Average Error: 5.9 → 1.2
Time: 10.7s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.39950285805274381 \cdot 10^{104}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;a \le 3.75740059193433464 \cdot 10^{65}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{z - t}{a} \cdot y\right) + \left(\frac{z - t}{a} \cdot y\right) \cdot 0\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.39950285805274381 \cdot 10^{104}:\\
\;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\

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

\mathbf{else}:\\
\;\;\;\;\left(x - \frac{z - t}{a} \cdot y\right) + \left(\frac{z - t}{a} \cdot y\right) \cdot 0\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r316810 = x;
        double r316811 = y;
        double r316812 = z;
        double r316813 = t;
        double r316814 = r316812 - r316813;
        double r316815 = r316811 * r316814;
        double r316816 = a;
        double r316817 = r316815 / r316816;
        double r316818 = r316810 - r316817;
        return r316818;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r316819 = a;
        double r316820 = -1.3995028580527438e+104;
        bool r316821 = r316819 <= r316820;
        double r316822 = x;
        double r316823 = y;
        double r316824 = z;
        double r316825 = t;
        double r316826 = r316824 - r316825;
        double r316827 = r316819 / r316826;
        double r316828 = r316823 / r316827;
        double r316829 = r316822 - r316828;
        double r316830 = 3.7574005919343346e+65;
        bool r316831 = r316819 <= r316830;
        double r316832 = r316823 * r316826;
        double r316833 = r316832 / r316819;
        double r316834 = r316822 - r316833;
        double r316835 = r316826 / r316819;
        double r316836 = r316835 * r316823;
        double r316837 = r316822 - r316836;
        double r316838 = 0.0;
        double r316839 = r316836 * r316838;
        double r316840 = r316837 + r316839;
        double r316841 = r316831 ? r316834 : r316840;
        double r316842 = r316821 ? r316829 : r316841;
        return r316842;
}

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

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. Using strategy rm

    3. Applied associate-/l*0.7

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

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

    1. Initial program 1.6

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

    if 3.7574005919343346e+65 < a

    1. Initial program 10.0

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt10.3

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

    4. Applied add-sqr-sqrt37.1

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

    5. Applied prod-diff37.1

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

    6. Simplified10.6

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

    7. Simplified0.7

      \[\leadsto \left(x - \frac{z - t}{a} \cdot y\right) + \color{blue}{\left(\frac{z - t}{a} \cdot y\right) \cdot 0}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.39950285805274381 \cdot 10^{104}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;a \le 3.75740059193433464 \cdot 10^{65}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{z - t}{a} \cdot y\right) + \left(\frac{z - t}{a} \cdot y\right) \cdot 0\\ \end{array}\]

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, 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)))