Average Error: 2.8 → 1.7
Time: 13.3s
Precision: 64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \le -1.4187362230985416 \cdot 10^{-273}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\frac{y}{\sin y} \cdot \sqrt[3]{z}}\\ \end{array}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot \frac{\sin y}{y} \le -1.4187362230985416 \cdot 10^{-273}:\\
\;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\frac{y}{\sin y} \cdot \sqrt[3]{z}}\\

\end{array}
double code(double x, double y, double z) {
	return ((x * (sin(y) / y)) / z);
}
↓
double code(double x, double y, double z) {
	double temp;
	if (((x * (sin(y) / y)) <= -1.4187362230985416e-273)) {
		temp = ((x * (sin(y) / y)) / z);
	} else {
		temp = ((x / (cbrt(z) * cbrt(z))) / ((y / sin(y)) * cbrt(z)));
	}
	return temp;
}

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

Original2.8
Target0.3
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;z \lt -4.21737202034271466 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* x (/ (sin y) y)) < -1.4187362230985416e-273

    1. Initial program 0.2

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

    if -1.4187362230985416e-273 < (* x (/ (sin y) y))

    1. Initial program 4.7

      \[\frac{x \cdot \frac{\sin y}{y}}{z}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity4.7

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

    4. Applied *-commutative4.7

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

    5. Applied times-frac2.6

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

    6. Simplified2.6

      \[\leadsto \color{blue}{\frac{\sin y}{y}} \cdot \frac{x}{z}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt3.4

      \[\leadsto \frac{\sin y}{y} \cdot \frac{x}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

    9. Applied associate-/r*3.4

      \[\leadsto \frac{\sin y}{y} \cdot \color{blue}{\frac{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\sqrt[3]{z}}}\]

    10. Applied clear-num3.4

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{\sin y}}} \cdot \frac{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\sqrt[3]{z}}\]

    11. Applied frac-times2.8

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\frac{y}{\sin y} \cdot \sqrt[3]{z}}}\]

    12. Simplified2.8

      \[\leadsto \frac{\color{blue}{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}}{\frac{y}{\sin y} \cdot \sqrt[3]{z}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \le -1.4187362230985416 \cdot 10^{-273}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\frac{y}{\sin y} \cdot \sqrt[3]{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(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 (/ y (sin y)))) z) (if (< z 4.446702369113811e+64) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1 (/ y (sin y)))) z)))

  (/ (* x (/ (sin y) y)) z))