Average Error: 10.8 → 1.0
Time: 18.0s
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 r374230 = x;
        double r374231 = y;
        double r374232 = z;
        double r374233 = t;
        double r374234 = r374232 - r374233;
        double r374235 = r374231 * r374234;
        double r374236 = a;
        double r374237 = r374232 - r374236;
        double r374238 = r374235 / r374237;
        double r374239 = r374230 + r374238;
        return r374239;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r374240 = z;
        double r374241 = t;
        double r374242 = r374240 - r374241;
        double r374243 = cbrt(r374242);
        double r374244 = r374243 * r374243;
        double r374245 = a;
        double r374246 = r374240 - r374245;
        double r374247 = cbrt(r374246);
        double r374248 = r374247 * r374247;
        double r374249 = r374244 / r374248;
        double r374250 = y;
        double r374251 = r374247 / r374250;
        double r374252 = r374243 / r374251;
        double r374253 = r374249 * r374252;
        double r374254 = x;
        double r374255 = r374253 + r374254;
        return r374255;
}

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))))