Average Error: 2.6 → 1.4
Time: 9.8s
Precision: binary64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\frac{\sqrt[3]{\frac{\sin y}{y}}}{\sqrt[3]{z}} \cdot \left(x \cdot \left(\frac{\sqrt[3]{\frac{\sin y}{y}}}{\sqrt[3]{z}} \cdot \frac{\sqrt[3]{\frac{\sin y}{y}}}{\sqrt[3]{z}}\right)\right)\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\frac{\sqrt[3]{\frac{\sin y}{y}}}{\sqrt[3]{z}} \cdot \left(x \cdot \left(\frac{\sqrt[3]{\frac{\sin y}{y}}}{\sqrt[3]{z}} \cdot \frac{\sqrt[3]{\frac{\sin y}{y}}}{\sqrt[3]{z}}\right)\right)
(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))
↓
(FPCore (x y z)
 :precision binary64
 (*
  (/ (cbrt (/ (sin y) y)) (cbrt z))
  (*
   x
   (* (/ (cbrt (/ (sin y) y)) (cbrt z)) (/ (cbrt (/ (sin y) y)) (cbrt z))))))
double code(double x, double y, double z) {
	return (x * (sin(y) / y)) / z;
}
↓
double code(double x, double y, double z) {
	return (cbrt(sin(y) / y) / cbrt(z)) * (x * ((cbrt(sin(y) / y) / cbrt(z)) * (cbrt(sin(y) / y) / cbrt(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
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

  1. Initial program 2.6

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

  3. Applied *-un-lft-identity_binary64_82622.6

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

  4. Applied times-frac_binary64_82683.0

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

  5. Simplified3.0

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

  7. Applied add-cube-cbrt_binary64_82973.7

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

  8. Applied add-cube-cbrt_binary64_82973.8

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

  9. Applied times-frac_binary64_82683.8

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

  10. Applied associate-*r*_binary64_82021.4

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

  11. Simplified1.4

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

  12. Final simplification1.4

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

Reproduce

herbie shell --seed 2021176 
(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))