Average Error: 1.4 → 1.2
Time: 7.7s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\frac{\sqrt[3]{y}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y}}} \cdot \frac{\sqrt[3]{y}}{\frac{\sqrt[3]{a - t}}{z - t}} + x\]
x + y \cdot \frac{z - t}{a - t}
↓
\frac{\sqrt[3]{y}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y}}} \cdot \frac{\sqrt[3]{y}}{\frac{\sqrt[3]{a - t}}{z - t}} + x
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 (((cbrt(y) / ((cbrt((a - t)) * cbrt((a - t))) / cbrt(y))) * (cbrt(y) / (cbrt((a - t)) / (z - t)))) + x);
}

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
Target0.4
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;y \lt -8.50808486055124107 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Initial program 1.4

    \[x + y \cdot \frac{z - t}{a - t}\]

  2. Simplified1.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)}\]
  3. Using strategy rm

  4. Applied clear-num1.4

    \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{\frac{a - t}{z - t}}}, x\right)\]
  5. Using strategy rm

  6. Applied fma-udef1.4

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

  7. Simplified1.3

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

  9. Applied *-un-lft-identity1.3

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

  10. Applied add-cube-cbrt1.8

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

  11. Applied times-frac1.8

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

  12. Applied add-cube-cbrt1.9

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

  13. Applied times-frac1.2

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

  14. Simplified1.2

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

  15. Final simplification1.2

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

Reproduce

herbie shell --seed 2020100 +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)))))