Average Error: 10.3 → 0.4
Time: 2.8s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.41058353208740873 \cdot 10^{-42}:\\ \;\;\;\;\left(\frac{x}{z} + y\right) - \frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \le 1.09112860820187419 \cdot 10^{-112}:\\ \;\;\;\;\left(x + y \cdot \left(z - x\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} + y\right) - x \cdot \frac{y}{z}\\ \end{array}\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.41058353208740873 \cdot 10^{-42}:\\
\;\;\;\;\left(\frac{x}{z} + y\right) - \frac{x}{\frac{z}{y}}\\

\mathbf{elif}\;z \le 1.09112860820187419 \cdot 10^{-112}:\\
\;\;\;\;\left(x + y \cdot \left(z - x\right)\right) \cdot \frac{1}{z}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{x}{z} + y\right) - x \cdot \frac{y}{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 <= -1.4105835320874087e-42)) {
		VAR = ((double) (((double) (((double) (x / z)) + y)) - ((double) (x / ((double) (z / y))))));
	} else {
		double VAR_1;
		if ((z <= 1.0911286082018742e-112)) {
			VAR_1 = ((double) (((double) (x + ((double) (y * ((double) (z - x)))))) * ((double) (1.0 / z))));
		} else {
			VAR_1 = ((double) (((double) (((double) (x / z)) + y)) - ((double) (x * ((double) (y / z))))));
		}
		VAR = VAR_1;
	}
	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

Original10.3
Target0.0
Herbie0.4
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.4105835320874087e-42

    1. Initial program 14.8

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

    2. Taylor expanded around 0 5.0

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

    4. Applied associate-/l*0.1

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

    if -1.4105835320874087e-42 < z < 1.0911286082018742e-112

    1. Initial program 0.1

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

    3. Applied div-inv0.4

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

    if 1.0911286082018742e-112 < z

    1. Initial program 13.2

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

    2. Taylor expanded around 0 4.6

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

    4. Applied *-un-lft-identity4.6

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

    5. Applied times-frac0.6

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

    6. Simplified0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.41058353208740873 \cdot 10^{-42}:\\ \;\;\;\;\left(\frac{x}{z} + y\right) - \frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \le 1.09112860820187419 \cdot 10^{-112}:\\ \;\;\;\;\left(x + y \cdot \left(z - x\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} + y\right) - x \cdot \frac{y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 
(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))