Average Error: 6.4 → 1.0
Time: 20.6s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\mathsf{fma}\left(\frac{1}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{1}{\frac{\frac{\sqrt[3]{a}}{\sqrt[3]{y}}}{z - t}}, x\right)\]
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{1}{\frac{\frac{\sqrt[3]{a}}{\sqrt[3]{y}}}{z - t}}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r255156 = x;
        double r255157 = y;
        double r255158 = z;
        double r255159 = t;
        double r255160 = r255158 - r255159;
        double r255161 = r255157 * r255160;
        double r255162 = a;
        double r255163 = r255161 / r255162;
        double r255164 = r255156 + r255163;
        return r255164;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r255165 = 1.0;
        double r255166 = a;
        double r255167 = cbrt(r255166);
        double r255168 = r255167 * r255167;
        double r255169 = y;
        double r255170 = cbrt(r255169);
        double r255171 = r255170 * r255170;
        double r255172 = r255168 / r255171;
        double r255173 = r255165 / r255172;
        double r255174 = r255167 / r255170;
        double r255175 = z;
        double r255176 = t;
        double r255177 = r255175 - r255176;
        double r255178 = r255174 / r255177;
        double r255179 = r255165 / r255178;
        double r255180 = x;
        double r255181 = fma(r255173, r255179, r255180);
        return r255181;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.4
Target0.7
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 6.4

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

  2. Simplified6.0

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

  4. Applied fma-udef6.0

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

  5. Simplified5.7

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

  7. Applied clear-num5.7

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

  8. Simplified2.7

    \[\leadsto \frac{1}{\color{blue}{\frac{\frac{a}{y}}{z - t}}} + x\]
  9. Using strategy rm

  10. Applied *-un-lft-identity2.7

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

  11. Applied add-cube-cbrt3.1

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

  12. Applied add-cube-cbrt3.2

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

  13. Applied times-frac3.2

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

  14. Applied times-frac1.0

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

  15. Applied *-un-lft-identity1.0

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

  16. Applied times-frac1.0

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

  17. Applied fma-def1.0

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

  18. Final simplification1.0

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

Reproduce

herbie shell --seed 2019196 +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)))