Average Error: 6.5 → 1.7
Time: 3.1s
Precision: binary64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \leq -4.210519492156666 \cdot 10^{-109} \lor \neg \left(a \leq 9.098611020860329 \cdot 10^{-104}\right):\\ \;\;\;\;x + y \cdot \left(\frac{t}{a} - \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(t - z\right)\right) \cdot \frac{1}{a}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \leq -4.210519492156666 \cdot 10^{-109} \lor \neg \left(a \leq 9.098611020860329 \cdot 10^{-104}\right):\\
\;\;\;\;x + y \cdot \left(\frac{t}{a} - \frac{z}{a}\right)\\

\mathbf{else}:\\
\;\;\;\;x + \left(y \cdot \left(t - z\right)\right) \cdot \frac{1}{a}\\

\end{array}
double code(double x, double y, double z, double t, double a) {
	return ((double) (x - (((double) (y * ((double) (z - t)))) / a)));
}
↓
double code(double x, double y, double z, double t, double a) {
	double VAR;
	if (((a <= -4.210519492156666e-109) || !(a <= 9.098611020860329e-104))) {
		VAR = ((double) (x + ((double) (y * ((double) ((t / a) - (z / a)))))));
	} else {
		VAR = ((double) (x + ((double) (((double) (y * ((double) (t - z)))) * (1.0 / a)))));
	}
	return VAR;
}

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

Original6.5
Target0.8
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -4.2105194921566659e-109 or 9.0986110208603289e-104 < a

    1. Initial program 7.9

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

    2. Simplified1.7

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

    4. Applied div-sub1.7

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

    if -4.2105194921566659e-109 < a < 9.0986110208603289e-104

    1. Initial program 1.6

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

    2. Simplified21.1

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

    4. Applied div-inv21.1

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

    5. Applied associate-*r*1.6

      \[\leadsto x + \color{blue}{\left(y \cdot \left(t - z\right)\right) \cdot \frac{1}{a}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.210519492156666 \cdot 10^{-109} \lor \neg \left(a \leq 9.098611020860329 \cdot 10^{-104}\right):\\ \;\;\;\;x + y \cdot \left(\frac{t}{a} - \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(t - z\right)\right) \cdot \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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 (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

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