Average Error: 10.8 → 1.0
Time: 18.1s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}} \cdot \frac{\sqrt[3]{z - t}}{\frac{\sqrt[3]{z - a}}{y}} + x\]
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 \frac{\sqrt[3]{z - t}}{\frac{\sqrt[3]{z - a}}{y}} + x
double f(double x, double y, double z, double t, double a) {
        double r434149 = x;
        double r434150 = y;
        double r434151 = z;
        double r434152 = t;
        double r434153 = r434151 - r434152;
        double r434154 = r434150 * r434153;
        double r434155 = a;
        double r434156 = r434151 - r434155;
        double r434157 = r434154 / r434156;
        double r434158 = r434149 + r434157;
        return r434158;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r434159 = z;
        double r434160 = t;
        double r434161 = r434159 - r434160;
        double r434162 = cbrt(r434161);
        double r434163 = r434162 * r434162;
        double r434164 = a;
        double r434165 = r434159 - r434164;
        double r434166 = cbrt(r434165);
        double r434167 = r434166 * r434166;
        double r434168 = r434163 / r434167;
        double r434169 = y;
        double r434170 = r434166 / r434169;
        double r434171 = r434162 / r434170;
        double r434172 = r434168 * r434171;
        double r434173 = x;
        double r434174 = r434172 + r434173;
        return r434174;
}

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

Original10.8
Target1.3
Herbie1.0
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 10.8

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

  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 *-un-lft-identity2.9

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

  10. Applied add-cube-cbrt3.4

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

  11. Applied times-frac3.4

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

  12. Applied add-cube-cbrt3.3

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

  13. Applied times-frac1.0

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

  14. Simplified1.0

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

  15. Final simplification1.0

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

Reproduce

herbie shell --seed 2019326 +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))))