Average Error: 4.5 → 0.3
Time: 2.6s
Precision: binary64
\[x + \frac{y \cdot \left(z - t\right)}{a} \]
↓
\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t_1 \leq -4.253903478309208 \cdot 10^{+303} \lor \neg \left(t_1 \leq 1.715184077966658 \cdot 10^{+302}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, x, t_1\right)\\ \end{array} \]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
t_1 := \frac{y \cdot \left(z - t\right)}{a}\\
\mathbf{if}\;t_1 \leq -4.253903478309208 \cdot 10^{+303} \lor \neg \left(t_1 \leq 1.715184077966658 \cdot 10^{+302}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(1, x, t_1\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)) a)))
   (if (or (<= t_1 -4.253903478309208e+303)
           (not (<= t_1 1.715184077966658e+302)))
     (fma (/ y a) (- z t) x)
     (fma 1.0 x 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 = (y * (z - t)) / a;
	double tmp;
	if ((t_1 <= -4.253903478309208e+303) || !(t_1 <= 1.715184077966658e+302)) {
		tmp = fma((y / a), (z - t), x);
	} else {
		tmp = fma(1.0, x, 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

Original4.5
Target0.5
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -4.25390347830920782e303 or 1.7151840779666582e302 < (/.f64 (*.f64 y (-.f64 z t)) a)

    1. Initial program 13.4

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

    2. Applied +-commutative_binary6413.4

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

    3. Taylor expanded in y around 0 22.3

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

    4. Simplified0.1

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

    if -4.25390347830920782e303 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1.7151840779666582e302

    1. Initial program 0.3

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

    2. Applied *-un-lft-identity_binary640.3

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

    3. Applied fma-def_binary640.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, x, \frac{y \cdot \left(z - t\right)}{a}\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -4.253903478309208 \cdot 10^{+303} \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \leq 1.715184077966658 \cdot 10^{+302}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, x, \frac{y \cdot \left(z - t\right)}{a}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, 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)))