Average Error: 6.4 → 1.3
Time: 42.9s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -4.292867508096600443770196014802005652851 \cdot 10^{72}:\\ \;\;\;\;\left(\frac{z}{a} - \frac{t}{a}\right) \cdot y + x\\ \mathbf{elif}\;y \le 6.304490465562008580777481844716509272673 \cdot 10^{-114}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{a} - \frac{t}{a}\right) \cdot y + x\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \le -4.292867508096600443770196014802005652851 \cdot 10^{72}:\\
\;\;\;\;\left(\frac{z}{a} - \frac{t}{a}\right) \cdot y + x\\

\mathbf{elif}\;y \le 6.304490465562008580777481844716509272673 \cdot 10^{-114}:\\
\;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r11740115 = x;
        double r11740116 = y;
        double r11740117 = z;
        double r11740118 = t;
        double r11740119 = r11740117 - r11740118;
        double r11740120 = r11740116 * r11740119;
        double r11740121 = a;
        double r11740122 = r11740120 / r11740121;
        double r11740123 = r11740115 + r11740122;
        return r11740123;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r11740124 = y;
        double r11740125 = -4.2928675080966004e+72;
        bool r11740126 = r11740124 <= r11740125;
        double r11740127 = z;
        double r11740128 = a;
        double r11740129 = r11740127 / r11740128;
        double r11740130 = t;
        double r11740131 = r11740130 / r11740128;
        double r11740132 = r11740129 - r11740131;
        double r11740133 = r11740132 * r11740124;
        double r11740134 = x;
        double r11740135 = r11740133 + r11740134;
        double r11740136 = 6.3044904655620086e-114;
        bool r11740137 = r11740124 <= r11740136;
        double r11740138 = r11740127 - r11740130;
        double r11740139 = r11740124 * r11740138;
        double r11740140 = r11740139 / r11740128;
        double r11740141 = r11740140 + r11740134;
        double r11740142 = r11740137 ? r11740141 : r11740135;
        double r11740143 = r11740126 ? r11740135 : r11740142;
        return r11740143;
}

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

Original6.4
Target0.7
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -4.2928675080966004e+72 or 6.3044904655620086e-114 < y

    1. Initial program 13.3

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

    3. Applied associate-/l*1.6

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

    4. Taylor expanded around 0 13.3

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

    5. Simplified1.7

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

    if -4.2928675080966004e+72 < y < 6.3044904655620086e-114

    1. Initial program 1.0

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.292867508096600443770196014802005652851 \cdot 10^{72}:\\ \;\;\;\;\left(\frac{z}{a} - \frac{t}{a}\right) \cdot y + x\\ \mathbf{elif}\;y \le 6.304490465562008580777481844716509272673 \cdot 10^{-114}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{a} - \frac{t}{a}\right) \cdot y + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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)))