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

↓

\[\frac{\frac{(b \cdot d + \left(c \cdot a\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]

Error

The X axis uses an exponential scale — Bits error versus `a`

Target

Original	25.3
Target	0.4
Herbie	25.3

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

Derivation

Initial program 25.3
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
Applied simplify25.3
\[\leadsto \color{blue}{\frac{(b \cdot d + \left(c \cdot a\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
Using strategy rm
Applied add-sqr-sqrt25.3
\[\leadsto \frac{(b \cdot d + \left(c \cdot a\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Applied associate-/r*25.3
\[\leadsto \color{blue}{\frac{\frac{(b \cdot d + \left(c \cdot a\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]

Runtime

herbie shell --seed '#(1070227846 1561819246 480764335 4016816270 2602869839 2117310382)' +o rules:numerics
(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))))