Average Error: 10.3 → 0.1
Time: 3.3s
Precision: binary64
\[\frac{x + y \cdot \left(z - x\right)}{z} \]
↓
\[\begin{array}{l} t_0 := \mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\ \mathbf{if}\;y \leq -1.2596917542803703 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.9501817946579704 \cdot 10^{+39}:\\ \;\;\;\;\left(y + \frac{x}{z}\right) - \frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\begin{array}{l}
t_0 := \mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\
\mathbf{if}\;y \leq -1.2596917542803703 \cdot 10^{+36}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;y \leq 1.9501817946579704 \cdot 10^{+39}:\\
\;\;\;\;\left(y + \frac{x}{z}\right) - \frac{y \cdot x}{z}\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

(FPCore (x y z) :precision binary64 (/ (+ x (* y (- z x))) z))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- y) (/ x z) y)))
   (if (<= y -1.2596917542803703e+36)
     t_0
     (if (<= y 1.9501817946579704e+39) (- (+ y (/ x z)) (/ (* y x) z)) t_0))))
double code(double x, double y, double z) {
	return (x + (y * (z - x))) / z;
}
↓
double code(double x, double y, double z) {
	double t_0 = fma(-y, (x / z), y);
	double tmp;
	if (y <= -1.2596917542803703e+36) {
		tmp = t_0;
	} else if (y <= 1.9501817946579704e+39) {
		tmp = (y + (x / z)) - ((y * x) / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.3
Target0.0
Herbie0.1
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}} \]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.2596917542803703e36 or 1.95018179465797035e39 < y

    1. Initial program 25.6

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

    2. Simplified25.6

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

    3. Taylor expanded in y around inf 25.6

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

    4. Simplified0.1

      \[\leadsto \color{blue}{y - \frac{x}{z} \cdot y} \]

    5. Applied egg-rr0.1

      \[\leadsto y - \color{blue}{\frac{y}{\frac{z}{x}}} \]

    6. Applied egg-rr0.1

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

    if -1.2596917542803703e36 < y < 1.95018179465797035e39

    1. Initial program 0.5

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

    2. Simplified0.5

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

    3. Taylor expanded in y around 0 0.2

      \[\leadsto \color{blue}{\left(y + \frac{x}{z}\right) - \frac{y \cdot x}{z}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2596917542803703 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\ \mathbf{elif}\;y \leq 1.9501817946579704 \cdot 10^{+39}:\\ \;\;\;\;\left(y + \frac{x}{z}\right) - \frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{x}{z}, y\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

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

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