Average Error: 11.3 → 1.1
Time: 5.4s
Precision: binary64
\[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -1.2069852408039634 \cdot 10^{+146} \lor \neg \left(z \leq 1.9758692242271444 \cdot 10^{+183}\right):\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \end{array} \]
x + \frac{y \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -1.2069852408039634 \cdot 10^{+146} \lor \neg \left(z \leq 1.9758692242271444 \cdot 10^{+183}\right):\\
\;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a - t}, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\


\end{array}

(FPCore (x y z t a) :precision binary64 (+ x (/ (* y (- z t)) (- a t))))
↓
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.2069852408039634e+146) (not (<= z 1.9758692242271444e+183)))
   (fma (- z t) (/ y (- a t)) x)
   (fma y (/ (- z t) (- a t)) x)))
double code(double x, double y, double z, double t, double a) {
	return x + ((y * (z - t)) / (a - t));
}
↓
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.2069852408039634e+146) || !(z <= 1.9758692242271444e+183)) {
		tmp = fma((z - t), (y / (a - t)), x);
	} else {
		tmp = fma(y, ((z - t) / (a - 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

Original11.3
Target1.3
Herbie1.1
\[x + \frac{y}{\frac{a - t}{z - t}} \]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.20698524080396343e146 or 1.9758692242271444e183 < z

    1. Initial program 17.0

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

    2. Simplified4.9

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

    3. Taylor expanded in y around 0 17.0

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

    4. Simplified3.5

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

    5. Taylor expanded in x around 0 17.0

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

    6. Simplified3.5

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

    if -1.20698524080396343e146 < z < 1.9758692242271444e183

    1. Initial program 10.0

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

    2. Simplified0.5

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2069852408039634 \cdot 10^{+146} \lor \neg \left(z \leq 1.9758692242271444 \cdot 10^{+183}\right):\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- a t) (- z t))))

  (+ x (/ (* y (- z t)) (- a t))))