Average Error: 6.1 → 1.4
Time: 5.3s
Precision: binary64
\[x - \frac{y \cdot \left(z - t\right)}{a} \]
↓
\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ t_2 := \mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{if}\;t_1 \leq 5.679022948171 \cdot 10^{-310}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 4.920673586609806 \cdot 10^{+143}:\\ \;\;\;\;x - \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
t_1 := y \cdot \left(z - t\right)\\
t_2 := \mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\
\mathbf{if}\;t_1 \leq 5.679022948171 \cdot 10^{-310}:\\
\;\;\;\;t_2\\

\mathbf{elif}\;t_1 \leq 4.920673586609806 \cdot 10^{+143}:\\
\;\;\;\;x - \frac{t_1}{a}\\

\mathbf{else}:\\
\;\;\;\;t_2\\


\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))) (t_2 (fma (/ y a) (- t z) x)))
   (if (<= t_1 5.679022948171e-310)
     t_2
     (if (<= t_1 4.920673586609806e+143) (- x (/ t_1 a)) t_2))))
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 t_2 = fma((y / a), (t - z), x);
	double tmp;
	if (t_1 <= 5.679022948171e-310) {
		tmp = t_2;
	} else if (t_1 <= 4.920673586609806e+143) {
		tmp = x - (t_1 / a);
	} else {
		tmp = t_2;
	}
	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
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 y (-.f64 z t)) < 5.679022948171016e-310 or 4.92067358660980582e143 < (*.f64 y (-.f64 z t))

    1. Initial program 8.9

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

    2. Taylor expanded in x around 0 8.9

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

    3. Simplified2.0

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

    4. Taylor expanded in y around 0 8.9

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

    5. Simplified2.0

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

    if 5.679022948171016e-310 < (*.f64 y (-.f64 z t)) < 4.92067358660980582e143

    1. Initial program 0.1

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq 5.679022948171 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 4.920673586609806 \cdot 10^{+143}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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)))