Average Error: 12.7 → 3.1
Time: 2.6s
Precision: binary64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -2.1478886459578017 \cdot 10^{+264}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq -1.3406762049438543 \cdot 10^{+127}:\\ \;\;\;\;x + \left(x \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \frac{\sqrt[3]{y}}{z}\\ \mathbf{elif}\;y \leq 3.1872202114932396 \cdot 10^{+58} \lor \neg \left(y \leq 1.1972472554863538 \cdot 10^{+195}\right):\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot x\right) \cdot \frac{1}{z}\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -2.1478886459578017 \cdot 10^{+264}:\\
\;\;\;\;x + x \cdot \frac{y}{z}\\

\mathbf{elif}\;y \leq -1.3406762049438543 \cdot 10^{+127}:\\
\;\;\;\;x + \left(x \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \frac{\sqrt[3]{y}}{z}\\

\mathbf{elif}\;y \leq 3.1872202114932396 \cdot 10^{+58} \lor \neg \left(y \leq 1.1972472554863538 \cdot 10^{+195}\right):\\
\;\;\;\;x + x \cdot \frac{y}{z}\\

\mathbf{else}:\\
\;\;\;\;x + \left(y \cdot x\right) \cdot \frac{1}{z}\\

\end{array}
(FPCore (x y z) :precision binary64 (/ (* x (+ y z)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.1478886459578017e+264)
   (+ x (* x (/ y z)))
   (if (<= y -1.3406762049438543e+127)
     (+ x (* (* x (* (cbrt y) (cbrt y))) (/ (cbrt y) z)))
     (if (or (<= y 3.1872202114932396e+58)
             (not (<= y 1.1972472554863538e+195)))
       (+ x (* x (/ y z)))
       (+ x (* (* y x) (/ 1.0 z)))))))
double code(double x, double y, double z) {
	return (((double) (x * ((double) (y + z)))) / z);
}
↓
double code(double x, double y, double z) {
	double VAR;
	if ((y <= -2.1478886459578017e+264)) {
		VAR = ((double) (x + ((double) (x * (y / z)))));
	} else {
		double VAR_1;
		if ((y <= -1.3406762049438543e+127)) {
			VAR_1 = ((double) (x + ((double) (((double) (x * ((double) (((double) cbrt(y)) * ((double) cbrt(y)))))) * (((double) cbrt(y)) / z)))));
		} else {
			double VAR_2;
			if (((y <= 3.1872202114932396e+58) || !(y <= 1.1972472554863538e+195))) {
				VAR_2 = ((double) (x + ((double) (x * (y / z)))));
			} else {
				VAR_2 = ((double) (x + ((double) (((double) (y * x)) * (1.0 / z)))));
			}
			VAR_1 = VAR_2;
		}
		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

Original12.7
Target2.8
Herbie3.1
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -2.1478886459578017e264 or -1.3406762049438543e127 < y < 3.1872202114932396e58 or 1.19724725548635375e195 < y

    1. Initial program 12.8

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

    2. Simplified1.9

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

    if -2.1478886459578017e264 < y < -1.3406762049438543e127

    1. Initial program 11.7

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

    2. Simplified9.3

      \[\leadsto \color{blue}{x + x \cdot \frac{y}{z}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity9.3

      \[\leadsto x + x \cdot \frac{y}{\color{blue}{1 \cdot z}}\]

    5. Applied add-cube-cbrt9.9

      \[\leadsto x + x \cdot \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{1 \cdot z}\]

    6. Applied times-frac9.9

      \[\leadsto x + x \cdot \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \frac{\sqrt[3]{y}}{z}\right)}\]

    7. Applied associate-*r*8.4

      \[\leadsto x + \color{blue}{\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}\right) \cdot \frac{\sqrt[3]{y}}{z}}\]

    8. Simplified8.4

      \[\leadsto x + \color{blue}{\left(x \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)} \cdot \frac{\sqrt[3]{y}}{z}\]

    if 3.1872202114932396e58 < y < 1.19724725548635375e195

    1. Initial program 12.7

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

    2. Simplified7.3

      \[\leadsto \color{blue}{x + x \cdot \frac{y}{z}}\]
    3. Using strategy rm

    4. Applied div-inv7.3

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

    5. Applied associate-*r*7.7

      \[\leadsto x + \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{z}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1478886459578017 \cdot 10^{+264}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq -1.3406762049438543 \cdot 10^{+127}:\\ \;\;\;\;x + \left(x \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \frac{\sqrt[3]{y}}{z}\\ \mathbf{elif}\;y \leq 3.1872202114932396 \cdot 10^{+58} \lor \neg \left(y \leq 1.1972472554863538 \cdot 10^{+195}\right):\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot x\right) \cdot \frac{1}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(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))