Average Error: 6.1 → 0.9
Time: 5.5s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -3.890834236040981319165798918232586072344 \cdot 10^{-33} \lor \neg \left(a \le 8.531444755159096136469695842813758583876 \cdot 10^{-75}\right):\\ \;\;\;\;y \cdot \frac{z - t}{a} + x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\left(z - t\right) \cdot y}{a} + x\\ \end{array}\]
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{z - t}{a} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r271560 = x;
        double r271561 = y;
        double r271562 = z;
        double r271563 = t;
        double r271564 = r271562 - r271563;
        double r271565 = r271561 * r271564;
        double r271566 = a;
        double r271567 = r271565 / r271566;
        double r271568 = r271560 + r271567;
        return r271568;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r271569 = a;
        double r271570 = -3.890834236040981e-33;
        bool r271571 = r271569 <= r271570;
        double r271572 = 8.531444755159096e-75;
        bool r271573 = r271569 <= r271572;
        double r271574 = !r271573;
        bool r271575 = r271571 || r271574;
        double r271576 = y;
        double r271577 = z;
        double r271578 = t;
        double r271579 = r271577 - r271578;
        double r271580 = r271579 / r271569;
        double r271581 = r271576 * r271580;
        double r271582 = x;
        double r271583 = r271581 + r271582;
        double r271584 = 1.0;
        double r271585 = r271579 * r271576;
        double r271586 = r271585 / r271569;
        double r271587 = r271584 * r271586;
        double r271588 = r271587 + r271582;
        double r271589 = r271575 ? r271583 : r271588;
        return r271589;
}

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

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

    4. Applied fma-udef1.5

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

    6. Applied div-inv1.5

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

    7. Applied associate-*l*0.9

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

    8. Simplified0.9

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

    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

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

    4. Applied fma-udef4.0

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

    6. Applied *-un-lft-identity4.0

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

    7. Applied *-un-lft-identity4.0

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

    8. Applied times-frac4.0

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

    9. Applied associate-*l*4.0

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

    10. Simplified1.0

      \[\leadsto \frac{1}{1} \cdot \color{blue}{\frac{\left(z - t\right) \cdot y}{a}} + x\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -3.890834236040981319165798918232586072344 \cdot 10^{-33} \lor \neg \left(a \le 8.531444755159096136469695842813758583876 \cdot 10^{-75}\right):\\ \;\;\;\;y \cdot \frac{z - t}{a} + x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\left(z - t\right) \cdot y}{a} + x\\ \end{array}\]

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, 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)))