Average Error: 12.6 → 0.8
Time: 4.5s
Precision: binary64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -\infty:\\ \;\;\;\;\frac{x}{\sqrt[3]{{\left(\frac{z}{y + z}\right)}^{3}}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq -5.336206498650438 \cdot 10^{+99}:\\ \;\;\;\;x + \frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 7.381013021528955 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 2.802615316481958 \cdot 10^{+267}:\\ \;\;\;\;x + \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -\infty:\\
\;\;\;\;\frac{x}{\sqrt[3]{{\left(\frac{z}{y + z}\right)}^{3}}}\\

\mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq -5.336206498650438 \cdot 10^{+99}:\\
\;\;\;\;x + \frac{x \cdot y}{z}\\

\mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 7.381013021528955 \cdot 10^{-16}:\\
\;\;\;\;x \cdot \frac{y + z}{z}\\

\mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 2.802615316481958 \cdot 10^{+267}:\\
\;\;\;\;x + \frac{x \cdot y}{z}\\

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

\end{array}
(FPCore (x y z) :precision binary64 (/ (* x (+ y z)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= (/ (* x (+ y z)) z) (- INFINITY))
   (/ x (cbrt (pow (/ z (+ y z)) 3.0)))
   (if (<= (/ (* x (+ y z)) z) -5.336206498650438e+99)
     (+ x (/ (* x y) z))
     (if (<= (/ (* x (+ y z)) z) 7.381013021528955e-16)
       (* x (/ (+ y z) z))
       (if (<= (/ (* x (+ y z)) z) 2.802615316481958e+267)
         (+ x (/ (* x y) z))
         (/ x (/ z (+ y z))))))))
double code(double x, double y, double z) {
	return (x * (y + z)) / z;
}
↓
double code(double x, double y, double z) {
	double tmp;
	if (((x * (y + z)) / z) <= -((double) INFINITY)) {
		tmp = x / cbrt(pow((z / (y + z)), 3.0));
	} else if (((x * (y + z)) / z) <= -5.336206498650438e+99) {
		tmp = x + ((x * y) / z);
	} else if (((x * (y + z)) / z) <= 7.381013021528955e-16) {
		tmp = x * ((y + z) / z);
	} else if (((x * (y + z)) / z) <= 2.802615316481958e+267) {
		tmp = x + ((x * y) / z);
	} else {
		tmp = x / (z / (y + z));
	}
	return tmp;
}

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

Original12.6
Target3.3
Herbie0.8
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 4 regimes

  2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -inf.0

    1. Initial program 64.0

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

    3. Applied associate-/l*_binary64_102530.0

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

    4. Simplified0.0

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

    6. Applied add-cbrt-cube_binary64_103441.7

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

    7. Simplified1.7

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

    if -inf.0 < (/.f64 (*.f64 x (+.f64 y z)) z) < -5.3362064986504381e99 or 7.3810130215289555e-16 < (/.f64 (*.f64 x (+.f64 y z)) z) < 2.8026153164819579e267

    1. Initial program 0.2

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

    3. Applied *-un-lft-identity_binary64_103080.2

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

    4. Applied times-frac_binary64_103148.7

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

    5. Simplified8.7

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

    6. Simplified8.7

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

    7. Taylor expanded around 0 0.2

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

    if -5.3362064986504381e99 < (/.f64 (*.f64 x (+.f64 y z)) z) < 7.3810130215289555e-16

    1. Initial program 5.8

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

    3. Applied *-un-lft-identity_binary64_103085.8

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

    4. Applied times-frac_binary64_103140.6

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

    5. Simplified0.6

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

    6. Simplified0.6

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

    if 2.8026153164819579e267 < (/.f64 (*.f64 x (+.f64 y z)) z)

    1. Initial program 49.8

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

    3. Applied associate-/l*_binary64_102533.3

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y + z}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -\infty:\\ \;\;\;\;\frac{x}{\sqrt[3]{{\left(\frac{z}{y + z}\right)}^{3}}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq -5.336206498650438 \cdot 10^{+99}:\\ \;\;\;\;x + \frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 7.381013021528955 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 2.802615316481958 \cdot 10^{+267}:\\ \;\;\;\;x + \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(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))