Average Error: 2.5 → 0.7
Time: 7.6s
Precision: binary64
\[\frac{x \cdot \frac{\sin y}{y}}{z} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -9.975770992237415 \cdot 10^{+179}:\\ \;\;\;\;\frac{\frac{x \cdot \sin y}{y}}{z}\\ \mathbf{elif}\;x \leq 5.5093964453865166 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \end{array} \]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -9.975770992237415 \cdot 10^{+179}:\\
\;\;\;\;\frac{\frac{x \cdot \sin y}{y}}{z}\\

\mathbf{elif}\;x \leq 5.5093964453865166 \cdot 10^{+23}:\\
\;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\


\end{array}

(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= x -9.975770992237415e+179)
   (/ (/ (* x (sin y)) y) z)
   (if (<= x 5.5093964453865166e+23)
     (/ x (/ z (/ (sin y) y)))
     (/ (/ x (/ y (sin 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 <= -9.975770992237415e+179) {
		tmp = ((x * sin(y)) / y) / z;
	} else if (x <= 5.5093964453865166e+23) {
		tmp = x / (z / (sin(y) / y));
	} else {
		tmp = (x / (y / sin(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

Original2.5
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if x < -9.97577099223741516e179

    1. Initial program 0.2

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

    2. Taylor expanded in x around 0 0.2

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

    if -9.97577099223741516e179 < x < 5.5093964453865166e23

    1. Initial program 3.6

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

    2. Applied associate-/l*_binary640.9

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

    if 5.5093964453865166e23 < x

    1. Initial program 0.2

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

    2. Applied clear-num_binary640.3

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

    3. Applied un-div-inv_binary640.3

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.975770992237415 \cdot 10^{+179}:\\ \;\;\;\;\frac{\frac{x \cdot \sin y}{y}}{z}\\ \mathbf{elif}\;x \leq 5.5093964453865166 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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))