Average Error: 6.3 → 1.0
Time: 15.8s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\mathsf{fma}\left(\frac{1}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{t - z}{\frac{\sqrt[3]{a}}{\sqrt[3]{y}}}, x\right)\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\mathsf{fma}\left(\frac{1}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{t - z}{\frac{\sqrt[3]{a}}{\sqrt[3]{y}}}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r212348 = x;
        double r212349 = y;
        double r212350 = z;
        double r212351 = t;
        double r212352 = r212350 - r212351;
        double r212353 = r212349 * r212352;
        double r212354 = a;
        double r212355 = r212353 / r212354;
        double r212356 = r212348 - r212355;
        return r212356;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r212357 = 1.0;
        double r212358 = a;
        double r212359 = cbrt(r212358);
        double r212360 = r212359 * r212359;
        double r212361 = y;
        double r212362 = cbrt(r212361);
        double r212363 = r212362 * r212362;
        double r212364 = r212360 / r212363;
        double r212365 = r212357 / r212364;
        double r212366 = t;
        double r212367 = z;
        double r212368 = r212366 - r212367;
        double r212369 = r212359 / r212362;
        double r212370 = r212368 / r212369;
        double r212371 = x;
        double r212372 = fma(r212365, r212370, r212371);
        return r212372;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.3
Target0.6
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Initial program 6.3

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

  2. Simplified2.6

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

  4. Applied fma-udef2.6

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

  5. Simplified2.6

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

  7. Applied add-cube-cbrt3.1

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

  8. Applied add-cube-cbrt3.2

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

  9. Applied times-frac3.2

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

  10. Applied *-un-lft-identity3.2

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

  11. Applied times-frac1.0

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

  12. Applied fma-def1.0

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

  13. Final simplification1.0

    \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{t - z}{\frac{\sqrt[3]{a}}{\sqrt[3]{y}}}, x\right)\]

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64

  :herbie-target
  (if (< y -1.07612662163899753e-10) (- x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.8944268627920891e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

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