Average Error: 6.1 → 0.8
Time: 8.2s
Precision: binary64
\[x + \frac{y \cdot \left(z - t\right)}{a} \]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -8.821147173561544 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;y \leq 4.361135130640675 \cdot 10^{-57}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{a}\right) - \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(z - t\right) \cdot \frac{1}{a}, x\right)\\ \end{array} \]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -8.821147173561544 \cdot 10^{+21}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\

\mathbf{elif}\;y \leq 4.361135130640675 \cdot 10^{-57}:\\
\;\;\;\;\left(x + \frac{y \cdot z}{a}\right) - \frac{y \cdot t}{a}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, \left(z - t\right) \cdot \frac{1}{a}, x\right)\\


\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 -8.821147173561544e+21)
   (fma y (/ (- z t) a) x)
   (if (<= y 4.361135130640675e-57)
     (- (+ x (/ (* y z) a)) (/ (* y t) a))
     (fma y (* (- z t) (/ 1.0 a)) x))))
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 <= -8.821147173561544e+21) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (y <= 4.361135130640675e-57) {
		tmp = (x + ((y * z) / a)) - ((y * t) / a);
	} else {
		tmp = fma(y, ((z - t) * (1.0 / a)), x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.1
Target0.7
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if y < -8.8211471735615442e21

    1. Initial program 17.9

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

    2. Simplified0.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)} \]

    if -8.8211471735615442e21 < y < 4.3611351306406748e-57

    1. Initial program 0.5

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

    2. Simplified9.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)} \]

    3. Taylor expanded in y around 0 0.5

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

    if 4.3611351306406748e-57 < y

    1. Initial program 11.1

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

    2. Simplified1.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)} \]

    3. Applied div-inv_binary641.3

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(z - t\right) \cdot \frac{1}{a}}, x\right) \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.821147173561544 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;y \leq 4.361135130640675 \cdot 10^{-57}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{a}\right) - \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(z - t\right) \cdot \frac{1}{a}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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)))