Average Error: 25.7 → 17.3
Time: 1.0m
Precision: 64
Internal Precision: 384
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;c \le -1.2672125587297843 \cdot 10^{+154}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{if}\;c \le 2.5884533571120104 \cdot 10^{+107}:\\ \;\;\;\;\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.7
Target0.5
Herbie17.3
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if c < -1.2672125587297843e+154 or 2.5884533571120104e+107 < c

    1. Initial program 42.0

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

    3. Applied add-cube-cbrt42.1

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

    4. Taylor expanded around inf 50.9

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{c}\right) - \log \left(\frac{1}{a}\right)}}\]

    5. Applied simplify14.8

      \[\leadsto \color{blue}{\frac{a}{c}}\]

    if -1.2672125587297843e+154 < c < 2.5884533571120104e+107

    1. Initial program 18.5

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

    3. Applied add-sqr-sqrt18.5

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

    4. Applied associate-/r*18.4

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed '#(1070386091 2509006183 1430610344 1025408621 36622005 1425925650)' 
(FPCore (a b c d)
  :name "Complex division, real part"

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))