Average Error: 10.6 → 0.5
Time: 3.5s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{z - a}} \cdot y\right) + x\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{z - a}} \cdot y\right) + x
double f(double x, double y, double z, double t, double a) {
        double r676802 = x;
        double r676803 = y;
        double r676804 = z;
        double r676805 = t;
        double r676806 = r676804 - r676805;
        double r676807 = r676803 * r676806;
        double r676808 = a;
        double r676809 = r676804 - r676808;
        double r676810 = r676807 / r676809;
        double r676811 = r676802 + r676810;
        return r676811;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r676812 = z;
        double r676813 = t;
        double r676814 = r676812 - r676813;
        double r676815 = cbrt(r676814);
        double r676816 = r676815 * r676815;
        double r676817 = a;
        double r676818 = r676812 - r676817;
        double r676819 = cbrt(r676818);
        double r676820 = r676819 * r676819;
        double r676821 = r676816 / r676820;
        double r676822 = r676815 / r676819;
        double r676823 = y;
        double r676824 = r676822 * r676823;
        double r676825 = r676821 * r676824;
        double r676826 = x;
        double r676827 = r676825 + r676826;
        return r676827;
}

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

Derivation

  1. Initial program 10.6

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

  2. Simplified2.9

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

  4. Applied clear-num3.1

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

  6. Applied fma-udef3.1

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

  7. Simplified2.9

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

  9. Applied associate-/r/1.3

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

  11. Applied add-cube-cbrt1.9

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

  12. Applied add-cube-cbrt1.7

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

  13. Applied times-frac1.7

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

  14. Applied associate-*l*0.5

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

  15. Final simplification0.5

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

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, A"
  :precision binary64

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

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