Average Error: 25.9 → 24.2
Time: 22.4s
Precision: 64
Internal Precision: 576
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;c \le 1.371158448840453 \cdot 10^{+123}:\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d} - a \cdot \frac{d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{c}{\sqrt{c \cdot c + d \cdot d}} - \frac{a \cdot d}{c \cdot c + d \cdot d}\\ \end{array}\]

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

Original25.9
Target0.4
Herbie24.2
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if c < 1.371158448840453e+123

    1. Initial program 22.7

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

    2. Initial simplification22.7

      \[\leadsto \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
    3. Using strategy rm

    4. Applied div-sub22.7

      \[\leadsto \color{blue}{\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a \cdot d}{c \cdot c + d \cdot d}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity22.7

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

    7. Applied times-frac21.1

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

    8. Simplified21.1

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

    if 1.371158448840453e+123 < c

    1. Initial program 42.5

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

    2. Initial simplification42.5

      \[\leadsto \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
    3. Using strategy rm

    4. Applied div-sub42.5

      \[\leadsto \color{blue}{\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a \cdot d}{c \cdot c + d \cdot d}}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt42.5

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

    7. Applied times-frac40.3

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

  4. Final simplification24.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \le 1.371158448840453 \cdot 10^{+123}:\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d} - a \cdot \frac{d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{c}{\sqrt{c \cdot c + d \cdot d}} - \frac{a \cdot d}{c \cdot c + d \cdot d}\\ \end{array}\]

Runtime

Time bar (total: 22.4s)Debug logProfile

herbie shell --seed 2018248 
(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))))