Complex division, imag part

Average Error: 26.5 → 15.6

Time: 15.2s

Precision: 64

Math

\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]

↓

\[\begin{array}{l} \mathbf{if}\;c \le 6.76697629171698933790750666516049735136 \cdot 10^{58}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(c, d\right)}{\frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(c, d\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r101212 = b;
        double r101213 = c;
        double r101214 = r101212 * r101213;
        double r101215 = a;
        double r101216 = d;
        double r101217 = r101215 * r101216;
        double r101218 = r101214 - r101217;
        double r101219 = r101213 * r101213;
        double r101220 = r101216 * r101216;
        double r101221 = r101219 + r101220;
        double r101222 = r101218 / r101221;
        return r101222;
}

↓

double f(double a, double b, double c, double d) {
        double r101223 = c;
        double r101224 = 6.766976291716989e+58;
        bool r101225 = r101223 <= r101224;
        double r101226 = 1.0;
        double r101227 = d;
        double r101228 = hypot(r101223, r101227);
        double r101229 = b;
        double r101230 = a;
        double r101231 = r101227 * r101230;
        double r101232 = -r101231;
        double r101233 = fma(r101229, r101223, r101232);
        double r101234 = r101233 / r101228;
        double r101235 = r101228 / r101234;
        double r101236 = r101226 / r101235;
        double r101237 = r101229 / r101228;
        double r101238 = r101225 ? r101236 : r101237;
        return r101238;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.5
Target	0.4
Herbie	15.6

\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if c < 6.766976291716989e+58

   1. Initial program 23.7

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 23.7

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

   4. Applied *-un-lft-identity 23.7

      \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]

   5. Applied times-frac 23.7

      \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]

   6. Simplified 23.7

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]

   7. Simplified 14.9

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(c, d\right)}}\]
   8. Using strategy rm

   9. Applied associate-*r/ 14.8

      \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(c, d\right)}}\]

   10. Simplified 14.7

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(c, d\right)}}}{\mathsf{hypot}\left(c, d\right)}\]
   11. Using strategy rm

   12. Applied clear-num 15.2

       \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(c, d\right)}{\frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(c, d\right)}}}}\]

   if 6.766976291716989e+58 < c

   1. Initial program 36.6

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 36.6

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

   4. Applied *-un-lft-identity 36.6

      \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]

   5. Applied times-frac 36.6

      \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]

   6. Simplified 36.6

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]

   7. Simplified 25.1

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(c, d\right)}}\]
   8. Using strategy rm

   9. Applied associate-*r/ 25.1

      \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(c, d\right)}}\]

   10. Simplified 25.0

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(c, d\right)}}}{\mathsf{hypot}\left(c, d\right)}\]

   11. Taylor expanded around inf 17.0

       \[\leadsto \frac{\color{blue}{b}}{\mathsf{hypot}\left(c, d\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 15.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le 6.76697629171698933790750666516049735136 \cdot 10^{58}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(c, d\right)}{\frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(c, d\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d)))))

  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))