Average Error: 25.4 → 24.1
Time: 28.3s
Precision: 64
Internal Precision: 128
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -2.847478627180154 \cdot 10^{-261}:\\ \;\;\;\;(\left(\frac{b}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\right) \cdot \left(\frac{c}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\right) + \left(\frac{-d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{a}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{(b \cdot c + \left(\left(-d\right) \cdot a\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.4
Target0.4
Herbie24.1
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if d < -2.847478627180154e-261

    1. Initial program 25.6

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

    2. Simplified25.6

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

    4. Applied div-sub25.6

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

    6. Applied add-sqr-sqrt25.6

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

    7. Applied times-frac23.8

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

    8. Applied add-sqr-sqrt23.8

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

    9. Applied times-frac22.8

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

    10. Applied prod-diff22.8

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

    11. Simplified22.8

      \[\leadsto (\left(\frac{b}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\right) \cdot \left(\frac{c}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\right) + \left(-\frac{d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{a}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\right))_* + \color{blue}{0}\]

    if -2.847478627180154e-261 < d

    1. Initial program 25.3

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

    2. Simplified25.3

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

    4. Applied add-sqr-sqrt25.3

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

    5. Applied associate-/r*25.2

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

    7. Applied fma-neg25.2

      \[\leadsto \frac{\frac{\color{blue}{(b \cdot c + \left(-a \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification24.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -2.847478627180154 \cdot 10^{-261}:\\ \;\;\;\;(\left(\frac{b}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\right) \cdot \left(\frac{c}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\right) + \left(\frac{-d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{a}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{(b \cdot c + \left(\left(-d\right) \cdot a\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019068 +o rules:numerics
(FPCore (a b c d)
  :name "Complex division, imag part"

  :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))))