Complex division, imag part

Average Error: 26.3 → 13.1

Time: 5.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -1.92809702332723783 \cdot 10^{94}:\\ \;\;\;\;\frac{-1 \cdot b}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \mathbf{elif}\;c \le 3.85849841677322835 \cdot 10^{39}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \mathbf{elif}\;c \le 4.0740961827290516 \cdot 10^{154}:\\ \;\;\;\;\frac{b}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{c}} - \frac{a}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{d}}\\ \mathbf{elif}\;c \le 5.21434504503365728 \cdot 10^{171}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r105179 = b;
        double r105180 = c;
        double r105181 = r105179 * r105180;
        double r105182 = a;
        double r105183 = d;
        double r105184 = r105182 * r105183;
        double r105185 = r105181 - r105184;
        double r105186 = r105180 * r105180;
        double r105187 = r105183 * r105183;
        double r105188 = r105186 + r105187;
        double r105189 = r105185 / r105188;
        return r105189;
}

↓

double f(double a, double b, double c, double d) {
        double r105190 = c;
        double r105191 = -1.928097023327238e+94;
        bool r105192 = r105190 <= r105191;
        double r105193 = -1.0;
        double r105194 = b;
        double r105195 = r105193 * r105194;
        double r105196 = d;
        double r105197 = hypot(r105190, r105196);
        double r105198 = 1.0;
        double r105199 = r105197 * r105198;
        double r105200 = r105195 / r105199;
        double r105201 = 3.8584984167732284e+39;
        bool r105202 = r105190 <= r105201;
        double r105203 = r105194 * r105190;
        double r105204 = a;
        double r105205 = r105204 * r105196;
        double r105206 = r105203 - r105205;
        double r105207 = r105206 / r105197;
        double r105208 = r105207 / r105199;
        double r105209 = 4.0740961827290516e+154;
        bool r105210 = r105190 <= r105209;
        double r105211 = r105196 * r105196;
        double r105212 = fma(r105190, r105190, r105211);
        double r105213 = r105212 / r105190;
        double r105214 = r105194 / r105213;
        double r105215 = r105212 / r105196;
        double r105216 = r105204 / r105215;
        double r105217 = r105214 - r105216;
        double r105218 = 5.214345045033657e+171;
        bool r105219 = r105190 <= r105218;
        double r105220 = r105194 / r105199;
        double r105221 = r105219 ? r105208 : r105220;
        double r105222 = r105210 ? r105217 : r105221;
        double r105223 = r105202 ? r105208 : r105222;
        double r105224 = r105192 ? r105200 : r105223;
        return r105224;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.3
Target	0.4
Herbie	13.1

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

Derivation

1. Split input into 4 regimes

2. if c < -1.928097023327238e+94

   1. Initial program 38.9

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

   3. Applied add-sqr-sqrt 38.9

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

   4. Applied *-un-lft-identity 38.9

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

   5. Applied times-frac 38.9

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

   6. Simplified 38.9

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

   7. Simplified 27.2

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

   9. Applied associate-*l/ 27.2

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

   10. Simplified 27.2

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

   11. Taylor expanded around -inf 17.1

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

   if -1.928097023327238e+94 < c < 3.8584984167732284e+39 or 4.0740961827290516e+154 < c < 5.214345045033657e+171

   1. Initial program 19.2

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

   3. Applied add-sqr-sqrt 19.2

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

   4. Applied *-un-lft-identity 19.2

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

   5. Applied times-frac 19.3

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

   6. Simplified 19.3

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

   7. Simplified 11.7

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

   9. Applied associate-*l/ 11.5

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

   10. Simplified 11.5

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

   if 3.8584984167732284e+39 < c < 4.0740961827290516e+154

   1. Initial program 22.2

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

   3. Applied div-sub 22.2

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

   4. Simplified 14.3

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

   5. Simplified 14.5

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

   if 5.214345045033657e+171 < c

   1. Initial program 45.5

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

   3. Applied add-sqr-sqrt 45.5

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

   4. Applied *-un-lft-identity 45.5

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

   5. Applied times-frac 45.5

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

   6. Simplified 45.5

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

   7. Simplified 31.4

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

   9. Applied associate-*l/ 31.4

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

   10. Simplified 31.4

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

   11. Taylor expanded around inf 13.6

       \[\leadsto \frac{\color{blue}{b}}{\mathsf{hypot}\left(c, d\right) \cdot 1}\]
3. Recombined 4 regimes into one program.

4. Final simplification 13.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.92809702332723783 \cdot 10^{94}:\\ \;\;\;\;\frac{-1 \cdot b}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \mathbf{elif}\;c \le 3.85849841677322835 \cdot 10^{39}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \mathbf{elif}\;c \le 4.0740961827290516 \cdot 10^{154}:\\ \;\;\;\;\frac{b}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{c}} - \frac{a}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{d}}\\ \mathbf{elif}\;c \le 5.21434504503365728 \cdot 10^{171}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +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))))