Complex division, real part

Average Error: 25.3 → 12.0

Time: 1.5m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -3.712028832851069 \cdot 10^{+161}:\\ \;\;\;\;\frac{-a}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;c \le 9.529062653614642 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{d^2 + c^2}^*}{(a \cdot c + \left(b \cdot d\right))_*}}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r10748057 = a;
        double r10748058 = c;
        double r10748059 = r10748057 * r10748058;
        double r10748060 = b;
        double r10748061 = d;
        double r10748062 = r10748060 * r10748061;
        double r10748063 = r10748059 + r10748062;
        double r10748064 = r10748058 * r10748058;
        double r10748065 = r10748061 * r10748061;
        double r10748066 = r10748064 + r10748065;
        double r10748067 = r10748063 / r10748066;
        return r10748067;
}

↓

double f(double a, double b, double c, double d) {
        double r10748068 = c;
        double r10748069 = -3.712028832851069e+161;
        bool r10748070 = r10748068 <= r10748069;
        double r10748071 = a;
        double r10748072 = -r10748071;
        double r10748073 = d;
        double r10748074 = hypot(r10748073, r10748068);
        double r10748075 = r10748072 / r10748074;
        double r10748076 = 9.529062653614642e+142;
        bool r10748077 = r10748068 <= r10748076;
        double r10748078 = 1.0;
        double r10748079 = b;
        double r10748080 = r10748079 * r10748073;
        double r10748081 = fma(r10748071, r10748068, r10748080);
        double r10748082 = r10748074 / r10748081;
        double r10748083 = r10748078 / r10748082;
        double r10748084 = r10748083 / r10748074;
        double r10748085 = r10748071 / r10748074;
        double r10748086 = r10748077 ? r10748084 : r10748085;
        double r10748087 = r10748070 ? r10748075 : r10748086;
        return r10748087;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.3
Target	0.5
Herbie	12.0

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

Derivation

1. Split input into 3 regimes

2. if c < -3.712028832851069e+161

   1. Initial program 44.3

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

   2. Simplified 44.3

      \[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 44.3

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

   5. Applied associate-/r* 44.3

      \[\leadsto \color{blue}{\frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 44.3

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

   8. Applied *-un-lft-identity 44.3

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

   9. Applied *-un-lft-identity 44.3

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

   10. Applied times-frac 44.3

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

   11. Applied times-frac 44.3

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

   12. Simplified 44.3

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

   13. Simplified 30.0

       \[\leadsto 1 \cdot \color{blue}{\frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]

   14. Taylor expanded around -inf 12.2

       \[\leadsto 1 \cdot \frac{\color{blue}{-1 \cdot a}}{\sqrt{d^2 + c^2}^*}\]

   15. Simplified 12.2

       \[\leadsto 1 \cdot \frac{\color{blue}{-a}}{\sqrt{d^2 + c^2}^*}\]

   if -3.712028832851069e+161 < c < 9.529062653614642e+142

   1. Initial program 18.8

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

   2. Simplified 18.8

      \[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 18.8

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

   5. Applied associate-/r* 18.7

      \[\leadsto \color{blue}{\frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 18.7

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

   8. Applied *-un-lft-identity 18.7

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

   9. Applied *-un-lft-identity 18.7

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

   10. Applied times-frac 18.7

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

   11. Applied times-frac 18.7

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

   12. Simplified 18.7

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

   13. Simplified 11.7

       \[\leadsto 1 \cdot \color{blue}{\frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
   14. Using strategy rm

   15. Applied clear-num 11.8

       \[\leadsto 1 \cdot \frac{\color{blue}{\frac{1}{\frac{\sqrt{d^2 + c^2}^*}{(a \cdot c + \left(b \cdot d\right))_*}}}}{\sqrt{d^2 + c^2}^*}\]

   if 9.529062653614642e+142 < c

   1. Initial program 43.3

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

   2. Simplified 43.3

      \[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 43.3

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

   5. Applied associate-/r* 43.3

      \[\leadsto \color{blue}{\frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 43.3

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

   8. Applied *-un-lft-identity 43.3

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

   9. Applied *-un-lft-identity 43.3

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

   10. Applied times-frac 43.3

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

   11. Applied times-frac 43.3

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

   12. Simplified 43.3

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

   13. Simplified 26.7

       \[\leadsto 1 \cdot \color{blue}{\frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]

   14. Taylor expanded around inf 12.8

       \[\leadsto 1 \cdot \frac{\color{blue}{a}}{\sqrt{d^2 + c^2}^*}\]
3. Recombined 3 regimes into one program.

4. Final simplification 12.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -3.712028832851069 \cdot 10^{+161}:\\ \;\;\;\;\frac{-a}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;c \le 9.529062653614642 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{d^2 + c^2}^*}{(a \cdot c + \left(b \cdot d\right))_*}}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (a b c d)
  :name "Complex division, real part"

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

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