Average Error: 6.0 → 0.7
Time: 9.1s
Precision: binary64
\[x + \frac{y \cdot \left(z - t\right)}{a} \]
↓
\[\begin{array}{l} \mathbf{if}\;a \leq -9.764712075529109 \cdot 10^{+47}:\\ \;\;\;\;x + \frac{y}{a \cdot \frac{1}{z - t}}\\ \mathbf{elif}\;a \leq 1.706055111709416 \cdot 10^{-29}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array} \]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \leq -9.764712075529109 \cdot 10^{+47}:\\
\;\;\;\;x + \frac{y}{a \cdot \frac{1}{z - t}}\\

\mathbf{elif}\;a \leq 1.706055111709416 \cdot 10^{-29}:\\
\;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\

\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 (<= a -9.764712075529109e+47)
   (+ x (/ y (* a (/ 1.0 (- z t)))))
   (if (<= a 1.706055111709416e-29)
     (+ x (* (* y (- z t)) (/ 1.0 a)))
     (+ 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 (a <= -9.764712075529109e+47) {
		tmp = x + (y / (a * (1.0 / (z - t))));
	} else if (a <= 1.706055111709416e-29) {
		tmp = x + ((y * (z - t)) * (1.0 / a));
	} 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

Original6.0
Target0.6
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if a < -9.76471207552910879e47

    1. Initial program 10.1

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

    2. Applied associate-/l*_binary640.4

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

    3. Applied div-inv_binary640.4

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

    if -9.76471207552910879e47 < a < 1.7060551117094161e-29

    1. Initial program 1.1

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

    2. Applied div-inv_binary641.2

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

    if 1.7060551117094161e-29 < a

    1. Initial program 8.5

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

    2. Applied *-un-lft-identity_binary648.5

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

    3. Applied times-frac_binary640.5

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

    4. Simplified0.5

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.764712075529109 \cdot 10^{+47}:\\ \;\;\;\;x + \frac{y}{a \cdot \frac{1}{z - t}}\\ \mathbf{elif}\;a \leq 1.706055111709416 \cdot 10^{-29}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array} \]

Reproduce

herbie shell --seed 2021280 
(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)))