Average Error: 6.1 → 1.1
Time: 15.9s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{a} \le -5.283688091587645217512420048322496768378 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{z}{\frac{a}{y}} - \frac{t}{\frac{a}{y}}\right) + x\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{a} \le 7.074426723200414422907289769364886021349 \cdot 10^{268}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\frac{a}{y}} - \frac{t}{\frac{a}{y}}\right) + x\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\left(z - t\right) \cdot y}{a} \le -5.283688091587645217512420048322496768378 \cdot 10^{-6}:\\
\;\;\;\;\left(\frac{z}{\frac{a}{y}} - \frac{t}{\frac{a}{y}}\right) + x\\

\mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{a} \le 7.074426723200414422907289769364886021349 \cdot 10^{268}:\\
\;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r13295019 = x;
        double r13295020 = y;
        double r13295021 = z;
        double r13295022 = t;
        double r13295023 = r13295021 - r13295022;
        double r13295024 = r13295020 * r13295023;
        double r13295025 = a;
        double r13295026 = r13295024 / r13295025;
        double r13295027 = r13295019 + r13295026;
        return r13295027;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r13295028 = z;
        double r13295029 = t;
        double r13295030 = r13295028 - r13295029;
        double r13295031 = y;
        double r13295032 = r13295030 * r13295031;
        double r13295033 = a;
        double r13295034 = r13295032 / r13295033;
        double r13295035 = -5.283688091587645e-06;
        bool r13295036 = r13295034 <= r13295035;
        double r13295037 = r13295033 / r13295031;
        double r13295038 = r13295028 / r13295037;
        double r13295039 = r13295029 / r13295037;
        double r13295040 = r13295038 - r13295039;
        double r13295041 = x;
        double r13295042 = r13295040 + r13295041;
        double r13295043 = 7.074426723200414e+268;
        bool r13295044 = r13295034 <= r13295043;
        double r13295045 = r13295034 + r13295041;
        double r13295046 = r13295044 ? r13295045 : r13295042;
        double r13295047 = r13295036 ? r13295042 : r13295046;
        return r13295047;
}

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
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 2 regimes

  2. if (/ (* y (- z t)) a) < -5.283688091587645e-06 or 7.074426723200414e+268 < (/ (* y (- z t)) a)

    1. Initial program 17.7

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

    2. Simplified11.0

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

    4. Applied *-un-lft-identity11.0

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

    5. Applied add-cube-cbrt11.7

      \[\leadsto \mathsf{fma}\left(y, \frac{\color{blue}{\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \sqrt[3]{z - t}}}{1 \cdot a}, x\right)\]

    6. Applied times-frac11.7

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{1} \cdot \frac{\sqrt[3]{z - t}}{a}}, x\right)\]

    7. Simplified11.7

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

    8. Taylor expanded around 0 17.8

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

    9. Simplified3.0

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

    10. Taylor expanded around 0 17.7

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

    11. Simplified2.6

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

    if -5.283688091587645e-06 < (/ (* y (- z t)) a) < 7.074426723200414e+268

    1. Initial program 0.4

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

    2. Simplified3.7

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

    4. Applied fma-udef3.7

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

    6. Applied associate-*r/0.4

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{a} \le -5.283688091587645217512420048322496768378 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{z}{\frac{a}{y}} - \frac{t}{\frac{a}{y}}\right) + x\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{a} \le 7.074426723200414422907289769364886021349 \cdot 10^{268}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\frac{a}{y}} - \frac{t}{\frac{a}{y}}\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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