Average Error: 10.6 → 0.9
Time: 6.2s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a - t}\]
↓
\[\frac{\sqrt[3]{z - t}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{z - t}}} \cdot \frac{\sqrt[3]{z - t}}{\frac{\sqrt[3]{a - t}}{y}} + x\]
x + \frac{y \cdot \left(z - t\right)}{a - t}
↓
\frac{\sqrt[3]{z - t}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{z - t}}} \cdot \frac{\sqrt[3]{z - t}}{\frac{\sqrt[3]{a - t}}{y}} + x
double f(double x, double y, double z, double t, double a) {
        double r534431 = x;
        double r534432 = y;
        double r534433 = z;
        double r534434 = t;
        double r534435 = r534433 - r534434;
        double r534436 = r534432 * r534435;
        double r534437 = a;
        double r534438 = r534437 - r534434;
        double r534439 = r534436 / r534438;
        double r534440 = r534431 + r534439;
        return r534440;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r534441 = z;
        double r534442 = t;
        double r534443 = r534441 - r534442;
        double r534444 = cbrt(r534443);
        double r534445 = a;
        double r534446 = r534445 - r534442;
        double r534447 = cbrt(r534446);
        double r534448 = r534447 * r534447;
        double r534449 = r534448 / r534444;
        double r534450 = r534444 / r534449;
        double r534451 = y;
        double r534452 = r534447 / r534451;
        double r534453 = r534444 / r534452;
        double r534454 = r534450 * r534453;
        double r534455 = x;
        double r534456 = r534454 + r534455;
        return r534456;
}

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

Original10.6
Target1.2
Herbie0.9
\[x + \frac{y}{\frac{a - t}{z - t}}\]

Derivation

  1. Initial program 10.6

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

  2. Simplified2.9

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

  4. Applied clear-num3.1

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

  6. Applied fma-udef3.1

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

  7. Simplified2.9

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

  9. Applied *-un-lft-identity2.9

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

  10. Applied add-cube-cbrt3.4

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

  11. Applied times-frac3.4

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

  12. Applied add-cube-cbrt3.3

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

  13. Applied times-frac0.9

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

  14. Simplified0.9

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

  15. Final simplification0.9

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

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"
  :precision binary64

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

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