Average Error: 6.1 → 0.7
Time: 4.0s
Precision: binary64
\[x - \frac{y \cdot \left(z - t\right)}{a} \]
↓
\[\begin{array}{l} \mathbf{if}\;a \leq -280431850242306000:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;a \leq 9.357780273859336 \cdot 10^{+52}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \end{array} \]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \leq -280431850242306000:\\
\;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\

\mathbf{elif}\;a \leq 9.357780273859336 \cdot 10^{+52}:\\
\;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{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 (<= a -280431850242306000.0)
   (- x (/ y (/ a (- z t))))
   (if (<= a 9.357780273859336e+52)
     (- x (/ (* y (- z t)) a))
     (fma y (/ (- t z) 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 (a <= -280431850242306000.0) {
		tmp = x - (y / (a / (z - t)));
	} else if (a <= 9.357780273859336e+52) {
		tmp = x - ((y * (z - t)) / a);
	} else {
		tmp = fma(y, ((t - z) / 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
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if a < -280431850242305980

    1. Initial program 9.5

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

    2. Applied associate-/l*_binary640.5

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

    if -280431850242305980 < a < 9.3577802738593356e52

    1. Initial program 0.9

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

    if 9.3577802738593356e52 < a

    1. Initial program 10.6

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

    2. Simplified0.6

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -280431850242306000:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;a \leq 9.357780273859336 \cdot 10^{+52}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021275 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :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)))