Average Error: 25.3 → 0.8
Time: 1.9s
Precision: 64
\[x \cdot \sqrt{y \cdot y - z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -4.17286658231481322 \cdot 10^{-249}:\\ \;\;\;\;x \cdot \left(-1 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
x \cdot \sqrt{y \cdot y - z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -4.17286658231481322 \cdot 10^{-249}:\\
\;\;\;\;x \cdot \left(-1 \cdot y\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r702960 = x;
        double r702961 = y;
        double r702962 = r702961 * r702961;
        double r702963 = z;
        double r702964 = r702963 * r702963;
        double r702965 = r702962 - r702964;
        double r702966 = sqrt(r702965);
        double r702967 = r702960 * r702966;
        return r702967;
}
↓
double f(double x, double y, double __attribute__((unused)) z) {
        double r702968 = y;
        double r702969 = -4.172866582314813e-249;
        bool r702970 = r702968 <= r702969;
        double r702971 = x;
        double r702972 = -1.0;
        double r702973 = r702972 * r702968;
        double r702974 = r702971 * r702973;
        double r702975 = r702971 * r702968;
        double r702976 = r702970 ? r702974 : r702975;
        return r702976;
}

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
Target0.6
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;y \lt 2.58160964882516951 \cdot 10^{-278}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -4.172866582314813e-249

    1. Initial program 24.8

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

    2. Taylor expanded around -inf 0.6

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

    if -4.172866582314813e-249 < y

    1. Initial program 25.8

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

    2. Taylor expanded around inf 1.1

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.17286658231481322 \cdot 10^{-249}:\\ \;\;\;\;x \cdot \left(-1 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(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)))))