Average Error: 25.8 → 12.6
Time: 29.5s
Precision: 64
Internal Precision: 128
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;c \le -2.519475376326208 \cdot 10^{+165}:\\ \;\;\;\;\frac{-a}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;c \le 1.1071365722430642 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{(a \cdot c + \left(d \cdot b\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.8
Target0.6
Herbie12.6
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if c < -2.519475376326208e+165

    1. Initial program 42.9

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

    2. Simplified42.9

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

    4. Applied add-sqr-sqrt42.9

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

    5. Applied associate-/r*42.9

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

    7. Applied fma-udef42.9

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

    8. Applied hypot-def42.9

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

    10. Applied fma-udef42.9

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

    11. Applied hypot-def27.3

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

    12. Taylor expanded around -inf 12.2

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

    13. Simplified12.2

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

    if -2.519475376326208e+165 < c < 1.1071365722430642e+161

    1. Initial program 20.0

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

    2. Simplified20.0

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

    4. Applied add-sqr-sqrt20.1

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

    5. Applied associate-/r*20.0

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

    7. Applied fma-udef20.0

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

    8. Applied hypot-def20.0

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

    10. Applied fma-udef20.0

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

    11. Applied hypot-def12.6

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

    if 1.1071365722430642e+161 < c

    1. Initial program 44.0

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

    2. Simplified44.0

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

    4. Applied add-sqr-sqrt44.0

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

    5. Applied associate-/r*44.0

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

    7. Applied fma-udef44.0

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

    8. Applied hypot-def44.0

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

    9. Taylor expanded around inf 13.2

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

  4. Final simplification12.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -2.519475376326208 \cdot 10^{+165}:\\ \;\;\;\;\frac{-a}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;c \le 1.1071365722430642 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{(a \cdot c + \left(d \cdot b\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019091 +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))))