Average Error: 25.9 → 13.1
Time: 1.2m
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;c \le -1.292761992654777 \cdot 10^{+76}:\\ \;\;\;\;\frac{-a}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;c \le 1.5418057255620893 \cdot 10^{+182}:\\ \;\;\;\;\frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.9
Target0.4
Herbie13.1
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if c < -1.292761992654777e+76

    1. Initial program 37.5

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

    2. Simplified37.5

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

    4. Applied add-sqr-sqrt37.5

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

    5. Applied associate-/r*37.5

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

    7. Applied *-un-lft-identity37.5

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

    8. Applied *-un-lft-identity37.5

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

    9. Applied sqrt-prod37.5

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

    10. Applied *-un-lft-identity37.5

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

    11. Applied times-frac37.5

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

    12. Applied times-frac37.5

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

    13. Simplified37.5

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

    14. Simplified25.3

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

    15. Taylor expanded around -inf 17.9

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

    16. Simplified17.9

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

    if -1.292761992654777e+76 < c < 1.5418057255620893e+182

    1. Initial program 19.7

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

    2. Simplified19.7

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

    4. Applied add-sqr-sqrt19.7

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

    5. Applied associate-/r*19.6

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

    7. Applied *-un-lft-identity19.6

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

    8. Applied *-un-lft-identity19.6

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

    9. Applied sqrt-prod19.6

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

    10. Applied *-un-lft-identity19.6

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

    11. Applied times-frac19.6

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

    12. Applied times-frac19.6

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

    13. Simplified19.6

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

    14. Simplified11.9

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

    if 1.5418057255620893e+182 < c

    1. Initial program 43.5

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

    2. Simplified43.5

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

    4. Applied add-sqr-sqrt43.5

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

    5. Applied associate-/r*43.5

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

    7. Applied *-un-lft-identity43.5

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

    8. Applied *-un-lft-identity43.5

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

    9. Applied sqrt-prod43.5

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

    10. Applied *-un-lft-identity43.5

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

    11. Applied times-frac43.5

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

    12. Applied times-frac43.5

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

    13. Simplified43.5

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

    14. Simplified30.5

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

    15. Taylor expanded around inf 11.5

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

  4. Final simplification13.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.292761992654777 \cdot 10^{+76}:\\ \;\;\;\;\frac{-a}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;c \le 1.5418057255620893 \cdot 10^{+182}:\\ \;\;\;\;\frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019100 +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))))