \frac{x}{y \cdot z}}{y}\\ \mathbf{elif}\;y \leq 2.3:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x} \cdot \frac{y \cdot y}{6}}\\ \end{array} \]

Alternative 9
Error	35.62%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -21500:\\ \;\;\;\;\frac{6 \cdot \frac{x}{y \cdot z}}{y}\\ \mathbf{elif}\;y \leq 75:\\ \;\;\;\;\frac{x}{\frac{z}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x} \cdot \frac{y \cdot y}{6}}\\ \end{array} \]

Alternative 10
Error	35.79%
Cost	841

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

Alternative 11
Error	35.85%
Cost	840

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

Alternative 12
Error	35.84%
Cost	840

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

Alternative 13
Error	35.83%
Cost	840

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

Alternative 14
Error	36.55%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+84} \lor \neg \left(y \leq 4.6 \cdot 10^{+31}\right):\\ \;\;\;\;\left(\frac{x}{z} + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 15
Error	35.72%
Cost	704

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

Alternative 16
Error	44.53%
Cost	192

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

Reproduce

herbie shell --seed 2023121 
(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))