Average Error: 6.1 → 1.5
Time: 5.9s
Precision: binary64
\[x + \frac{y \cdot \left(z - t\right)}{a} \]
↓
\[\begin{array}{l} t_1 := x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t_1 \leq -9.28850039489529 \cdot 10^{-259}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{y}}{z - t}}\\ \mathbf{elif}\;t_1 \leq 5.444881623987164 \cdot 10^{+304}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
t_1 := x + \frac{y \cdot \left(z - t\right)}{a}\\
\mathbf{if}\;t_1 \leq -9.28850039489529 \cdot 10^{-259}:\\
\;\;\;\;x + \frac{1}{\frac{\frac{a}{y}}{z - t}}\\

\mathbf{elif}\;t_1 \leq 5.444881623987164 \cdot 10^{+304}:\\
\;\;\;\;t_1\\

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


\end{array}

(FPCore (x y z t a) :precision binary64 (+ x (/ (* y (- z t)) a)))
↓
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y (- z t)) a))))
   (if (<= t_1 -9.28850039489529e-259)
     (+ x (/ 1.0 (/ (/ a y) (- z t))))
     (if (<= t_1 5.444881623987164e+304) t_1 (+ x (/ y (/ a (- z t))))))))
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 t_1 = x + ((y * (z - t)) / a);
	double tmp;
	if (t_1 <= -9.28850039489529e-259) {
		tmp = x + (1.0 / ((a / y) / (z - t)));
	} else if (t_1 <= 5.444881623987164e+304) {
		tmp = t_1;
	} else {
		tmp = x + (y / (a / (z - t)));
	}
	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
Target1.0
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (+.f64 x (/.f64 (*.f64 y (-.f64 z t)) a)) < -9.288500394895289e-259

    1. Initial program 5.9

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

    2. Applied clear-num_binary645.9

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

    3. Applied associate-/r*_binary642.5

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

    if -9.288500394895289e-259 < (+.f64 x (/.f64 (*.f64 y (-.f64 z t)) a)) < 5.4448816239871644e304

    1. Initial program 0.6

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

    if 5.4448816239871644e304 < (+.f64 x (/.f64 (*.f64 y (-.f64 z t)) a))

    1. Initial program 58.3

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

    2. Applied associate-/l*_binary641.3

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - t\right)}{a} \leq -9.28850039489529 \cdot 10^{-259}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{y}}{z - t}}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - t\right)}{a} \leq 5.444881623987164 \cdot 10^{+304}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(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)))