Average Error: 12.3 → 3.1
Time: 3.0s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\frac{x}{\frac{z}{y + z}}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\frac{x}{\frac{z}{y + z}}
double f(double x, double y, double z) {
        double r277937 = x;
        double r277938 = y;
        double r277939 = z;
        double r277940 = r277938 + r277939;
        double r277941 = r277937 * r277940;
        double r277942 = r277941 / r277939;
        return r277942;
}
↓
double f(double x, double y, double z) {
        double r277943 = x;
        double r277944 = z;
        double r277945 = y;
        double r277946 = r277945 + r277944;
        double r277947 = r277944 / r277946;
        double r277948 = r277943 / r277947;
        return r277948;
}

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

Original12.3
Target3.1
Herbie3.1
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 3 regimes

  2. if z < 1.277387360816594e-224

    1. Initial program 12.3

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

    3. Applied associate-/l*3.9

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

    if 1.277387360816594e-224 < z < 9.073765212035872e-20

    1. Initial program 4.8

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

    2. Taylor expanded around 0 1.5

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

    if 9.073765212035872e-20 < z

    1. Initial program 16.0

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

    3. Applied *-un-lft-identity16.0

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

    4. Applied times-frac0.1

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

    5. Simplified0.1

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

  4. Final simplification3.1

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

Reproduce

herbie shell --seed 1978988140 
(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))