Average Error: 2.6 → 2.0
Time: 3.7s
Precision: binary64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\left(\sqrt[3]{x} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{\frac{y}{\sin y}} \cdot \sqrt[3]{\frac{y}{\sin y}}}\right) \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{\frac{y}{\sin y}} \cdot z}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\left(\sqrt[3]{x} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{\frac{y}{\sin y}} \cdot \sqrt[3]{\frac{y}{\sin y}}}\right) \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{\frac{y}{\sin y}} \cdot z}
double code(double x, double y, double z) {
	return ((double) (((double) (x * ((double) (((double) sin(y)) / y)))) / z));
}
↓
double code(double x, double y, double z) {
	return ((double) (((double) (((double) cbrt(x)) * ((double) (((double) cbrt(x)) / ((double) (((double) cbrt(((double) (y / ((double) sin(y)))))) * ((double) cbrt(((double) (y / ((double) sin(y)))))))))))) * ((double) (((double) cbrt(x)) / ((double) (((double) cbrt(((double) (y / ((double) sin(y)))))) * z))))));
}

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
Herbie2.0
\[\begin{array}{l} \mathbf{if}\;z \lt -4.21737202034271466 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

  1. Initial program 2.6

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

  3. Applied clear-num2.6

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

  5. Applied un-div-inv2.6

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

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

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

  8. Applied add-cube-cbrt2.9

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

  9. Applied add-cube-cbrt3.6

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

  10. Applied times-frac3.6

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

  11. Applied times-frac1.4

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

  12. Simplified1.4

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

  13. Simplified2.0

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

  14. Final simplification2.0

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

Reproduce

herbie shell --seed 2020179 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< z -4.2173720203427147e-29) (/ (* x (/ 1.0 (/ y (sin y)))) z) (if (< z 4.446702369113811e+64) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1.0 (/ y (sin y)))) z)))

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