Average Error: 24.7 → 23.3
Time: 26.0s
Precision: 64
Internal Precision: 384
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -4.853005009166494 \cdot 10^{-283}:\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a}{1} \cdot \frac{d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;a \le 4.9595495246039594 \cdot 10^{+63}:\\ \;\;\;\;\frac{b}{\sqrt[3]{c \cdot c + d \cdot d} \cdot \sqrt[3]{c \cdot c + d \cdot d}} \cdot \frac{c}{\sqrt[3]{c \cdot c + d \cdot d}} - \frac{a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;a \le +\infty:\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{d}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{d}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original24.7
Target0.4
Herbie23.3
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -4.853005009166494e-283

    1. Initial program 24.8

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

    3. Applied div-sub24.8

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

    5. Applied *-un-lft-identity24.8

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

    6. Applied times-frac23.3

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

    if -4.853005009166494e-283 < a < 4.9595495246039594e+63

    1. Initial program 19.8

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

    3. Applied div-sub19.9

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

    5. Applied add-cube-cbrt20.2

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

    6. Applied times-frac19.4

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

    if 4.9595495246039594e+63 < a

    1. Initial program 32.7

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

    3. Applied div-sub32.7

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

    5. Applied add-sqr-sqrt32.7

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

    6. Applied times-frac29.7

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

Runtime

Time bar (total: 26.0s)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' 
(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))))