Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Accuracy: 96.0% → 99.7%

Time: 12.8s

Precision: binary64

Cost: 20425

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \frac{\sin y}{y}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-288} \lor \neg \left(t_0 \leq 2 \cdot 10^{-247}\right):\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ (sin y) y))))
   (if (or (<= t_0 -1e-288) (not (<= t_0 2e-247)))
     (/ t_0 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 t_0 = x * (sin(y) / y);
	double tmp;
	if ((t_0 <= -1e-288) || !(t_0 <= 2e-247)) {
		tmp = t_0 / 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) :: t_0
    real(8) :: tmp
    t_0 = x * (sin(y) / y)
    if ((t_0 <= (-1d-288)) .or. (.not. (t_0 <= 2d-247))) then
        tmp = t_0 / 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 t_0 = x * (Math.sin(y) / y);
	double tmp;
	if ((t_0 <= -1e-288) || !(t_0 <= 2e-247)) {
		tmp = t_0 / 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):
	t_0 = x * (math.sin(y) / y)
	tmp = 0
	if (t_0 <= -1e-288) or not (t_0 <= 2e-247):
		tmp = t_0 / 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)
	t_0 = Float64(x * Float64(sin(y) / y))
	tmp = 0.0
	if ((t_0 <= -1e-288) || !(t_0 <= 2e-247))
		tmp = Float64(t_0 / 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)
	t_0 = x * (sin(y) / y);
	tmp = 0.0;
	if ((t_0 <= -1e-288) || ~((t_0 <= 2e-247)))
		tmp = t_0 / 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_] := Block[{t$95$0 = N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-288], N[Not[LessEqual[t$95$0, 2e-247]], $MachinePrecision]], N[(t$95$0 / z), $MachinePrecision], N[(x / N[(y * N[(z / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	96.0%
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 (*.f64 x (/.f64 (sin.f64 y) y)) < -1.00000000000000006e-288 or 2e-247 < (*.f64 x (/.f64 (sin.f64 y) y))

   1. Initial program 99.8%

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

   if -1.00000000000000006e-288 < (*.f64 x (/.f64 (sin.f64 y) y)) < 2e-247

   1. Initial program 82.0%

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

   2. Simplified 99.7%

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

      [Start] 82.0	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      associate-/l* [=>] 99.8	\[ \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
      associate-/r/ [=>] 99.7	\[ \frac{x}{\color{blue}{\frac{z}{\sin y} \cdot y}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-288} \lor \neg \left(x \cdot \frac{\sin y}{y} \leq 2 \cdot 10^{-247}\right):\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	95.7%
Cost	13508

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \]

Alternative 2
Accuracy	95.3%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-8} \lor \neg \left(y \leq 2.6 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot \frac{\sin y}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 3
Accuracy	96.0%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-8} \lor \neg \left(y \leq 2.2 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 4
Accuracy	64.8%
Cost	841

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

Alternative 5
Accuracy	64.9%
Cost	841

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

Alternative 6
Accuracy	64.9%
Cost	840

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

Alternative 7
Accuracy	64.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+57} \lor \neg \left(y \leq 2 \cdot 10^{-14}\right):\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 8
Accuracy	64.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+54}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+75}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} + 1\right) + -1\\ \end{array} \]

Alternative 9
Accuracy	65.2%
Cost	704

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

Alternative 10
Accuracy	55.6%
Cost	192

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

Reproduce

herbie shell --seed 2023133 
(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))