Average Error: 1.4 → 0.5
Time: 6.6s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[x + \frac{1}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}} \cdot \frac{\frac{\sqrt[3]{a - t}}{\sqrt[3]{z - t}}}{y}}\]
x + y \cdot \frac{z - t}{a - t}
↓
x + \frac{1}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}} \cdot \frac{\frac{\sqrt[3]{a - t}}{\sqrt[3]{z - t}}}{y}}
double code(double x, double y, double z, double t, double a) {
	return (x + (y * ((z - t) / (a - t))));
}
↓
double code(double x, double y, double z, double t, double a) {
	return (x + (1.0 / (((cbrt((a - t)) * cbrt((a - t))) / (cbrt((z - t)) * cbrt((z - t)))) * ((cbrt((a - t)) / cbrt((z - t))) / y))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original1.4
Target0.5
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -8.50808486055124107 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Initial program 1.4

    \[x + y \cdot \frac{z - t}{a - t}\]
  2. Using strategy rm

  3. Applied clear-num1.5

    \[\leadsto x + y \cdot \color{blue}{\frac{1}{\frac{a - t}{z - t}}}\]
  4. Using strategy rm

  5. Applied un-div-inv1.4

    \[\leadsto x + \color{blue}{\frac{y}{\frac{a - t}{z - t}}}\]
  6. Using strategy rm

  7. Applied clear-num1.4

    \[\leadsto x + \color{blue}{\frac{1}{\frac{\frac{a - t}{z - t}}{y}}}\]
  8. Using strategy rm

  9. Applied *-un-lft-identity1.4

    \[\leadsto x + \frac{1}{\frac{\frac{a - t}{z - t}}{\color{blue}{1 \cdot y}}}\]

  10. Applied add-cube-cbrt1.9

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

  11. Applied add-cube-cbrt1.8

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

  12. Applied times-frac1.7

    \[\leadsto x + \frac{1}{\frac{\color{blue}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}} \cdot \frac{\sqrt[3]{a - t}}{\sqrt[3]{z - t}}}}{1 \cdot y}}\]

  13. Applied times-frac0.5

    \[\leadsto x + \frac{1}{\color{blue}{\frac{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}}{1} \cdot \frac{\frac{\sqrt[3]{a - t}}{\sqrt[3]{z - t}}}{y}}}\]

  14. Simplified0.5

    \[\leadsto x + \frac{1}{\color{blue}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}} \cdot \frac{\frac{\sqrt[3]{a - t}}{\sqrt[3]{z - t}}}{y}}\]

  15. Final simplification0.5

    \[\leadsto x + \frac{1}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}} \cdot \frac{\frac{\sqrt[3]{a - t}}{\sqrt[3]{z - t}}}{y}}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

  (+ x (* y (/ (- z t) (- a t)))))