Average Error: 6.1 → 1.6
Time: 18.0s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;z - t \le -8.381032267864286848111598473235001808927 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{elif}\;z - t \le 2.979324857395300632013474840610844551682 \cdot 10^{-117}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{z - t}{\frac{a}{\sqrt[3]{y}}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{\frac{a}{y}} + x\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;z - t \le -8.381032267864286848111598473235001808927 \cdot 10^{-13}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\

\mathbf{elif}\;z - t \le 2.979324857395300632013474840610844551682 \cdot 10^{-117}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{z - t}{\frac{a}{\sqrt[3]{y}}}, x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{z - t}{\frac{a}{y}} + x\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r232855 = x;
        double r232856 = y;
        double r232857 = z;
        double r232858 = t;
        double r232859 = r232857 - r232858;
        double r232860 = r232856 * r232859;
        double r232861 = a;
        double r232862 = r232860 / r232861;
        double r232863 = r232855 + r232862;
        return r232863;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r232864 = z;
        double r232865 = t;
        double r232866 = r232864 - r232865;
        double r232867 = -8.381032267864287e-13;
        bool r232868 = r232866 <= r232867;
        double r232869 = y;
        double r232870 = a;
        double r232871 = r232869 / r232870;
        double r232872 = x;
        double r232873 = fma(r232871, r232866, r232872);
        double r232874 = 2.9793248573953006e-117;
        bool r232875 = r232866 <= r232874;
        double r232876 = 1.0;
        double r232877 = cbrt(r232869);
        double r232878 = r232877 * r232877;
        double r232879 = r232876 / r232878;
        double r232880 = r232876 / r232879;
        double r232881 = r232870 / r232877;
        double r232882 = r232866 / r232881;
        double r232883 = fma(r232880, r232882, r232872);
        double r232884 = r232870 / r232869;
        double r232885 = r232866 / r232884;
        double r232886 = r232885 + r232872;
        double r232887 = r232875 ? r232883 : r232886;
        double r232888 = r232868 ? r232873 : r232887;
        return r232888;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.1
Target0.7
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. Split input into 3 regimes

  2. if (- z t) < -8.381032267864287e-13

    1. Initial program 8.2

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

    2. Simplified1.6

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

    if -8.381032267864287e-13 < (- z t) < 2.9793248573953006e-117

    1. Initial program 1.2

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

    2. Simplified5.6

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

    4. Applied fma-udef5.6

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

    5. Simplified5.2

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

    7. Applied add-cube-cbrt5.5

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

    8. Applied *-un-lft-identity5.5

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

    9. Applied times-frac5.5

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

    10. Applied *-un-lft-identity5.5

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

    11. Applied times-frac1.3

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

    12. Applied fma-def1.3

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

    if 2.9793248573953006e-117 < (- z t)

    1. Initial program 6.8

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

    2. Simplified1.5

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

    4. Applied fma-udef1.5

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

    5. Simplified1.6

      \[\leadsto \color{blue}{\frac{z - t}{\frac{a}{y}}} + x\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z - t \le -8.381032267864286848111598473235001808927 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{elif}\;z - t \le 2.979324857395300632013474840610844551682 \cdot 10^{-117}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{z - t}{\frac{a}{\sqrt[3]{y}}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{\frac{a}{y}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :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)))