Average Error: 5.9 → 1.2
Time: 8.2s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.39950285805274381 \cdot 10^{104}:\\ \;\;\;\;\frac{y}{\frac{a}{z - t}} + x\\ \mathbf{elif}\;a \le 3.75740059193433464 \cdot 10^{65}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.39950285805274381 \cdot 10^{104}:\\
\;\;\;\;\frac{y}{\frac{a}{z - t}} + x\\

\mathbf{elif}\;a \le 3.75740059193433464 \cdot 10^{65}:\\
\;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r325000 = x;
        double r325001 = y;
        double r325002 = z;
        double r325003 = t;
        double r325004 = r325002 - r325003;
        double r325005 = r325001 * r325004;
        double r325006 = a;
        double r325007 = r325005 / r325006;
        double r325008 = r325000 + r325007;
        return r325008;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r325009 = a;
        double r325010 = -1.3995028580527438e+104;
        bool r325011 = r325009 <= r325010;
        double r325012 = y;
        double r325013 = z;
        double r325014 = t;
        double r325015 = r325013 - r325014;
        double r325016 = r325009 / r325015;
        double r325017 = r325012 / r325016;
        double r325018 = x;
        double r325019 = r325017 + r325018;
        double r325020 = 3.7574005919343346e+65;
        bool r325021 = r325009 <= r325020;
        double r325022 = r325012 * r325015;
        double r325023 = r325022 / r325009;
        double r325024 = r325023 + r325018;
        double r325025 = r325015 / r325009;
        double r325026 = fma(r325012, r325025, r325018);
        double r325027 = r325021 ? r325024 : r325026;
        double r325028 = r325011 ? r325019 : r325027;
        return r325028;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original5.9
Target0.7
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -1.3995028580527438e+104

    1. Initial program 12.4

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

    2. Simplified2.3

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

    4. Applied fma-udef2.3

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

    5. Simplified12.4

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

    7. Applied associate-/l*0.7

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

    if -1.3995028580527438e+104 < a < 3.7574005919343346e+65

    1. Initial program 1.6

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

    2. Simplified3.0

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

    4. Applied fma-udef3.0

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

    5. Simplified1.6

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

    if 3.7574005919343346e+65 < a

    1. Initial program 10.0

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

    2. Simplified2.0

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

    4. Applied fma-udef2.0

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

    5. Simplified10.0

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

    7. Applied *-un-lft-identity10.0

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

    8. Applied times-frac0.7

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

    9. Applied fma-def0.7

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.39950285805274381 \cdot 10^{104}:\\ \;\;\;\;\frac{y}{\frac{a}{z - t}} + x\\ \mathbf{elif}\;a \le 3.75740059193433464 \cdot 10^{65}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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)))