Average Error: 25.5 → 1.8
Time: 37.7s
Precision: 64
Internal Precision: 128
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \cdot c \le -2.8997370585315772 \cdot 10^{+301}:\\ \;\;\;\;\frac{\frac{c}{\frac{\sqrt{d^2 + c^2}^*}{b}}}{\sqrt{d^2 + c^2}^*} - \frac{\frac{d \cdot a}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;b \cdot c \le 5.856336305743566 \cdot 10^{+295}:\\ \;\;\;\;\frac{\frac{b \cdot c}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*} - \frac{\frac{d}{\sqrt{d^2 + c^2}^*} \cdot a}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\sqrt{\sqrt{d^2 + c^2}^*}} \cdot \frac{b}{\sqrt{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*} - \frac{\frac{d \cdot a}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

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

Original25.5
Target0.6
Herbie1.8
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* b c) < -2.8997370585315772e+301

    1. Initial program 62.5

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

    2. Initial simplification62.5

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

    4. Applied add-sqr-sqrt62.5

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

    5. Applied *-un-lft-identity62.5

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

    6. Applied times-frac62.5

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

    7. Simplified62.5

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

    8. Simplified60.1

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

    10. Applied associate-*l/60.1

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

    11. Simplified60.1

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

    13. Applied div-sub60.1

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

    14. Applied div-sub60.1

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

    16. Applied associate-/l*6.1

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

    if -2.8997370585315772e+301 < (* b c) < 5.856336305743566e+295

    1. Initial program 19.7

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

    2. Initial simplification19.7

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

    4. Applied add-sqr-sqrt19.7

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

    5. Applied *-un-lft-identity19.7

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

    6. Applied times-frac19.7

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

    7. Simplified19.7

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

    8. Simplified9.4

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

    10. Applied associate-*l/9.3

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

    11. Simplified9.3

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

    13. Applied div-sub9.3

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

    14. Applied div-sub9.3

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

    16. Applied *-un-lft-identity9.3

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

    17. Applied times-frac1.0

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

    18. Simplified1.0

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

    if 5.856336305743566e+295 < (* b c)

    1. Initial program 60.7

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

    2. Initial simplification60.7

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

    4. Applied add-sqr-sqrt60.7

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

    5. Applied *-un-lft-identity60.7

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

    6. Applied times-frac60.7

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

    7. Simplified60.7

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

    8. Simplified58.7

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

    10. Applied associate-*l/58.7

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

    11. Simplified58.7

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

    13. Applied div-sub58.7

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

    14. Applied div-sub58.7

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

    16. Applied add-sqr-sqrt58.7

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

    17. Applied times-frac8.3

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot c \le -2.8997370585315772 \cdot 10^{+301}:\\ \;\;\;\;\frac{\frac{c}{\frac{\sqrt{d^2 + c^2}^*}{b}}}{\sqrt{d^2 + c^2}^*} - \frac{\frac{d \cdot a}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;b \cdot c \le 5.856336305743566 \cdot 10^{+295}:\\ \;\;\;\;\frac{\frac{b \cdot c}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*} - \frac{\frac{d}{\sqrt{d^2 + c^2}^*} \cdot a}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\sqrt{\sqrt{d^2 + c^2}^*}} \cdot \frac{b}{\sqrt{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*} - \frac{\frac{d \cdot a}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Runtime

Time bar (total: 37.7s)Debug logProfile

herbie shell --seed 2018274 +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))))