Average Error: 2.6 → 0.5
Time: 33.4s
Precision: 64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -685445088654725504 \lor \neg \left(z \le 9.777982240122365731670702998885673597927 \cdot 10^{44}\right):\\ \;\;\;\;\frac{\frac{1}{\frac{z}{x}}}{\frac{1}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{y}}{z}\\ \end{array}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -685445088654725504 \lor \neg \left(z \le 9.777982240122365731670702998885673597927 \cdot 10^{44}\right):\\
\;\;\;\;\frac{\frac{1}{\frac{z}{x}}}{\frac{1}{\frac{\sin y}{y}}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r434963 = x;
        double r434964 = y;
        double r434965 = sin(r434964);
        double r434966 = r434965 / r434964;
        double r434967 = r434963 * r434966;
        double r434968 = z;
        double r434969 = r434967 / r434968;
        return r434969;
}
↓
double f(double x, double y, double z) {
        double r434970 = z;
        double r434971 = -6.854450886547255e+17;
        bool r434972 = r434970 <= r434971;
        double r434973 = 9.777982240122366e+44;
        bool r434974 = r434970 <= r434973;
        double r434975 = !r434974;
        bool r434976 = r434972 || r434975;
        double r434977 = 1.0;
        double r434978 = x;
        double r434979 = r434970 / r434978;
        double r434980 = r434977 / r434979;
        double r434981 = y;
        double r434982 = sin(r434981);
        double r434983 = r434982 / r434981;
        double r434984 = r434977 / r434983;
        double r434985 = r434980 / r434984;
        double r434986 = r434983 / r434970;
        double r434987 = r434978 * r434986;
        double r434988 = r434976 ? r434985 : r434987;
        return r434988;
}

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

Original2.6
Target0.3
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;z \lt -4.217372020342714661850238929213415773451 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.446702369113811028051510715777703865332 \cdot 10^{64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -6.854450886547255e+17 or 9.777982240122366e+44 < z

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.9

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

    4. Applied add-cube-cbrt0.4

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

    5. Applied times-frac0.4

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

    7. Applied clear-num1.2

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

    8. Simplified0.9

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

    10. Applied div-inv0.9

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

    11. Applied associate-/r*0.5

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

    if -6.854450886547255e+17 < z < 9.777982240122366e+44

    1. Initial program 4.9

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

    3. Applied *-un-lft-identity4.9

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

    4. Applied times-frac0.4

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

    5. Simplified0.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -685445088654725504 \lor \neg \left(z \le 9.777982240122365731670702998885673597927 \cdot 10^{44}\right):\\ \;\;\;\;\frac{\frac{1}{\frac{z}{x}}}{\frac{1}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{y}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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))