Average Error: 6.2 → 0.4
Time: 5.7s
Precision: binary64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -3.348057335323633 \cdot 10^{+243}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 3.6307988486536463 \cdot 10^{+245}:\\ \;\;\;\;x + \left(\frac{y \cdot z}{a} - \frac{y \cdot t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - t}{\frac{a}{y}}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \leq -3.348057335323633 \cdot 10^{+243}:\\
\;\;\;\;x + y \cdot \frac{z - t}{a}\\

\mathbf{elif}\;y \cdot \left(z - t\right) \leq 3.6307988486536463 \cdot 10^{+245}:\\
\;\;\;\;x + \left(\frac{y \cdot z}{a} - \frac{y \cdot t}{a}\right)\\

\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)) -3.348057335323633e+243)
   (+ x (* y (/ (- z t) a)))
   (if (<= (* y (- z t)) 3.6307988486536463e+245)
     (+ x (- (/ (* y z) a) (/ (* y 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)) <= -3.348057335323633e+243) {
		tmp = x + (y * ((z - t) / a));
	} else if ((y * (z - t)) <= 3.6307988486536463e+245) {
		tmp = x + (((y * z) / a) - ((y * 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
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 y (-.f64 z t)) < -3.34805733532363309e243

    1. Initial program 37.4

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

    3. Applied *-un-lft-identity_binary6437.4

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

    4. Applied times-frac_binary640.3

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

    5. Simplified0.3

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

    if -3.34805733532363309e243 < (*.f64 y (-.f64 z t)) < 3.63079884865364626e245

    1. Initial program 0.4

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

    2. Taylor expanded around 0 0.4

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

    if 3.63079884865364626e245 < (*.f64 y (-.f64 z t))

    1. Initial program 37.1

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

    3. Applied clear-num_binary6437.1

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

    4. Simplified37.1

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

    6. Applied *-un-lft-identity_binary6437.1

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

    7. Applied times-frac_binary640.4

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

    8. Applied associate-/r*_binary640.3

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

    9. Simplified0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -3.348057335323633 \cdot 10^{+243}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 3.6307988486536463 \cdot 10^{+245}:\\ \;\;\;\;x + \left(\frac{y \cdot z}{a} - \frac{y \cdot t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - t}{\frac{a}{y}}\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(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)))