Average Error: 12.4 → 2.6
Time: 2.2s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -6.90740619564049756 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{elif}\;z \le 1.62589878773930302 \cdot 10^{-80}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -6.90740619564049756 \cdot 10^{-60}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\

\mathbf{elif}\;z \le 1.62589878773930302 \cdot 10^{-80}:\\
\;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\frac{z}{y + z}}\\

\end{array}
double code(double x, double y, double z) {
	return ((x * (y + z)) / z);
}
↓
double code(double x, double y, double z) {
	double VAR;
	if ((z <= -6.907406195640498e-60)) {
		VAR = fma((y / z), x, x);
	} else {
		double VAR_1;
		if ((z <= 1.625898787739303e-80)) {
			VAR_1 = ((x * (y + z)) / z);
		} else {
			VAR_1 = (x / (z / (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

Original12.4
Target2.9
Herbie2.6
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -6.907406195640498e-60

    1. Initial program 14.6

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

    2. Simplified0.5

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

    if -6.907406195640498e-60 < z < 1.625898787739303e-80

    1. Initial program 7.7

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

    if 1.625898787739303e-80 < z

    1. Initial program 14.1

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

    3. Applied associate-/l*0.3

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

  4. Final simplification2.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.90740619564049756 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{elif}\;z \le 1.62589878773930302 \cdot 10^{-80}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

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

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