Average Error: 9.8 → 0.1
Time: 2.3s
Precision: binary64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.65301397789678598 \cdot 10^{-37} \lor \neg \left(z \le 2.20000429689763433 \cdot 10^{-45}\right):\\ \;\;\;\;\left(\frac{x}{z} + y\right) - x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array}\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.65301397789678598 \cdot 10^{-37} \lor \neg \left(z \le 2.20000429689763433 \cdot 10^{-45}\right):\\
\;\;\;\;\left(\frac{x}{z} + y\right) - x \cdot \frac{y}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\

\end{array}
double code(double x, double y, double z) {
	return ((double) (((double) (x + ((double) (y * ((double) (z - x)))))) / z));
}
↓
double code(double x, double y, double z) {
	double VAR;
	if (((z <= -3.653013977896786e-37) || !(z <= 2.2000042968976343e-45))) {
		VAR = ((double) (((double) (((double) (x / z)) + y)) - ((double) (x * ((double) (y / z))))));
	} else {
		VAR = ((double) (((double) (x + ((double) (y * ((double) (z - x)))))) / z));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.8
Target0.0
Herbie0.1
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.65301397789678598e-37 or 2.20000429689763433e-45 < z

    1. Initial program 14.6

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

    2. Taylor expanded around 0 4.8

      \[\leadsto \color{blue}{\left(\frac{x}{z} + y\right) - \frac{x \cdot y}{z}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity4.8

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

    5. Applied times-frac0.2

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

    6. Simplified0.2

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

    if -3.65301397789678598e-37 < z < 2.20000429689763433e-45

    1. Initial program 0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.65301397789678598 \cdot 10^{-37} \lor \neg \left(z \le 2.20000429689763433 \cdot 10^{-45}\right):\\ \;\;\;\;\left(\frac{x}{z} + y\right) - x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(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))