Average Error: 6.1 → 1.5
Time: 5.1s
Precision: binary64
\[x - \frac{y \cdot \left(z - t\right)}{a} \]
↓
\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -1.1974734501407108 \cdot 10^{+89} \lor \neg \left(t_1 \leq 9.158950484639035 \cdot 10^{+53}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, \mathsf{fma}\left(-1, z, t\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t_1}{a}\\ \end{array} \]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
t_1 := y \cdot \left(z - t\right)\\
\mathbf{if}\;t_1 \leq -1.1974734501407108 \cdot 10^{+89} \lor \neg \left(t_1 \leq 9.158950484639035 \cdot 10^{+53}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, \mathsf{fma}\left(-1, z, t\right), x\right)\\

\mathbf{else}:\\
\;\;\;\;x - \frac{t_1}{a}\\


\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 (* y (- z t))))
   (if (or (<= t_1 -1.1974734501407108e+89)
           (not (<= t_1 9.158950484639035e+53)))
     (fma (/ y a) (fma -1.0 z t) x)
     (- x (/ t_1 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 t_1 = y * (z - t);
	double tmp;
	if ((t_1 <= -1.1974734501407108e+89) || !(t_1 <= 9.158950484639035e+53)) {
		tmp = fma((y / a), fma(-1.0, z, t), x);
	} else {
		tmp = x - (t_1 / a);
	}
	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
Target0.9
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 2 regimes

  2. if (*.f64 y (-.f64 z t)) < -1.1974734501407108e89 or 9.158950484639035e53 < (*.f64 y (-.f64 z t))

    1. Initial program 13.8

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

    2. Taylor expanded in x around 0 13.8

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

    3. Simplified2.6

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

    if -1.1974734501407108e89 < (*.f64 y (-.f64 z t)) < 9.158950484639035e53

    1. Initial program 0.6

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -1.1974734501407108 \cdot 10^{+89} \lor \neg \left(y \cdot \left(z - t\right) \leq 9.158950484639035 \cdot 10^{+53}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, \mathsf{fma}\left(-1, z, t\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(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)))