Average Error: 11.0 → 0.4
Time: 10.4s
Precision: binary64
\[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
↓
\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{if}\;t_1 \leq -2.5690390576883054 \cdot 10^{+262}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{\frac{z - a}{z - t}}, x\right)\\ \mathbf{elif}\;t_1 \leq 3.850864673207466 \cdot 10^{+236}:\\ \;\;\;\;t_1 + x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \end{array} \]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\begin{array}{l}
t_1 := \frac{y \cdot \left(z - t\right)}{z - a}\\
\mathbf{if}\;t_1 \leq -2.5690390576883054 \cdot 10^{+262}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{1}{\frac{z - a}{z - t}}, x\right)\\

\mathbf{elif}\;t_1 \leq 3.850864673207466 \cdot 10^{+236}:\\
\;\;\;\;t_1 + x\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \frac{z - t}{z - a}\\


\end{array}

(FPCore (x y z t a) :precision binary64 (+ x (/ (* y (- z t)) (- z a))))
↓
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) (- z a))))
   (if (<= t_1 -2.5690390576883054e+262)
     (fma y (/ 1.0 (/ (- z a) (- z t))) x)
     (if (<= t_1 3.850864673207466e+236)
       (+ t_1 x)
       (+ x (* y (/ (- z t) (- z a))))))))
double code(double x, double y, double z, double t, double a) {
	return x + ((y * (z - t)) / (z - a));
}
↓
double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / (z - a);
	double tmp;
	if (t_1 <= -2.5690390576883054e+262) {
		tmp = fma(y, (1.0 / ((z - a) / (z - t))), x);
	} else if (t_1 <= 3.850864673207466e+236) {
		tmp = t_1 + x;
	} else {
		tmp = x + (y * ((z - t) / (z - a)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original11.0
Target1.3
Herbie0.4
\[x + \frac{y}{\frac{z - a}{z - t}} \]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < -2.5690390576883054e262

    1. Initial program 57.4

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

    2. Simplified1.0

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

    3. Applied clear-num_binary641.1

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

    if -2.5690390576883054e262 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < 3.8508646732074662e236

    1. Initial program 0.2

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

    if 3.8508646732074662e236 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))

    1. Initial program 54.1

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

    2. Simplified1.8

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

    3. Applied fma-udef_binary641.8

      \[\leadsto \color{blue}{y \cdot \frac{z - t}{z - a} + x} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} \leq -2.5690390576883054 \cdot 10^{+262}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{\frac{z - a}{z - t}}, x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \leq 3.850864673207466 \cdot 10^{+236}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

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

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