Average Error: 25.0 → 4.8
Time: 48.1s
Precision: 64
Internal Precision: 128
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -9538347154527392.0 \lor \neg \left(d \le 3.40271269432902 \cdot 10^{+126}\right):\\ \;\;\;\;\frac{\frac{c \cdot b}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*} - \frac{a}{\sqrt{d^2 + c^2}^*} \cdot \frac{d}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{\sqrt{\sqrt{d^2 + c^2}^*}} \cdot \frac{c}{\sqrt{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*} - \frac{\frac{a \cdot d}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original25.0
Target0.4
Herbie4.8
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if d < -9538347154527392.0 or 3.40271269432902e+126 < d

    1. Initial program 35.3

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

    2. Initial simplification35.3

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

    4. Applied add-sqr-sqrt35.3

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

    5. Applied *-un-lft-identity35.3

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

    6. Applied times-frac35.3

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

    7. Simplified35.3

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

    8. Simplified23.4

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

    10. Applied associate-*l/23.3

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

    11. Simplified23.3

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

    13. Applied div-sub23.3

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

    14. Applied div-sub23.3

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

    16. Applied add-sqr-sqrt23.4

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

    17. Applied add-sqr-sqrt23.5

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

    18. Applied times-frac8.3

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

    19. Applied times-frac8.3

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

    20. Simplified8.1

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

    21. Simplified7.9

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

    if -9538347154527392.0 < d < 3.40271269432902e+126

    1. Initial program 18.2

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

    2. Initial simplification18.2

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

    4. Applied add-sqr-sqrt18.2

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

    5. Applied *-un-lft-identity18.2

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

    6. Applied times-frac18.2

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

    7. Simplified18.2

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

    8. Simplified11.1

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

    10. Applied associate-*l/11.0

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

    11. Simplified11.0

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

    13. Applied div-sub11.0

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

    14. Applied div-sub11.0

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

    16. Applied add-sqr-sqrt11.2

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

    17. Applied times-frac2.7

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

  4. Final simplification4.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -9538347154527392.0 \lor \neg \left(d \le 3.40271269432902 \cdot 10^{+126}\right):\\ \;\;\;\;\frac{\frac{c \cdot b}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*} - \frac{a}{\sqrt{d^2 + c^2}^*} \cdot \frac{d}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{\sqrt{\sqrt{d^2 + c^2}^*}} \cdot \frac{c}{\sqrt{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*} - \frac{\frac{a \cdot d}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Runtime

Time bar (total: 48.1s)Debug logProfile

herbie shell --seed 2018296 +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))))