Average Error: 2.8 → 0.6
Time: 2.6s
Precision: 64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.35421204550298263 \cdot 10^{-138} \lor \neg \left(z \le 1.21824188071821231 \cdot 10^{-60}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\frac{y}{\sin y} \cdot \left(-z\right)}\\ \end{array}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.35421204550298263 \cdot 10^{-138} \lor \neg \left(z \le 1.21824188071821231 \cdot 10^{-60}\right):\\
\;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{-x}{\frac{y}{\sin y} \cdot \left(-z\right)}\\

\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 VAR;
	if (((z <= -5.3542120455029826e-138) || !(z <= 1.2182418807182123e-60))) {
		VAR = ((sin(y) / y) * (x / z));
	} else {
		VAR = (-x / ((y / sin(y)) * -z));
	}
	return VAR;
}

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
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;z \lt -4.21737202034271466 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -5.3542120455029826e-138 or 1.2182418807182123e-60 < z

    1. Initial program 0.7

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

    3. Applied *-commutative0.7

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

    4. Applied associate-/l*1.0

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

    6. Applied div-inv1.1

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

    7. Simplified0.8

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

    if -5.3542120455029826e-138 < z < 1.2182418807182123e-60

    1. Initial program 8.1

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

    3. Applied *-commutative8.1

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

    4. Applied associate-/l*7.9

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

    6. Applied div-inv8.5

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

    7. Simplified8.3

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

    9. Applied frac-2neg8.3

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

    10. Applied clear-num8.4

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

    11. Applied frac-times0.2

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

    12. Simplified0.2

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.35421204550298263 \cdot 10^{-138} \lor \neg \left(z \le 1.21824188071821231 \cdot 10^{-60}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\frac{y}{\sin y} \cdot \left(-z\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(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))