Average Error: 6.1 → 1.1
Time: 11.9s
Precision: binary64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -3.251553770096906 \cdot 10^{+267}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{y}}{z - t}}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 5.8611512384324076 \cdot 10^{+259}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -3.251553770096906 \cdot 10^{+267}:\\
\;\;\;\;x + \frac{1}{\frac{\frac{a}{y}}{z - t}}\\

\mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 5.8611512384324076 \cdot 10^{+259}:\\
\;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\

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

\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)) a) -3.251553770096906e+267)
   (+ x (/ 1.0 (/ (/ a y) (- z t))))
   (if (<= (/ (* y (- z t)) a) 5.8611512384324076e+259)
     (+ (/ (* y (- z t)) a) x)
     (+ x (* y (/ (- z t) a))))))
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)) / a) <= -3.251553770096906e+267) {
		tmp = x + (1.0 / ((a / y) / (z - t)));
	} else if (((y * (z - t)) / a) <= 5.8611512384324076e+259) {
		tmp = ((y * (z - t)) / a) + x;
	} else {
		tmp = x + (y * ((z - t) / a));
	}
	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

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

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -3.251553770096906e267

    1. Initial program 42.3

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

    2. Simplified3.5

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

    4. Applied add-cube-cbrt_binary64_168224.5

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

    5. Applied associate-*l*_binary64_167284.5

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

    6. Simplified4.5

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

    7. Taylor expanded around 0 42.3

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

    8. Simplified3.2

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

    10. Applied clear-num_binary64_167863.3

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

    if -3.251553770096906e267 < (/.f64 (*.f64 y (-.f64 z t)) a) < 5.86115123843240756e259

    1. Initial program 0.4

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

    if 5.86115123843240756e259 < (/.f64 (*.f64 y (-.f64 z t)) a)

    1. Initial program 41.6

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

    2. Simplified7.9

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -3.251553770096906 \cdot 10^{+267}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{y}}{z - t}}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 5.8611512384324076 \cdot 10^{+259}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2021176 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-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)))