Average Error: 2.9 → 1.3
Time: 4.8s
Precision: binary64
\[\frac{x \cdot \frac{\sin y}{y}}{z} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 0.0012779817648914992:\\ \;\;\;\;\frac{x}{\frac{z}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin y}{y}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin y \cdot \frac{1}{y}\right)\right)}{z}\\ \end{array} \]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 0.0012779817648914992:\\
\;\;\;\;\frac{x}{\frac{z}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin y}{y}\right)\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin y \cdot \frac{1}{y}\right)\right)}{z}\\


\end{array}

(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= x 0.0012779817648914992)
   (/ x (/ z (expm1 (log1p (/ (sin y) y)))))
   (/ (* x (expm1 (log1p (* (sin y) (/ 1.0 y))))) 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 (x <= 0.0012779817648914992) {
		tmp = x / (z / expm1(log1p((sin(y) / y))));
	} else {
		tmp = (x * expm1(log1p((sin(y) * (1.0 / y))))) / z;
	}
	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

Original2.9
Target0.3
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if x < 0.0012779817648914992

    1. Initial program 3.7

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

    2. Applied div-inv_binary643.8

      \[\leadsto \frac{x \cdot \color{blue}{\left(\sin y \cdot \frac{1}{y}\right)}}{z} \]

    3. Applied associate-/l*_binary641.7

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\sin y \cdot \frac{1}{y}}}} \]

    4. Applied expm1-log1p-u_binary641.7

      \[\leadsto \frac{x}{\frac{z}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin y \cdot \frac{1}{y}\right)\right)}}} \]

    5. Simplified1.7

      \[\leadsto \frac{x}{\frac{z}{\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{\sin y}{y}\right)}\right)}} \]

    if 0.0012779817648914992 < x

    1. Initial program 0.2

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

    2. Applied div-inv_binary640.3

      \[\leadsto \frac{x \cdot \color{blue}{\left(\sin y \cdot \frac{1}{y}\right)}}{z} \]

    3. Applied expm1-log1p-u_binary640.2

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin y \cdot \frac{1}{y}\right)\right)}}{z} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0012779817648914992:\\ \;\;\;\;\frac{x}{\frac{z}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin y}{y}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin y \cdot \frac{1}{y}\right)\right)}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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))