Average Error: 6.3 → 1.6
Time: 21.2s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[x + \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a}} \cdot y\right) \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a}} \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a}}\right)\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
x + \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a}} \cdot y\right) \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a}} \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a}}\right)
double f(double x, double y, double z, double t, double a) {
        double r15807242 = x;
        double r15807243 = y;
        double r15807244 = z;
        double r15807245 = t;
        double r15807246 = r15807244 - r15807245;
        double r15807247 = r15807243 * r15807246;
        double r15807248 = a;
        double r15807249 = r15807247 / r15807248;
        double r15807250 = r15807242 + r15807249;
        return r15807250;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r15807251 = x;
        double r15807252 = z;
        double r15807253 = t;
        double r15807254 = r15807252 - r15807253;
        double r15807255 = cbrt(r15807254);
        double r15807256 = a;
        double r15807257 = cbrt(r15807256);
        double r15807258 = r15807255 / r15807257;
        double r15807259 = y;
        double r15807260 = r15807258 * r15807259;
        double r15807261 = r15807258 * r15807258;
        double r15807262 = r15807260 * r15807261;
        double r15807263 = r15807251 + r15807262;
        return r15807263;
}

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

Original6.3
Target0.8
Herbie1.6
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Initial program 6.3

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

  2. Simplified2.6

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

  4. Applied fma-udef2.6

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

  5. Taylor expanded around 0 6.3

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

  6. Simplified2.5

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

  8. Applied *-un-lft-identity2.5

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

  9. Applied add-cube-cbrt3.1

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

  10. Applied times-frac3.1

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

  11. Applied add-cube-cbrt3.2

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

  12. Applied times-frac1.8

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

  13. Simplified1.8

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

  14. Simplified1.6

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

  15. Final simplification1.6

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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