Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.7 → 0.6

Time: 11.6s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;z \leq -4 \cdot 10^{+79}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \mathbf{elif}\;z \leq 10^{-108}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y}{\sin y}}\\ \end{array} \]

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 (<= z -4e+79)
     (/ (* x t_0) z)
     (if (<= z 1e-108) (/ x (/ z t_0)) (/ (/ x z) (/ y (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 t_0 = sin(y) / y;
	double tmp;
	if (z <= -4e+79) {
		tmp = (x * t_0) / z;
	} else if (z <= 1e-108) {
		tmp = x / (z / t_0);
	} else {
		tmp = (x / z) / (y / 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) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (z <= (-4d+79)) then
        tmp = (x * t_0) / z
    else if (z <= 1d-108) then
        tmp = x / (z / t_0)
    else
        tmp = (x / z) / (y / 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 t_0 = Math.sin(y) / y;
	double tmp;
	if (z <= -4e+79) {
		tmp = (x * t_0) / z;
	} else if (z <= 1e-108) {
		tmp = x / (z / t_0);
	} else {
		tmp = (x / z) / (y / Math.sin(y));
	}
	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 <= -4e+79:
		tmp = (x * t_0) / z
	elif z <= 1e-108:
		tmp = x / (z / t_0)
	else:
		tmp = (x / z) / (y / 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)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (z <= -4e+79)
		tmp = Float64(Float64(x * t_0) / z);
	elseif (z <= 1e-108)
		tmp = Float64(x / Float64(z / t_0));
	else
		tmp = Float64(Float64(x / z) / Float64(y / 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)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (z <= -4e+79)
		tmp = (x * t_0) / z;
	elseif (z <= 1e-108)
		tmp = x / (z / t_0);
	else
		tmp = (x / z) / (y / 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_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[z, -4e+79], N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1e-108], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	2.7
Target	0.3
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if z < -3.99999999999999987e79

   1. Initial program 0.1

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

   if -3.99999999999999987e79 < z < 1.00000000000000004e-108

   1. Initial program 5.8

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

   2. Simplified 0.5

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

      [Start] 5.8	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      associate-/l* [=>] 0.5	\[ \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

   if 1.00000000000000004e-108 < z

   1. Initial program 0.6

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

   2. Simplified 11.2

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

      [Start] 0.6	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      associate-/l* [=>] 4.2	\[ \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
      associate-/r/ [=>] 11.2	\[ \frac{x}{\color{blue}{\frac{z}{\sin y} \cdot y}} \]

   3. Applied egg-rr 8.9

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

   4. Applied egg-rr 0.8

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+79}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{elif}\;z \leq 10^{-108}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y}{\sin y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	20425

\[\begin{array}{l} t_0 := x \cdot \frac{\sin y}{y}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-186} \lor \neg \left(t_0 \leq 10^{-177}\right):\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

Alternative 2
Error	3.1
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -0.00032 \lor \neg \left(y \leq 3.6 \cdot 10^{-15}\right):\\ \;\;\;\;x \cdot \frac{\sin y}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + y \cdot \left(y \cdot -0.16666666666666666\right)}}\\ \end{array} \]

Alternative 3
Error	1.6
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+205} \lor \neg \left(z \leq 5.6 \cdot 10^{+126}\right):\\ \;\;\;\;\frac{\sin y}{y \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array} \]

Alternative 4
Error	3.0
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -0.00032:\\ \;\;\;\;x \cdot \frac{\sin y}{z \cdot y}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + y \cdot \left(y \cdot -0.16666666666666666\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{z \cdot y}\\ \end{array} \]

Alternative 5
Error	3.1
Cost	6848

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

Alternative 6
Error	22.4
Cost	969

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

Alternative 7
Error	22.6
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45 \lor \neg \left(y \leq 2.5\right):\\ \;\;\;\;6 \cdot \frac{\frac{x}{z}}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 8
Error	22.5
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+32} \lor \neg \left(y \leq 3.1\right):\\ \;\;\;\;x \cdot \frac{-6}{y \cdot \left(z \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 9
Error	22.5
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -9.4 \cdot 10^{+32} \lor \neg \left(y \leq 3.1\right):\\ \;\;\;\;\frac{x}{z \cdot y} \cdot \frac{-6}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 10
Error	22.8
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+33} \lor \neg \left(y \leq 5 \cdot 10^{+44}\right):\\ \;\;\;\;\left(\frac{x}{z} + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 11
Error	22.4
Cost	704

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

Alternative 12
Error	25.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot y}\\ \end{array} \]

Alternative 13
Error	28.4
Cost	192

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

Reproduce

herbie shell --seed 2022356 
(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))