Average Error: 12.4 → 4.6
Time: 8.8s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\frac{x \cdot y}{z} + x\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\frac{x \cdot y}{z} + x
double f(double x, double y, double z) {
        double r380724 = x;
        double r380725 = y;
        double r380726 = z;
        double r380727 = r380725 + r380726;
        double r380728 = r380724 * r380727;
        double r380729 = r380728 / r380726;
        return r380729;
}
↓
double f(double x, double y, double z) {
        double r380730 = x;
        double r380731 = y;
        double r380732 = r380730 * r380731;
        double r380733 = z;
        double r380734 = r380732 / r380733;
        double r380735 = r380734 + r380730;
        return r380735;
}

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
Target2.8
Herbie4.6
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -2.0176262665767197e+157 or 1.0279121045400385e-44 < x

    1. Initial program 23.5

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

    2. Taylor expanded around 0 7.6

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

    4. Applied associate-/l*0.2

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

    6. Applied clear-num0.2

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{z}{y}}{x}}} + x\]
    7. Using strategy rm

    8. Applied div-inv0.3

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

    9. Applied add-sqr-sqrt0.3

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

    10. Applied times-frac0.3

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

    11. Simplified0.3

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

    12. Simplified0.3

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

    if -2.0176262665767197e+157 < x < 1.0279121045400385e-44

    1. Initial program 6.8

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

    2. Taylor expanded around 0 3.1

      \[\leadsto \color{blue}{\frac{x \cdot y}{z} + x}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.6

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

Reproduce

herbie shell --seed 2019291 
(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))