Average Error: 6.4 → 1.1
Time: 2.7s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.96605357789947992 \cdot 10^{-23}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(z - t\right)\\ \mathbf{elif}\;a \le 1.23690458140457419 \cdot 10^{-16}:\\ \;\;\;\;1 \cdot \frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.96605357789947992 \cdot 10^{-23}:\\
\;\;\;\;x + \frac{y}{a} \cdot \left(z - t\right)\\

\mathbf{elif}\;a \le 1.23690458140457419 \cdot 10^{-16}:\\
\;\;\;\;1 \cdot \frac{\left(z - t\right) \cdot y}{a} + x\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r280845 = x;
        double r280846 = y;
        double r280847 = z;
        double r280848 = t;
        double r280849 = r280847 - r280848;
        double r280850 = r280846 * r280849;
        double r280851 = a;
        double r280852 = r280850 / r280851;
        double r280853 = r280845 + r280852;
        return r280853;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r280854 = a;
        double r280855 = -1.96605357789948e-23;
        bool r280856 = r280854 <= r280855;
        double r280857 = x;
        double r280858 = y;
        double r280859 = r280858 / r280854;
        double r280860 = z;
        double r280861 = t;
        double r280862 = r280860 - r280861;
        double r280863 = r280859 * r280862;
        double r280864 = r280857 + r280863;
        double r280865 = 1.2369045814045742e-16;
        bool r280866 = r280854 <= r280865;
        double r280867 = 1.0;
        double r280868 = r280862 * r280858;
        double r280869 = r280868 / r280854;
        double r280870 = r280867 * r280869;
        double r280871 = r280870 + r280857;
        double r280872 = r280862 / r280854;
        double r280873 = fma(r280872, r280858, r280857);
        double r280874 = r280867 * r280873;
        double r280875 = r280866 ? r280871 : r280874;
        double r280876 = r280856 ? r280864 : r280875;
        return r280876;
}

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
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -1.96605357789948e-23

    1. Initial program 9.3

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

    2. Simplified1.9

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

    4. Applied fma-udef1.9

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

    6. Applied +-commutative1.9

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

    if -1.96605357789948e-23 < a < 1.2369045814045742e-16

    1. Initial program 0.9

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

    2. Simplified3.7

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

    4. Applied fma-udef3.7

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

    6. Applied *-un-lft-identity3.7

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

    7. Applied associate-*l*3.7

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

    8. Simplified0.9

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

    if 1.2369045814045742e-16 < a

    1. Initial program 9.4

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

    2. Simplified1.6

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

    4. Applied fma-udef1.6

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

    6. Applied *-un-lft-identity1.6

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

    7. Applied associate-*l*1.6

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

    8. Simplified9.4

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

    10. Applied *-un-lft-identity9.4

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

    11. Applied distribute-lft-out9.4

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

    12. Simplified0.5

      \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.96605357789947992 \cdot 10^{-23}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(z - t\right)\\ \mathbf{elif}\;a \le 1.23690458140457419 \cdot 10^{-16}:\\ \;\;\;\;1 \cdot \frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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 -1.0761266216389975e-10) (+ x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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