Average Error: 6.4 → 1.4
Time: 30.5s
Precision: binary64
Cost: 1472
\[\frac{x \cdot y}{z}\]
↓
\[\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z}} \cdot \left(x \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\right)\right) \cdot \frac{\sqrt[3]{-y} \cdot \sqrt[3]{-1}}{\sqrt[3]{z}}\]
\frac{x \cdot y}{z}
↓
\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z}} \cdot \left(x \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\right)\right) \cdot \frac{\sqrt[3]{-y} \cdot \sqrt[3]{-1}}{\sqrt[3]{z}}
(FPCore (x y z) :precision binary64 (/ (* x y) z))
↓
(FPCore (x y z)
 :precision binary64
 (*
  (* (/ (cbrt y) (cbrt z)) (* x (/ (cbrt y) (cbrt z))))
  (/ (* (cbrt (- y)) (cbrt -1.0)) (cbrt z))))
double code(double x, double y, double z) {
	return (x * y) / z;
}
↓
double code(double x, double y, double z) {
	return ((cbrt(y) / cbrt(z)) * (x * (cbrt(y) / cbrt(z)))) * ((cbrt(-y) * cbrt(-1.0)) / 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

Original6.4
Target6.0
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;z < -4.262230790519429 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z < 1.7042130660650472 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

  1. Initial program 6.4

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

  3. Applied *-un-lft-identity_binary64_181516.4

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

  4. Applied times-frac_binary64_181576.3

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

  5. Simplified6.3

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

  7. Applied add-cube-cbrt_binary64_181867.1

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

  8. Applied add-cube-cbrt_binary64_181867.3

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

  9. Applied times-frac_binary64_181577.3

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

  10. Applied associate-*r*_binary64_180912.0

    \[\leadsto \color{blue}{\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}}\]
  11. Using strategy rm

  12. Applied times-frac_binary64_181572.0

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

  13. Applied associate-*r*_binary64_180911.4

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

  14. Taylor expanded around -inf 33.9

    \[\leadsto \left(\left(x \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\right) \cdot \frac{\color{blue}{{\left(-1 \cdot y\right)}^{0.3333333333333333} \cdot \sqrt[3]{-1}}}{\sqrt[3]{z}}\]

  15. Simplified1.4

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

  16. Final simplification1.4

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

Reproduce

herbie shell --seed 2020322 
(FPCore (x y z)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -4.262230790519429e-138) (/ (* x y) z) (if (< z 1.7042130660650472e-164) (/ x (/ z y)) (* (/ x z) y)))

  (/ (* x y) z))