Average Error: 31.2 → 17.5
Time: 8.2s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.953163933293596454341424469878526728026 \cdot 10^{119}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 2.93160876788335701324895973715263720284 \cdot 10^{138}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -8.953163933293596454341424469878526728026 \cdot 10^{119}:\\
\;\;\;\;-1 \cdot 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 r43151 = re;
        double r43152 = r43151 * r43151;
        double r43153 = im;
        double r43154 = r43153 * r43153;
        double r43155 = r43152 + r43154;
        double r43156 = sqrt(r43155);
        return r43156;
}
↓
double f(double re, double im) {
        double r43157 = re;
        double r43158 = -8.953163933293596e+119;
        bool r43159 = r43157 <= r43158;
        double r43160 = -1.0;
        double r43161 = r43160 * r43157;
        double r43162 = 2.931608767883357e+138;
        bool r43163 = r43157 <= r43162;
        double r43164 = r43157 * r43157;
        double r43165 = im;
        double r43166 = r43165 * r43165;
        double r43167 = r43164 + r43166;
        double r43168 = sqrt(r43167);
        double r43169 = r43163 ? r43168 : r43157;
        double r43170 = r43159 ? r43161 : r43169;
        return r43170;
}

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

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.953163933293596454341424469878526728026 \cdot 10^{119}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 2.93160876788335701324895973715263720284 \cdot 10^{138}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

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