Average Error: 1.2 → 1.1
Time: 12.4s
Precision: binary64
\[x + y \cdot \frac{z - t}{a - t} \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -1.2741771487626443 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \end{array} \]
x + y \cdot \frac{z - t}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -1.2741771487626443 \cdot 10^{+52}:\\
\;\;\;\;\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 (<= z -1.2741771487626443e+52)
   (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 <= -1.2741771487626443e+52) {
		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

Original1.2
Target0.4
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;y < -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.2741771487626443e52

    1. Initial program 2.5

      \[x + y \cdot \frac{z - t}{a - t} \]

    2. Simplified2.5

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

    3. Taylor expanded in y around 0 12.4

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

    4. Simplified2.1

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

    5. Taylor expanded in x around 0 12.4

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

    6. Simplified2.1

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

    if -1.2741771487626443e52 < z

    1. Initial program 0.9

      \[x + y \cdot \frac{z - t}{a - t} \]

    2. Simplified0.9

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2741771487626443 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1.0 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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