Average Error: 10.6 → 1.0
Time: 19.0s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a - t}\]
↓
\[\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \left(z - t\right)\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} + x\]
x + \frac{y \cdot \left(z - t\right)}{a - t}
↓
\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \left(z - t\right)\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} + x
double f(double x, double y, double z, double t, double a) {
        double r355853 = x;
        double r355854 = y;
        double r355855 = z;
        double r355856 = t;
        double r355857 = r355855 - r355856;
        double r355858 = r355854 * r355857;
        double r355859 = a;
        double r355860 = r355859 - r355856;
        double r355861 = r355858 / r355860;
        double r355862 = r355853 + r355861;
        return r355862;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r355863 = y;
        double r355864 = cbrt(r355863);
        double r355865 = r355864 * r355864;
        double r355866 = a;
        double r355867 = t;
        double r355868 = r355866 - r355867;
        double r355869 = cbrt(r355868);
        double r355870 = r355869 * r355869;
        double r355871 = r355865 / r355870;
        double r355872 = z;
        double r355873 = r355872 - r355867;
        double r355874 = r355871 * r355873;
        double r355875 = r355864 / r355869;
        double r355876 = r355874 * r355875;
        double r355877 = x;
        double r355878 = r355876 + r355877;
        return r355878;
}

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
Target1.3
Herbie1.0
\[x + \frac{y}{\frac{a - t}{z - t}}\]

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 div-inv2.9

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

  6. Applied fma-udef2.9

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

  7. Simplified2.9

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

  9. Applied add-cube-cbrt3.3

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

  10. Applied add-cube-cbrt3.4

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

  11. Applied times-frac3.4

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

  12. Applied associate-*r*1.0

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

  13. Simplified1.0

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

  14. Final simplification1.0

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

Reproduce

herbie shell --seed 2019208 +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))))