Average Error: 2.5 → 0.5
Time: 12.3s
Precision: binary64
\[\frac{x \cdot \frac{\sin y}{y}}{z} \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -3.4387833264871806 \cdot 10^{+50}:\\ \;\;\;\;\frac{\frac{\sin y \cdot x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;z \leq 8.570453442873267 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \end{array}\\ \end{array} \]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -3.4387833264871806 \cdot 10^{+50}:\\
\;\;\;\;\frac{\frac{\sin y \cdot x}{y}}{z}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_0 := \frac{\sin y}{y}\\
\mathbf{if}\;z \leq 8.570453442873267 \cdot 10^{-33}:\\
\;\;\;\;\frac{x}{\frac{z}{t_0}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x \cdot t_0}{z}\\


\end{array}\\


\end{array}

(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= z -3.4387833264871806e+50)
   (/ (/ (* (sin y) x) y) z)
   (let* ((t_0 (/ (sin y) y)))
     (if (<= z 8.570453442873267e-33) (/ x (/ z t_0)) (/ (* x t_0) z)))))
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 <= -3.4387833264871806e+50) {
		tmp = ((sin(y) * x) / y) / z;
	} else {
		double t_0 = sin(y) / y;
		double tmp_1;
		if (z <= 8.570453442873267e-33) {
			tmp_1 = x / (z / t_0);
		} else {
			tmp_1 = (x * t_0) / z;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original2.5
Target0.3
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if z < -3.4387833264871806e50

    1. Initial program 0.1

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

    2. Taylor expanded in x around 0 1.6

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

    if -3.4387833264871806e50 < z < 8.57045344287326717e-33

    1. Initial program 4.8

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

    2. Applied associate-/l*_binary640.2

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

    if 8.57045344287326717e-33 < z

    1. Initial program 0.2

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4387833264871806 \cdot 10^{+50}:\\ \;\;\;\;\frac{\frac{\sin y \cdot x}{y}}{z}\\ \mathbf{elif}\;z \leq 8.570453442873267 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(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))