Average Error: 2.6 → 3.1
Time: 13.4s
Precision: 64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\frac{x}{\frac{z}{\frac{\sin y}{y}}}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\frac{x}{\frac{z}{\frac{\sin y}{y}}}
double f(double x, double y, double z) {
        double r395685 = x;
        double r395686 = y;
        double r395687 = sin(r395686);
        double r395688 = r395687 / r395686;
        double r395689 = r395685 * r395688;
        double r395690 = z;
        double r395691 = r395689 / r395690;
        return r395691;
}
↓
double f(double x, double y, double z) {
        double r395692 = x;
        double r395693 = z;
        double r395694 = y;
        double r395695 = sin(r395694);
        double r395696 = r395695 / r395694;
        double r395697 = r395693 / r395696;
        double r395698 = r395692 / r395697;
        return r395698;
}

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

Derivation

  1. Split input into 2 regimes

  2. if z < -4.5702531938853563e-132 or 7.563073643537452e-27 < z

    1. Initial program 0.6

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

    3. Applied clear-num0.6

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

    5. Applied *-un-lft-identity0.6

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

    6. Applied *-un-lft-identity0.6

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

    7. Applied times-frac0.6

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

    8. Applied add-cube-cbrt0.6

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

    9. Applied times-frac0.6

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

    10. Applied associate-*r*0.6

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

    11. Simplified0.6

      \[\leadsto \frac{\color{blue}{x} \cdot \frac{\sqrt[3]{1}}{\frac{y}{\sin y}}}{z}\]
    12. Using strategy rm

    13. Applied associate-*r/0.6

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

    14. Simplified0.6

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

    if -4.5702531938853563e-132 < z < 7.563073643537452e-27

    1. Initial program 6.7

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

    3. Applied associate-/l*0.2

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

  4. Final simplification3.1

    \[\leadsto \frac{x}{\frac{z}{\frac{\sin y}{y}}}\]

Reproduce

herbie shell --seed 2019303 
(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))