Average Error: 25.5 → 4.9
Time: 36.3s
Precision: 64
Internal Precision: 128
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -7.337353623269386 \cdot 10^{+26} \lor \neg \left(a \le 1.9108902209969745 \cdot 10^{+25}\right):\\ \;\;\;\;\frac{\frac{c \cdot b}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*} - \frac{\frac{a}{\frac{\sqrt{d^2 + c^2}^*}{d}}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{\sqrt{\sqrt{d^2 + c^2}^*}} \cdot \frac{c}{\sqrt{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*} - \frac{\frac{a \cdot d}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

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

Original25.5
Target0.5
Herbie4.9
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -7.337353623269386e+26 or 1.9108902209969745e+25 < a

    1. Initial program 32.1

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

    2. Initial simplification32.1

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

    4. Applied add-sqr-sqrt32.1

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

    5. Applied *-un-lft-identity32.1

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

    6. Applied times-frac32.1

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

    7. Simplified32.1

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

    8. Simplified23.8

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

    10. Applied associate-*l/23.7

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

    11. Simplified23.7

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

    13. Applied div-sub23.7

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

    14. Applied div-sub23.7

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

    16. Applied associate-/l*9.0

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

    if -7.337353623269386e+26 < a < 1.9108902209969745e+25

    1. Initial program 20.2

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

    2. Initial simplification20.2

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

    4. Applied add-sqr-sqrt20.2

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

    5. Applied *-un-lft-identity20.2

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

    6. Applied times-frac20.2

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

    7. Simplified20.2

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

    8. Simplified10.2

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

    10. Applied associate-*l/10.0

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

    11. Simplified10.0

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

    13. Applied div-sub10.0

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

    14. Applied div-sub10.0

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

    16. Applied add-sqr-sqrt10.2

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

    17. Applied times-frac1.6

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

  4. Final simplification4.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -7.337353623269386 \cdot 10^{+26} \lor \neg \left(a \le 1.9108902209969745 \cdot 10^{+25}\right):\\ \;\;\;\;\frac{\frac{c \cdot b}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*} - \frac{\frac{a}{\frac{\sqrt{d^2 + c^2}^*}{d}}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{\sqrt{\sqrt{d^2 + c^2}^*}} \cdot \frac{c}{\sqrt{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*} - \frac{\frac{a \cdot d}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Runtime

Time bar (total: 36.3s)Debug logProfile

herbie shell --seed 2018349 +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))))