Average Error: 1.3 → 0.6
Time: 4.9s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{z - t}}} + x\]
x + y \cdot \frac{z - t}{a - t}
↓
\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{z - t}}} + x
double f(double x, double y, double z, double t, double a) {
        double r539247 = x;
        double r539248 = y;
        double r539249 = z;
        double r539250 = t;
        double r539251 = r539249 - r539250;
        double r539252 = a;
        double r539253 = r539252 - r539250;
        double r539254 = r539251 / r539253;
        double r539255 = r539248 * r539254;
        double r539256 = r539247 + r539255;
        return r539256;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r539257 = z;
        double r539258 = t;
        double r539259 = r539257 - r539258;
        double r539260 = cbrt(r539259);
        double r539261 = r539260 * r539260;
        double r539262 = a;
        double r539263 = r539262 - r539258;
        double r539264 = cbrt(r539263);
        double r539265 = r539264 * r539264;
        double r539266 = r539261 / r539265;
        double r539267 = y;
        double r539268 = r539264 / r539260;
        double r539269 = r539267 / r539268;
        double r539270 = r539266 * r539269;
        double r539271 = x;
        double r539272 = r539270 + r539271;
        return r539272;
}

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
Target0.5
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;y \lt -8.50808486055124107 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Initial program 1.3

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

  2. Simplified1.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)}\]
  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. Simplified1.2

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

  9. Applied add-cube-cbrt1.7

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

  10. Applied add-cube-cbrt1.6

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

  11. Applied times-frac1.6

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

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

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

  13. Applied times-frac0.6

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

  14. Simplified0.6

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

  15. Final simplification0.6

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

Reproduce

herbie shell --seed 2020033 +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)))))