Average Error: 6.1 → 1.1
Time: 4.2s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -757188198512776.875:\\ \;\;\;\;y \cdot \frac{t - z}{a} + x\\ \mathbf{elif}\;a \le 32027701517384002043904:\\ \;\;\;\;\left(\frac{t \cdot y}{a} - \frac{z \cdot y}{a}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right) + x\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -757188198512776.875:\\
\;\;\;\;y \cdot \frac{t - z}{a} + x\\

\mathbf{elif}\;a \le 32027701517384002043904:\\
\;\;\;\;\left(\frac{t \cdot y}{a} - \frac{z \cdot y}{a}\right) + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r367579 = x;
        double r367580 = y;
        double r367581 = z;
        double r367582 = t;
        double r367583 = r367581 - r367582;
        double r367584 = r367580 * r367583;
        double r367585 = a;
        double r367586 = r367584 / r367585;
        double r367587 = r367579 - r367586;
        return r367587;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r367588 = a;
        double r367589 = -757188198512776.9;
        bool r367590 = r367588 <= r367589;
        double r367591 = y;
        double r367592 = t;
        double r367593 = z;
        double r367594 = r367592 - r367593;
        double r367595 = r367594 / r367588;
        double r367596 = r367591 * r367595;
        double r367597 = x;
        double r367598 = r367596 + r367597;
        double r367599 = 3.2027701517384e+22;
        bool r367600 = r367588 <= r367599;
        double r367601 = r367592 * r367591;
        double r367602 = r367601 / r367588;
        double r367603 = r367593 * r367591;
        double r367604 = r367603 / r367588;
        double r367605 = r367602 - r367604;
        double r367606 = r367605 + r367597;
        double r367607 = r367591 / r367588;
        double r367608 = r367607 * r367594;
        double r367609 = r367608 + r367597;
        double r367610 = r367600 ? r367606 : r367609;
        double r367611 = r367590 ? r367598 : r367610;
        return r367611;
}

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
Target0.8
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -757188198512776.9

    1. Initial program 9.5

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

    2. Simplified1.6

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

    4. Applied fma-udef1.6

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

    6. Applied div-inv1.7

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

    7. Applied associate-*l*0.7

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

    8. Simplified0.6

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

    if -757188198512776.9 < a < 3.2027701517384e+22

    1. Initial program 0.9

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

    2. Simplified3.2

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

    4. Applied fma-udef3.2

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

    6. Applied add-cube-cbrt3.9

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

    7. Applied associate-*l*3.9

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

    8. Taylor expanded around 0 0.9

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

    if 3.2027701517384e+22 < a

    1. Initial program 10.1

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

    2. Simplified1.8

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

    4. Applied fma-udef1.8

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -757188198512776.875:\\ \;\;\;\;y \cdot \frac{t - z}{a} + x\\ \mathbf{elif}\;a \le 32027701517384002043904:\\ \;\;\;\;\left(\frac{t \cdot y}{a} - \frac{z \cdot y}{a}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +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)))