Average Error: 31.2 → 17.5
Time: 16.3s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.953163933293596454341424469878526728026 \cdot 10^{119}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 2.93160876788335701324895973715263720284 \cdot 10^{138}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -8.953163933293596454341424469878526728026 \cdot 10^{119}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 2.93160876788335701324895973715263720284 \cdot 10^{138}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{else}:\\
\;\;\;\;re\\

\end{array}
double f(double re, double im) {
        double r44549 = re;
        double r44550 = r44549 * r44549;
        double r44551 = im;
        double r44552 = r44551 * r44551;
        double r44553 = r44550 + r44552;
        double r44554 = sqrt(r44553);
        return r44554;
}
↓
double f(double re, double im) {
        double r44555 = re;
        double r44556 = -8.953163933293596e+119;
        bool r44557 = r44555 <= r44556;
        double r44558 = -r44555;
        double r44559 = 2.931608767883357e+138;
        bool r44560 = r44555 <= r44559;
        double r44561 = r44555 * r44555;
        double r44562 = im;
        double r44563 = r44562 * r44562;
        double r44564 = r44561 + r44563;
        double r44565 = sqrt(r44564);
        double r44566 = r44560 ? r44565 : r44555;
        double r44567 = r44557 ? r44558 : r44566;
        return r44567;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if re < -8.953163933293596e+119

    1. Initial program 55.7

      \[\sqrt{re \cdot re + im \cdot im}\]

    2. Taylor expanded around -inf 9.5

      \[\leadsto \color{blue}{-1 \cdot re}\]

    3. Simplified9.5

      \[\leadsto \color{blue}{-re}\]

    if -8.953163933293596e+119 < re < 2.931608767883357e+138

    1. Initial program 20.9

      \[\sqrt{re \cdot re + im \cdot im}\]

    if 2.931608767883357e+138 < re

    1. Initial program 59.0

      \[\sqrt{re \cdot re + im \cdot im}\]

    2. Taylor expanded around inf 8.0

      \[\leadsto \color{blue}{re}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.953163933293596454341424469878526728026 \cdot 10^{119}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 2.93160876788335701324895973715263720284 \cdot 10^{138}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))