Average Error: 2.6 → 1.5
Time: 4.3s
Precision: 64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\left(x \cdot \frac{\sqrt[3]{\frac{\sin y}{y}} \cdot \sqrt[3]{\frac{\sin y}{y}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{\frac{\sin y}{y}}}{\sqrt[3]{z}}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\left(x \cdot \frac{\sqrt[3]{\frac{\sin y}{y}} \cdot \sqrt[3]{\frac{\sin y}{y}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{\frac{\sin y}{y}}}{\sqrt[3]{z}}
double f(double x, double y, double z) {
        double r558373 = x;
        double r558374 = y;
        double r558375 = sin(r558374);
        double r558376 = r558375 / r558374;
        double r558377 = r558373 * r558376;
        double r558378 = z;
        double r558379 = r558377 / r558378;
        return r558379;
}
↓
double f(double x, double y, double z) {
        double r558380 = x;
        double r558381 = y;
        double r558382 = sin(r558381);
        double r558383 = r558382 / r558381;
        double r558384 = cbrt(r558383);
        double r558385 = r558384 * r558384;
        double r558386 = z;
        double r558387 = cbrt(r558386);
        double r558388 = r558387 * r558387;
        double r558389 = r558385 / r558388;
        double r558390 = r558380 * r558389;
        double r558391 = r558384 / r558387;
        double r558392 = r558390 * r558391;
        return r558392;
}

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
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;z \lt -4.21737202034271466 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

  1. Initial program 2.6

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

  3. Applied *-un-lft-identity2.6

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

  4. Applied times-frac2.8

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

  5. Simplified2.8

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

  7. Applied add-cube-cbrt3.6

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

  8. Applied add-cube-cbrt3.6

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

  9. Applied times-frac3.7

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

  10. Applied associate-*r*1.5

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

  11. Final simplification1.5

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

Reproduce

herbie shell --seed 2020024 +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))