Average Error: 6.2 → 0.9
Time: 4.3s
Precision: binary64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -4.619890674196386 \cdot 10^{+289}:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 1.1416188709729102 \cdot 10^{+65}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z - t}{\frac{a}{y}}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \leq -4.619890674196386 \cdot 10^{+289}:\\
\;\;\;\;x - y \cdot \frac{z - t}{a}\\

\mathbf{elif}\;y \cdot \left(z - t\right) \leq 1.1416188709729102 \cdot 10^{+65}:\\
\;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\

\mathbf{else}:\\
\;\;\;\;x - \frac{z - t}{\frac{a}{y}}\\

\end{array}
(FPCore (x y z t a) :precision binary64 (- x (/ (* y (- z t)) a)))
↓
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* y (- z t)) -4.619890674196386e+289)
   (- x (* y (/ (- z t) a)))
   (if (<= (* y (- z t)) 1.1416188709729102e+65)
     (- x (/ (* y (- z t)) a))
     (- x (/ (- z t) (/ a y))))))
double code(double x, double y, double z, double t, double a) {
	return x - ((y * (z - t)) / a);
}
↓
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y * (z - t)) <= -4.619890674196386e+289) {
		tmp = x - (y * ((z - t) / a));
	} else if ((y * (z - t)) <= 1.1416188709729102e+65) {
		tmp = x - ((y * (z - t)) / a);
	} else {
		tmp = x - ((z - t) / (a / y));
	}
	return tmp;
}

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

Original6.2
Target0.7
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 y (-.f64 z t)) < -4.61989067419638624e289

    1. Initial program 53.5

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

    3. Applied *-un-lft-identity_binary6453.5

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

    4. Applied times-frac_binary640.2

      \[\leadsto x - \color{blue}{\frac{y}{1} \cdot \frac{z - t}{a}}\]

    5. Simplified0.2

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

    if -4.61989067419638624e289 < (*.f64 y (-.f64 z t)) < 1.1416188709729102e65

    1. Initial program 0.4

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

    if 1.1416188709729102e65 < (*.f64 y (-.f64 z t))

    1. Initial program 14.5

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

    3. Applied clear-num_binary6414.6

      \[\leadsto x - \color{blue}{\frac{1}{\frac{a}{y \cdot \left(z - t\right)}}}\]
    4. Using strategy rm

    5. Applied associate-/r*_binary642.9

      \[\leadsto x - \frac{1}{\color{blue}{\frac{\frac{a}{y}}{z - t}}}\]

    6. Taylor expanded around 0 14.5

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

    7. Simplified2.9

      \[\leadsto x - \color{blue}{\frac{z - t}{\frac{a}{y}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -4.619890674196386 \cdot 10^{+289}:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 1.1416188709729102 \cdot 10^{+65}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z - t}{\frac{a}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020253 
(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)))