Average Error: 25.3 → 12.7
Time: 33.0s
Precision: 64
Internal Precision: 128
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;c \le -3.262760289755574 \cdot 10^{+177}:\\ \;\;\;\;\frac{-b}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;c \le 3.693264705983811 \cdot 10^{+75}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{d^2 + c^2}^*}{b \cdot c - a \cdot d}}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original25.3
Target0.6
Herbie12.7
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if c < -3.262760289755574e+177

    1. Initial program 43.9

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

    2. Initial simplification43.9

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

    4. Applied add-sqr-sqrt43.9

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

    5. Applied *-un-lft-identity43.9

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

    6. Applied times-frac43.9

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

    7. Simplified43.9

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

    8. Simplified29.1

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

    10. Applied associate-*l/29.1

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

    11. Simplified29.1

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

    12. Taylor expanded around -inf 29.1

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

    13. Taylor expanded around -inf 11.2

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

    14. Simplified11.2

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

    if -3.262760289755574e+177 < c < 3.693264705983811e+75

    1. Initial program 19.1

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

    2. Initial simplification19.1

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

    4. Applied add-sqr-sqrt19.1

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

    5. Applied *-un-lft-identity19.1

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

    6. Applied times-frac19.2

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

    7. Simplified19.1

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

    8. Simplified11.4

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

    10. Applied associate-*l/11.3

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

    11. Simplified11.3

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

    12. Taylor expanded around -inf 11.3

      \[\leadsto \frac{\frac{\color{blue}{b \cdot c - a \cdot d}}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\]
    13. Using strategy rm

    14. Applied *-un-lft-identity11.3

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

    15. Applied associate-/l*11.4

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

    if 3.693264705983811e+75 < c

    1. Initial program 37.2

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

    2. Initial simplification37.2

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

    4. Applied add-sqr-sqrt37.2

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

    5. Applied *-un-lft-identity37.2

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

    6. Applied times-frac37.2

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

    7. Simplified37.2

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

    8. Simplified25.8

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

    10. Applied associate-*l/25.7

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

    11. Simplified25.7

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

    12. Taylor expanded around inf 18.1

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

  4. Final simplification12.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -3.262760289755574 \cdot 10^{+177}:\\ \;\;\;\;\frac{-b}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;c \le 3.693264705983811 \cdot 10^{+75}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{d^2 + c^2}^*}{b \cdot c - a \cdot d}}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Runtime

Time bar (total: 33.0s)Debug logProfile

herbie shell --seed 2018277 +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))))