Average Error: 2.8 → 3.2
Time: 10.1s
Precision: 64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le 2.038405760172519767516186916830460470804 \cdot 10^{-223}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\sin y \cdot \frac{1}{y}\right)}{z}\\ \end{array}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le 2.038405760172519767516186916830460470804 \cdot 10^{-223}:\\
\;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r557159 = x;
        double r557160 = y;
        double r557161 = sin(r557160);
        double r557162 = r557161 / r557160;
        double r557163 = r557159 * r557162;
        double r557164 = z;
        double r557165 = r557163 / r557164;
        return r557165;
}
↓
double f(double x, double y, double z) {
        double r557166 = z;
        double r557167 = 2.0384057601725198e-223;
        bool r557168 = r557166 <= r557167;
        double r557169 = y;
        double r557170 = sin(r557169);
        double r557171 = r557170 / r557169;
        double r557172 = x;
        double r557173 = r557172 / r557166;
        double r557174 = r557171 * r557173;
        double r557175 = 1.0;
        double r557176 = r557175 / r557169;
        double r557177 = r557170 * r557176;
        double r557178 = r557172 * r557177;
        double r557179 = r557178 / r557166;
        double r557180 = r557168 ? r557174 : r557179;
        return r557180;
}

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
Herbie3.2
\[\begin{array}{l} \mathbf{if}\;z \lt -4.217372020342714661850238929213415773451 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.446702369113811028051510715777703865332 \cdot 10^{64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < 2.0384057601725198e-223

    1. Initial program 3.5

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

    3. Applied div-inv3.6

      \[\leadsto \frac{x \cdot \color{blue}{\left(\sin y \cdot \frac{1}{y}\right)}}{z}\]
    4. Using strategy rm

    5. Applied clear-num4.0

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

    6. Simplified3.9

      \[\leadsto \frac{1}{\color{blue}{\frac{z}{\frac{\sin y}{y} \cdot x}}}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity3.9

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

    9. Applied times-frac4.2

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

    10. Applied add-cube-cbrt4.2

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

    11. Applied times-frac4.3

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

    12. Simplified4.3

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

    13. Simplified4.0

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

    if 2.0384057601725198e-223 < z

    1. Initial program 2.1

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

    3. Applied div-inv2.1

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

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 2.038405760172519767516186916830460470804 \cdot 10^{-223}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\sin y \cdot \frac{1}{y}\right)}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(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))