Average Error: 31.6 → 18.3
Time: 1.1s
Precision: binary64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -9.051774626528414 \cdot 10^{+140}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -3.8634739304108896 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq -2.5654629043472667 \cdot 10^{-218}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 7.009893403905713 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -9.051774626528414 \cdot 10^{+140}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \leq -3.8634739304108896 \cdot 10^{-142}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \leq -2.5654629043472667 \cdot 10^{-218}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \leq 7.009893403905713 \cdot 10^{+36}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
(FPCore (re im) :precision binary64 (sqrt (+ (* re re) (* im im))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re -9.051774626528414e+140)
   (- re)
   (if (<= re -3.8634739304108896e-142)
     (sqrt (+ (* re re) (* im im)))
     (if (<= re -2.5654629043472667e-218)
       im
       (if (<= re 7.009893403905713e+36) (sqrt (+ (* re re) (* im im))) re)))))
double code(double re, double im) {
	return sqrt((re * re) + (im * im));
}
↓
double code(double re, double im) {
	double tmp;
	if (re <= -9.051774626528414e+140) {
		tmp = -re;
	} else if (re <= -3.8634739304108896e-142) {
		tmp = sqrt((re * re) + (im * im));
	} else if (re <= -2.5654629043472667e-218) {
		tmp = im;
	} else if (re <= 7.009893403905713e+36) {
		tmp = sqrt((re * re) + (im * im));
	} else {
		tmp = re;
	}
	return tmp;
}

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 4 regimes

  2. if re < -9.0517746265284142e140

    1. Initial program 61.0

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

    2. Taylor expanded around -inf 9.0

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

    3. Simplified9.0

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

    if -9.0517746265284142e140 < re < -3.8634739304108896e-142 or -2.5654629043472667e-218 < re < 7.009893403905713e36

    1. Initial program 20.4

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

    if -3.8634739304108896e-142 < re < -2.5654629043472667e-218

    1. Initial program 31.3

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

    2. Taylor expanded around 0 37.8

      \[\leadsto \color{blue}{im}\]

    if 7.009893403905713e36 < re

    1. Initial program 42.3

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

    2. Taylor expanded around inf 12.9

      \[\leadsto \color{blue}{re}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification18.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.051774626528414 \cdot 10^{+140}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -3.8634739304108896 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq -2.5654629043472667 \cdot 10^{-218}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 7.009893403905713 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020299 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))