Average Error: 2.8 → 2.9
Time: 3.6s
Precision: binary64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 1.1620883010817137 \cdot 10^{-236}:\\ \;\;\;\;x \cdot \frac{\sin y}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{\sin y}{y}} \cdot \left(x \cdot \sqrt{\frac{\sin y}{y}}\right)}{z}\\ \end{array}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\sin y}{y} \leq 1.1620883010817137 \cdot 10^{-236}:\\
\;\;\;\;x \cdot \frac{\sin y}{y \cdot z}\\

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

\end{array}
(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= (/ (sin y) y) 1.1620883010817137e-236)
   (* x (/ (sin y) (* y z)))
   (/ (* (sqrt (/ (sin y) y)) (* x (sqrt (/ (sin y) 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 ((sin(y) / y) <= 1.1620883010817137e-236) {
		tmp = x * (sin(y) / (y * z));
	} else {
		tmp = (sqrt(sin(y) / y) * (x * sqrt(sin(y) / 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

Original2.8
Target0.3
Herbie2.9
\[\begin{array}{l} \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (sin.f64 y) y) < 1.162088301081714e-236

    1. Initial program 5.8

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

    3. Applied *-un-lft-identity_binary64_172645.8

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

    4. Applied times-frac_binary64_172705.5

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

    5. Simplified5.5

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

    7. Applied associate-/l/_binary64_172135.9

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

    8. Simplified5.9

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

    if 1.162088301081714e-236 < (/.f64 (sin.f64 y) y)

    1. Initial program 1.3

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

    3. Applied add-sqr-sqrt_binary64_172851.4

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

    4. Applied associate-*r*_binary64_172061.4

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

  4. Final simplification2.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 1.1620883010817137 \cdot 10^{-236}:\\ \;\;\;\;x \cdot \frac{\sin y}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{\sin y}{y}} \cdot \left(x \cdot \sqrt{\frac{\sin y}{y}}\right)}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020281 
(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))