Average Error: 5.9 → 6.2
Time: 9.8s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[y \cdot \left(\frac{z}{a} - \frac{t}{a}\right) + x\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
y \cdot \left(\frac{z}{a} - \frac{t}{a}\right) + x
double f(double x, double y, double z, double t, double a) {
        double r267227 = x;
        double r267228 = y;
        double r267229 = z;
        double r267230 = t;
        double r267231 = r267229 - r267230;
        double r267232 = r267228 * r267231;
        double r267233 = a;
        double r267234 = r267232 / r267233;
        double r267235 = r267227 + r267234;
        return r267235;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r267236 = y;
        double r267237 = z;
        double r267238 = a;
        double r267239 = r267237 / r267238;
        double r267240 = t;
        double r267241 = r267240 / r267238;
        double r267242 = r267239 - r267241;
        double r267243 = r267236 * r267242;
        double r267244 = x;
        double r267245 = r267243 + r267244;
        return r267245;
}

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
Target0.8
Herbie6.2
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -3.079662010669181e+61 or 1.737882189922714e-20 < y

    1. Initial program 15.8

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

    2. Taylor expanded around 0 15.8

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

    3. Simplified3.9

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

    4. Taylor expanded around 0 15.8

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

    5. Simplified1.0

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

    if -3.079662010669181e+61 < y < 1.737882189922714e-20

    1. Initial program 0.7

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

    2. Taylor expanded around 0 0.7

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

  4. Final simplification6.2

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

Reproduce

herbie shell --seed 2019291 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.07612662163899753e-10) (+ x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.8944268627920891e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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