Average Error: 0.4 → 0.4
Time: 43.9s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{0.1111111111111111049432054187491303309798}{x}\right) - 1\right)\]
\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(y + \frac{0.1111111111111111049432054187491303309798}{x}\right) - 1\right)
double f(double x, double y) {
        double r374996 = 3.0;
        double r374997 = x;
        double r374998 = sqrt(r374997);
        double r374999 = r374996 * r374998;
        double r375000 = y;
        double r375001 = 1.0;
        double r375002 = 9.0;
        double r375003 = r374997 * r375002;
        double r375004 = r375001 / r375003;
        double r375005 = r375000 + r375004;
        double r375006 = r375005 - r375001;
        double r375007 = r374999 * r375006;
        return r375007;
}
↓
double f(double x, double y) {
        double r375008 = 3.0;
        double r375009 = x;
        double r375010 = sqrt(r375009);
        double r375011 = r375008 * r375010;
        double r375012 = y;
        double r375013 = 0.1111111111111111;
        double r375014 = r375013 / r375009;
        double r375015 = r375012 + r375014;
        double r375016 = 1.0;
        double r375017 = r375015 - r375016;
        double r375018 = r375011 * r375017;
        return r375018;
}

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

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (* 3 (+ (* y (sqrt x)) (* (- (/ 1 (* x 9)) 1) (sqrt x))))

  (* (* 3 (sqrt x)) (- (+ y (/ 1 (* x 9))) 1)))