Average Error: 25.6 → 15.0
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;c \le -5.03571375883534 \cdot 10^{+131} \lor \neg \left(c \le 1.5009880531967683 \cdot 10^{+116}\right):\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

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

Original25.6
Target0.5
Herbie15.0
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if c < -5.03571375883534e+131 or 1.5009880531967683e+116 < c

    1. Initial program 41.4

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

    3. Applied add-cube-cbrt41.5

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\left(\sqrt[3]{c \cdot c + d \cdot d} \cdot \sqrt[3]{c \cdot c + d \cdot d}\right) \cdot \sqrt[3]{c \cdot c + d \cdot d}}}\]

    4. Applied *-un-lft-identity41.5

      \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\left(\sqrt[3]{c \cdot c + d \cdot d} \cdot \sqrt[3]{c \cdot c + d \cdot d}\right) \cdot \sqrt[3]{c \cdot c + d \cdot d}}\]

    5. Applied times-frac41.5

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{c \cdot c + d \cdot d} \cdot \sqrt[3]{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt[3]{c \cdot c + d \cdot d}}}\]

    6. Taylor expanded around inf 15.5

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

    8. Applied unpow215.5

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

    9. Applied times-frac8.7

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

    if -5.03571375883534e+131 < c < 1.5009880531967683e+116

    1. Initial program 18.1

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

    3. Applied add-sqr-sqrt18.1

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

    4. Applied associate-/r*18.0

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

  4. Applied simplify15.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;c \le -5.03571375883534 \cdot 10^{+131} \lor \neg \left(c \le 1.5009880531967683 \cdot 10^{+116}\right):\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018199 
(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))))