Complex division, imag part

Average Error: 25.9 → 12.6

Time: 14.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -5.089145779830487536688267543405577993059 \cdot 10^{131}:\\ \;\;\;\;\frac{-b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le 1.608222828046577445825524961558493147213 \cdot 10^{194}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r67426 = b;
        double r67427 = c;
        double r67428 = r67426 * r67427;
        double r67429 = a;
        double r67430 = d;
        double r67431 = r67429 * r67430;
        double r67432 = r67428 - r67431;
        double r67433 = r67427 * r67427;
        double r67434 = r67430 * r67430;
        double r67435 = r67433 + r67434;
        double r67436 = r67432 / r67435;
        return r67436;
}

↓

double f(double a, double b, double c, double d) {
        double r67437 = c;
        double r67438 = -5.0891457798304875e+131;
        bool r67439 = r67437 <= r67438;
        double r67440 = b;
        double r67441 = -r67440;
        double r67442 = d;
        double r67443 = hypot(r67437, r67442);
        double r67444 = r67441 / r67443;
        double r67445 = 1.6082228280465774e+194;
        bool r67446 = r67437 <= r67445;
        double r67447 = a;
        double r67448 = r67442 * r67447;
        double r67449 = -r67448;
        double r67450 = fma(r67440, r67437, r67449);
        double r67451 = r67450 / r67443;
        double r67452 = r67451 / r67443;
        double r67453 = r67440 / r67443;
        double r67454 = r67446 ? r67452 : r67453;
        double r67455 = r67439 ? r67444 : r67454;
        return r67455;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.9
Target	0.4
Herbie	12.6

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

Derivation

1. Split input into 3 regimes

2. if c < -5.0891457798304875e+131

   1. Initial program 42.5

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

   3. Applied add-sqr-sqrt 42.5

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

   4. Applied *-un-lft-identity 42.5

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

   5. Applied times-frac 42.5

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

   6. Simplified 42.5

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

   7. Simplified 27.7

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

   9. Applied associate-*r/ 27.7

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

   10. Simplified 27.6

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

   11. Taylor expanded around -inf 13.4

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

   12. Simplified 13.4

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

   if -5.0891457798304875e+131 < c < 1.6082228280465774e+194

   1. Initial program 20.4

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

   3. Applied add-sqr-sqrt 20.4

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

   4. Applied *-un-lft-identity 20.4

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

   5. Applied times-frac 20.4

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

   6. Simplified 20.4

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

   7. Simplified 12.8

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

   9. Applied associate-*r/ 12.8

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

   10. Simplified 12.7

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

   if 1.6082228280465774e+194 < c

   1. Initial program 42.7

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

   3. Applied add-sqr-sqrt 42.7

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

   4. Applied *-un-lft-identity 42.7

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

   5. Applied times-frac 42.7

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

   6. Simplified 42.7

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

   7. Simplified 31.0

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

   9. Applied associate-*r/ 31.0

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

   10. Simplified 30.9

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

   11. Taylor expanded around inf 10.7

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

4. Final simplification 12.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -5.089145779830487536688267543405577993059 \cdot 10^{131}:\\ \;\;\;\;\frac{-b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le 1.608222828046577445825524961558493147213 \cdot 10^{194}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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))))