Average Error: 25.6 → 11.8
Time: 1.0m
Precision: 64
Internal Precision: 384
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;c \le -2.6822987150870495 \cdot 10^{+170}:\\ \;\;\;\;\frac{-b}{\sqrt{d^2 + c^2}^*}\\ \mathbf{if}\;c \le 7.88814037733874 \cdot 10^{+63}:\\ \;\;\;\;\frac{1}{\sqrt{d^2 + c^2}^*} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{a}{c}\right) \cdot \left(-d\right) + b)_*}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.6
Target0.4
Herbie11.8
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if c < -2.6822987150870495e+170

    1. Initial program 43.2

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

    2. Applied simplify43.2

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

    4. Applied add-sqr-sqrt43.2

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

    5. Applied *-un-lft-identity43.2

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

    6. Applied times-frac43.2

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

    7. Applied simplify43.2

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

    8. Applied simplify29.6

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

    9. Taylor expanded around -inf 12.1

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

    10. Applied simplify12.0

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

    if -2.6822987150870495e+170 < c < 7.88814037733874e+63

    1. Initial program 19.3

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

    2. Applied simplify19.3

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

    4. Applied add-sqr-sqrt19.3

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

    5. Applied *-un-lft-identity19.3

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

    6. Applied times-frac19.4

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

    7. Applied simplify19.3

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

    8. Applied simplify11.7

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

    if 7.88814037733874e+63 < c

    1. Initial program 35.5

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

    2. Applied simplify35.5

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

    4. Applied add-sqr-sqrt35.5

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

    5. Applied *-un-lft-identity35.5

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

    6. Applied times-frac35.5

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

    7. Applied simplify35.5

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

    8. Applied simplify24.2

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

    9. Taylor expanded around 0 29.4

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

    10. Applied simplify11.9

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' +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))))