Average Error: 9.0 → 0.1
Time: 11.7s
Precision: 64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[x \cdot \frac{1 + \frac{x}{y}}{1 + x}\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
x \cdot \frac{1 + \frac{x}{y}}{1 + x}
double f(double x, double y) {
        double r38845952 = x;
        double r38845953 = y;
        double r38845954 = r38845952 / r38845953;
        double r38845955 = 1.0;
        double r38845956 = r38845954 + r38845955;
        double r38845957 = r38845952 * r38845956;
        double r38845958 = r38845952 + r38845955;
        double r38845959 = r38845957 / r38845958;
        return r38845959;
}
↓
double f(double x, double y) {
        double r38845960 = x;
        double r38845961 = 1.0;
        double r38845962 = y;
        double r38845963 = r38845960 / r38845962;
        double r38845964 = r38845961 + r38845963;
        double r38845965 = r38845961 + r38845960;
        double r38845966 = r38845964 / r38845965;
        double r38845967 = r38845960 * r38845966;
        return r38845967;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.0
Target0.1
Herbie0.1
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Initial program 9.0

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity9.0

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

  4. Applied times-frac0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"

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

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