Average Error: 1.5 → 0.6
Time: 4.3s
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 f(double x, double y, double z, double t, double a) {
        double r663334 = x;
        double r663335 = y;
        double r663336 = z;
        double r663337 = t;
        double r663338 = r663336 - r663337;
        double r663339 = a;
        double r663340 = r663336 - r663339;
        double r663341 = r663338 / r663340;
        double r663342 = r663335 * r663341;
        double r663343 = r663334 + r663342;
        return r663343;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r663344 = z;
        double r663345 = t;
        double r663346 = r663344 - r663345;
        double r663347 = cbrt(r663346);
        double r663348 = r663347 * r663347;
        double r663349 = a;
        double r663350 = r663344 - r663349;
        double r663351 = cbrt(r663350);
        double r663352 = r663351 * r663351;
        double r663353 = r663348 / r663352;
        double r663354 = y;
        double r663355 = r663351 / r663347;
        double r663356 = r663354 / r663355;
        double r663357 = r663353 * r663356;
        double r663358 = x;
        double r663359 = r663357 + r663358;
        return r663359;
}

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

Original1.5
Target1.4
Herbie0.6
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 1.5

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

  2. Simplified1.5

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

  4. Applied clear-num1.6

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

  6. Applied fma-udef1.6

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

  7. Simplified1.4

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

  9. Applied add-cube-cbrt1.9

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

  10. Applied add-cube-cbrt1.8

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

  11. Applied times-frac1.8

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

  12. Applied *-un-lft-identity1.8

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

  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 2020060 +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)))))