Average Error: 6.1 → 0.9
Time: 3.2s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -3.890834236040981319165798918232586072344 \cdot 10^{-33} \lor \neg \left(a \le 8.531444755159096136469695842813758583876 \cdot 10^{-75}\right):\\ \;\;\;\;y \cdot \frac{t - z}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t - z\right) \cdot y}{a} + x\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -3.890834236040981319165798918232586072344 \cdot 10^{-33} \lor \neg \left(a \le 8.531444755159096136469695842813758583876 \cdot 10^{-75}\right):\\
\;\;\;\;y \cdot \frac{t - z}{a} + x\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(t - z\right) \cdot y}{a} + x\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r297508 = x;
        double r297509 = y;
        double r297510 = z;
        double r297511 = t;
        double r297512 = r297510 - r297511;
        double r297513 = r297509 * r297512;
        double r297514 = a;
        double r297515 = r297513 / r297514;
        double r297516 = r297508 - r297515;
        return r297516;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r297517 = a;
        double r297518 = -3.890834236040981e-33;
        bool r297519 = r297517 <= r297518;
        double r297520 = 8.531444755159096e-75;
        bool r297521 = r297517 <= r297520;
        double r297522 = !r297521;
        bool r297523 = r297519 || r297522;
        double r297524 = y;
        double r297525 = t;
        double r297526 = z;
        double r297527 = r297525 - r297526;
        double r297528 = r297527 / r297517;
        double r297529 = r297524 * r297528;
        double r297530 = x;
        double r297531 = r297529 + r297530;
        double r297532 = r297527 * r297524;
        double r297533 = r297532 / r297517;
        double r297534 = r297533 + r297530;
        double r297535 = r297523 ? r297531 : r297534;
        return r297535;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.1
Target0.7
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -3.890834236040981e-33 or 8.531444755159096e-75 < a

    1. Initial program 8.2

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

    2. Simplified1.5

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

    4. Applied fma-udef1.5

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

    6. Applied div-inv1.5

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

    7. Applied associate-*l*0.9

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

    8. Simplified0.9

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

    if -3.890834236040981e-33 < a < 8.531444755159096e-75

    1. Initial program 1.0

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

    2. Simplified4.0

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

    4. Applied fma-udef4.0

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

    5. Taylor expanded around 0 1.0

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

    6. Simplified1.0

      \[\leadsto \color{blue}{\frac{\left(t - z\right) \cdot y}{a}} + x\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -3.890834236040981319165798918232586072344 \cdot 10^{-33} \lor \neg \left(a \le 8.531444755159096136469695842813758583876 \cdot 10^{-75}\right):\\ \;\;\;\;y \cdot \frac{t - z}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t - z\right) \cdot y}{a} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 +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 -1.0761266216389975e-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)))