Average Error: 6.0 → 1.6
Time: 1.3min
Precision: binary64
\[x + \frac{y \cdot \left(z - t\right)}{a} \]
↓
\[\begin{array}{l} t_1 := \mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \mathbf{if}\;z - t \leq -2.544404770474763 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z - t \leq 6.825613452492701 \cdot 10^{-109}:\\ \;\;\;\;x + \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
t_1 := \mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\
\mathbf{if}\;z - t \leq -2.544404770474763 \cdot 10^{-65}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;z - t \leq 6.825613452492701 \cdot 10^{-109}:\\
\;\;\;\;x + \frac{\left(z - t\right) \cdot y}{a}\\

\mathbf{else}:\\
\;\;\;\;t_1\\


\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 (fma (- z t) (/ y a) x)))
   (if (<= (- z t) -2.544404770474763e-65)
     t_1
     (if (<= (- z t) 6.825613452492701e-109) (+ x (/ (* (- z t) y) a)) t_1))))
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 = fma((z - t), (y / a), x);
	double tmp;
	if ((z - t) <= -2.544404770474763e-65) {
		tmp = t_1;
	} else if ((z - t) <= 6.825613452492701e-109) {
		tmp = x + (((z - t) * y) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.0
Target0.7
Herbie1.6
\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (-.f64 z t) < -2.5444047704747631e-65 or 6.8256134524927015e-109 < (-.f64 z t)

    1. Initial program 7.1

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

    2. Simplified6.8

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

    3. Taylor expanded in y around 0 7.1

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

    4. Simplified1.8

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

    if -2.5444047704747631e-65 < (-.f64 z t) < 6.8256134524927015e-109

    1. Initial program 0.7

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z - t \leq -2.544404770474763 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;z - t \leq 6.825613452492701 \cdot 10^{-109}:\\ \;\;\;\;x + \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(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)))