Average Error: 6.0 → 1.2
Time: 5.6s
Precision: binary64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -\infty:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{y}}{z - t}}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq -1.2873438121665836 \cdot 10^{-209}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(\left(z - t\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right)}{\sqrt[3]{a}}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -\infty:\\
\;\;\;\;x + \frac{1}{\frac{\frac{a}{y}}{z - t}}\\

\mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq -1.2873438121665836 \cdot 10^{-209}:\\
\;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\

\mathbf{else}:\\
\;\;\;\;x + \frac{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(\left(z - t\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right)}{\sqrt[3]{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) (- INFINITY))
   (+ x (/ 1.0 (/ (/ a y) (- z t))))
   (if (<= (/ (* y (- z t)) a) -1.2873438121665836e-209)
     (+ (/ (* y (- z t)) a) x)
     (+
      x
      (/
       (* (/ (* (cbrt y) (cbrt y)) (cbrt a)) (* (- z t) (/ (cbrt y) (cbrt a))))
       (cbrt 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) <= -((double) INFINITY)) {
		tmp = x + (1.0 / ((a / y) / (z - t)));
	} else if (((y * (z - t)) / a) <= -1.2873438121665836e-209) {
		tmp = ((y * (z - t)) / a) + x;
	} else {
		tmp = x + ((((cbrt(y) * cbrt(y)) / cbrt(a)) * ((z - t) * (cbrt(y) / cbrt(a)))) / cbrt(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

Original6.0
Target0.7
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -inf.0

    1. Initial program 64.0

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

    3. Applied add-cube-cbrt_binary6464.0

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

    4. Applied associate-/r*_binary6464.0

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

    5. Simplified16.6

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

    7. Applied clear-num_binary6416.6

      \[\leadsto x + \color{blue}{\frac{1}{\frac{\sqrt[3]{a}}{\frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(z - t\right)}}}\]

    8. Simplified0.3

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

    if -inf.0 < (/.f64 (*.f64 y (-.f64 z t)) a) < -1.2873438121665836e-209

    1. Initial program 0.2

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

    if -1.2873438121665836e-209 < (/.f64 (*.f64 y (-.f64 z t)) a)

    1. Initial program 5.4

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

    3. Applied add-cube-cbrt_binary645.8

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

    4. Applied associate-/r*_binary645.8

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

    5. Simplified2.8

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

    7. Applied add-cube-cbrt_binary642.9

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

    8. Applied times-frac_binary642.9

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

    9. Applied associate-*l*_binary641.9

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

    10. Simplified1.9

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -\infty:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{y}}{z - t}}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq -1.2873438121665836 \cdot 10^{-209}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(\left(z - t\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right)}{\sqrt[3]{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021175 
(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)))