Average Error: 0.4 → 0.4
Time: 17.4s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(\frac{0.1111111111111111049432054187491303309798}{x} + y\right) - 1\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(\frac{0.1111111111111111049432054187491303309798}{x} + y\right) - 1\right)
double f(double x, double y) {
        double r24434189 = 3.0;
        double r24434190 = x;
        double r24434191 = sqrt(r24434190);
        double r24434192 = r24434189 * r24434191;
        double r24434193 = y;
        double r24434194 = 1.0;
        double r24434195 = 9.0;
        double r24434196 = r24434190 * r24434195;
        double r24434197 = r24434194 / r24434196;
        double r24434198 = r24434193 + r24434197;
        double r24434199 = r24434198 - r24434194;
        double r24434200 = r24434192 * r24434199;
        return r24434200;
}
↓
double f(double x, double y) {
        double r24434201 = 3.0;
        double r24434202 = x;
        double r24434203 = sqrt(r24434202);
        double r24434204 = r24434201 * r24434203;
        double r24434205 = 0.1111111111111111;
        double r24434206 = r24434205 / r24434202;
        double r24434207 = y;
        double r24434208 = r24434206 + r24434207;
        double r24434209 = 1.0;
        double r24434210 = r24434208 - r24434209;
        double r24434211 = r24434204 * r24434210;
        return r24434211;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.4
Target0.4
Herbie0.4
\[3 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \sqrt{x}\right)\]

Derivation

  1. Initial program 0.4

    \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]

  2. Taylor expanded around 0 0.4

    \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \color{blue}{\frac{0.1111111111111111049432054187491303309798}{x}}\right) - 1\right)\]

  3. Final simplification0.4

    \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(\frac{0.1111111111111111049432054187491303309798}{x} + y\right) - 1\right)\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (* 3.0 (+ (* y (sqrt x)) (* (- (/ 1.0 (* x 9.0)) 1.0) (sqrt x))))

  (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))