Average Error: 2.4 → 1.4
Time: 6.5s
Precision: binary64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\frac{y}{\sin y}} \cdot \sqrt[3]{\frac{y}{\sin y}}} \cdot \frac{\frac{\sqrt[3]{x}}{\sqrt[3]{\frac{y}{\sin y}}}}{z}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\frac{y}{\sin y}} \cdot \sqrt[3]{\frac{y}{\sin y}}} \cdot \frac{\frac{\sqrt[3]{x}}{\sqrt[3]{\frac{y}{\sin y}}}}{z}
(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))
↓
(FPCore (x y z)
 :precision binary64
 (*
  (/ (* (cbrt x) (cbrt x)) (* (cbrt (/ y (sin y))) (cbrt (/ y (sin y)))))
  (/ (/ (cbrt x) (cbrt (/ y (sin y)))) 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(x) * cbrt(x)) / (cbrt(y / sin(y)) * cbrt(y / sin(y)))) * ((cbrt(x) / cbrt(y / 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

Original2.4
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 2.4

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

  3. Applied clear-num_binary64_126942.4

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

  5. Applied un-div-inv_binary64_126932.4

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

  7. Applied *-un-lft-identity_binary64_126952.4

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

  8. Applied add-cube-cbrt_binary64_127302.7

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

  9. Applied add-cube-cbrt_binary64_127303.3

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

  10. Applied times-frac_binary64_127013.3

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

  11. Applied times-frac_binary64_127011.4

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

  12. Final simplification1.4

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

Reproduce

herbie shell --seed 2021015 
(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))