Average Error: 25.3 → 0.9
Time: 1.8s
Precision: 64
\[x \cdot \sqrt{y \cdot y - z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -4.1007788905905145 \cdot 10^{-238}:\\ \;\;\;\;x \cdot \left(-1 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
x \cdot \sqrt{y \cdot y - z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -4.1007788905905145 \cdot 10^{-238}:\\
\;\;\;\;x \cdot \left(-1 \cdot y\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot y\\

\end{array}
double f(double x, double y, double z) {
        double r675557 = x;
        double r675558 = y;
        double r675559 = r675558 * r675558;
        double r675560 = z;
        double r675561 = r675560 * r675560;
        double r675562 = r675559 - r675561;
        double r675563 = sqrt(r675562);
        double r675564 = r675557 * r675563;
        return r675564;
}
↓
double f(double x, double y, double __attribute__((unused)) z) {
        double r675565 = y;
        double r675566 = -4.1007788905905145e-238;
        bool r675567 = r675565 <= r675566;
        double r675568 = x;
        double r675569 = -1.0;
        double r675570 = r675569 * r675565;
        double r675571 = r675568 * r675570;
        double r675572 = r675568 * r675565;
        double r675573 = r675567 ? r675571 : r675572;
        return r675573;
}

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

Original25.3
Target0.7
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y \lt 2.58160964882516951 \cdot 10^{-278}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -4.1007788905905145e-238

    1. Initial program 25.5

      \[x \cdot \sqrt{y \cdot y - z \cdot z}\]

    2. Taylor expanded around -inf 0.7

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

    if -4.1007788905905145e-238 < y

    1. Initial program 25.1

      \[x \cdot \sqrt{y \cdot y - z \cdot z}\]

    2. Taylor expanded around inf 1.1

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.1007788905905145 \cdot 10^{-238}:\\ \;\;\;\;x \cdot \left(-1 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, B"
  :precision binary64

  :herbie-target
  (if (< y 2.5816096488251695e-278) (- (* x y)) (* x (* (sqrt (+ y z)) (sqrt (- y z)))))

  (* x (sqrt (- (* y y) (* z z)))))