Average Error: 26.1 → 26.1
Time: 1.4m
Precision: 64
Internal Precision: 384
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{a \cdot c + b \cdot d}{d \cdot d + c \cdot c}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original26.1
Target0.4
Herbie26.1
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.1

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

  3. Applied add-sqr-sqrt26.1

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

  4. Applied associate-/r*26.0

    \[\leadsto \color{blue}{\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
  5. Removed slow pow expressions.

  6. Applied simplify26.1

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

Runtime

Time bar (total: 1.4m)Debug log

herbie shell --seed '#(3622638036 3041702260 3649696288 21285302 1742518495 296600799)' 
(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))))