Average Error: 8.4 → 0.1
Time: 10.4s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}\]
↓
\[\frac{x}{1.0 + x} \cdot \left(1.0 + \frac{x}{y}\right)\]
\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}
↓
\frac{x}{1.0 + x} \cdot \left(1.0 + \frac{x}{y}\right)
double f(double x, double y) {
        double r43400203 = x;
        double r43400204 = y;
        double r43400205 = r43400203 / r43400204;
        double r43400206 = 1.0;
        double r43400207 = r43400205 + r43400206;
        double r43400208 = r43400203 * r43400207;
        double r43400209 = r43400203 + r43400206;
        double r43400210 = r43400208 / r43400209;
        return r43400210;
}
↓
double f(double x, double y) {
        double r43400211 = x;
        double r43400212 = 1.0;
        double r43400213 = r43400212 + r43400211;
        double r43400214 = r43400211 / r43400213;
        double r43400215 = y;
        double r43400216 = r43400211 / r43400215;
        double r43400217 = r43400212 + r43400216;
        double r43400218 = r43400214 * r43400217;
        return r43400218;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original8.4
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1.0}{x + 1.0}\]

Derivation

  1. Initial program 8.4

    \[\frac{x \cdot \left(\frac{x}{y} + 1.0\right)}{x + 1.0}\]
  2. Using strategy rm

  3. Applied associate-/l*0.1

    \[\leadsto \color{blue}{\frac{x}{\frac{x + 1.0}{\frac{x}{y} + 1.0}}}\]
  4. Using strategy rm

  5. Applied associate-/r/0.1

    \[\leadsto \color{blue}{\frac{x}{x + 1.0} \cdot \left(\frac{x}{y} + 1.0\right)}\]

  6. Final simplification0.1

    \[\leadsto \frac{x}{1.0 + x} \cdot \left(1.0 + \frac{x}{y}\right)\]

Reproduce

herbie shell --seed 2019164 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"

  :herbie-target
  (* (/ x 1) (/ (+ (/ x y) 1.0) (+ x 1.0)))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))