Average Error: 2.6 → 1.5
Time: 14.5s
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 r321424 = x;
        double r321425 = y;
        double r321426 = sin(r321425);
        double r321427 = r321426 / r321425;
        double r321428 = r321424 * r321427;
        double r321429 = z;
        double r321430 = r321428 / r321429;
        return r321430;
}
↓
double f(double x, double y, double z) {
        double r321431 = x;
        double r321432 = y;
        double r321433 = sin(r321432);
        double r321434 = r321433 / r321432;
        double r321435 = cbrt(r321434);
        double r321436 = r321435 * r321435;
        double r321437 = z;
        double r321438 = cbrt(r321437);
        double r321439 = r321438 * r321438;
        double r321440 = r321436 / r321439;
        double r321441 = r321431 * r321440;
        double r321442 = r321435 / r321438;
        double r321443 = r321441 * r321442;
        return r321443;
}

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
Target0.2
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;z \lt -4.217372020342714661850238929213415773451 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.446702369113811028051510715777703865332 \cdot 10^{64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

  1. 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-frac2.7

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

  5. Simplified2.7

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

  7. Applied add-cube-cbrt3.5

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

  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-frac3.6

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

  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 2019305 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< z -4.21737202034271466e-29) (/ (* x (/ 1 (/ y (sin y)))) z) (if (< z 4.44670236911381103e64) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1 (/ y (sin y)))) z)))

  (/ (* x (/ (sin y) y)) z))