Average Error: 6.5 → 0.7
Time: 7.1s
Precision: binary64
\[x + \frac{y \cdot \left(z - t\right)}{a} \]
↓
\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;t_1 \leq 1.4424983057985938 \cdot 10^{+118}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{a}\right) - \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \end{array} \]
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 -\infty:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\

\mathbf{elif}\;t_1 \leq 1.4424983057985938 \cdot 10^{+118}:\\
\;\;\;\;\left(x + \frac{y \cdot z}{a}\right) - \frac{y \cdot t}{a}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, 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
 (let* ((t_1 (* y (- z t))))
   (if (<= t_1 (- INFINITY))
     (fma y (/ (- z t) a) x)
     (if (<= t_1 1.4424983057985938e+118)
       (- (+ x (/ (* y z) a)) (/ (* y t) a))
       (fma (/ y a) (- z t) 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 t_1 = y * (z - t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (t_1 <= 1.4424983057985938e+118) {
		tmp = (x + ((y * z) / a)) - ((y * t) / a);
	} else {
		tmp = fma((y / a), (z - t), 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

Original6.5
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 (*.f64 y (-.f64 z t)) < -inf.0

    1. Initial program 64.0

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

    2. Simplified0.2

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

    if -inf.0 < (*.f64 y (-.f64 z t)) < 1.442498305798594e118

    1. Initial program 0.4

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

    2. Simplified7.0

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

    3. Taylor expanded in y around 0 0.4

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

    if 1.442498305798594e118 < (*.f64 y (-.f64 z t))

    1. Initial program 19.0

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

    2. Simplified2.6

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

    3. Taylor expanded in y around 0 19.0

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

    4. Simplified2.3

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 1.4424983057985938 \cdot 10^{+118}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{a}\right) - \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022077 
(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)))