Average Error: 1.3 → 0.6
Time: 4.2s
Precision: 64
\[x + y \cdot \frac{z - t}{z - a}\]
↓
\[\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}} \cdot \frac{y}{\frac{\sqrt[3]{z - a}}{\sqrt[3]{z - t}}} + x\]
x + y \cdot \frac{z - t}{z - a}
↓
\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}} \cdot \frac{y}{\frac{\sqrt[3]{z - a}}{\sqrt[3]{z - t}}} + x
double code(double x, double y, double z, double t, double a) {
	return ((double) (x + ((double) (y * ((double) (((double) (z - t)) / ((double) (z - a))))))));
}
↓
double code(double x, double y, double z, double t, double a) {
	return ((double) (((double) (((double) (((double) (((double) cbrt(((double) (z - t)))) * ((double) cbrt(((double) (z - t)))))) / ((double) (((double) cbrt(((double) (z - a)))) * ((double) cbrt(((double) (z - a)))))))) * ((double) (y / ((double) (((double) cbrt(((double) (z - a)))) / ((double) cbrt(((double) (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

Original1.3
Target1.2
Herbie0.6
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 1.3

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

  2. Simplified1.3

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

  4. Applied clear-num1.4

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

  6. Applied fma-udef1.4

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

  7. Simplified1.2

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

  9. Applied add-cube-cbrt1.7

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

  10. Applied add-cube-cbrt1.6

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

  11. Applied times-frac1.6

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

  12. Applied *-un-lft-identity1.6

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

  13. Applied times-frac0.6

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

  14. Simplified0.6

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

  15. Final simplification0.6

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

Reproduce

herbie shell --seed 2020123 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

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