Average Error: 26.5 → 26.5

Time: 16.6s

Precision: 64

Internal Precision: 128

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

↓

\[\frac{\frac{1}{\frac{1}{\frac{d \cdot b + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}}}{\sqrt{c \cdot c + d \cdot d}}\]

Error

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

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Target

Original	26.5
Target	0.4
Herbie	26.5

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

Derivation

Initial program 26.5
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
Initial simplification26.5
\[\leadsto \frac{b \cdot d + a \cdot c}{c \cdot c + d \cdot d}\]
Using strategy rm
Applied add-sqr-sqrt26.5
\[\leadsto \frac{b \cdot d + a \cdot c}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
Applied associate-/r*26.4
\[\leadsto \color{blue}{\frac{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
Using strategy rm
Applied *-un-lft-identity26.4
\[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(b \cdot d + a \cdot c\right)}}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\]
Applied associate-/l*26.5
\[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{b \cdot d + a \cdot c}}}}{\sqrt{c \cdot c + d \cdot d}}\]
Using strategy rm
Applied clear-num26.5
\[\leadsto \frac{\frac{1}{\color{blue}{\frac{1}{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}}}}{\sqrt{c \cdot c + d \cdot d}}\]
Final simplification26.5
\[\leadsto \frac{\frac{1}{\frac{1}{\frac{d \cdot b + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}}}{\sqrt{c \cdot c + d \cdot d}}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	26.5	26.5	23.6	2.8	0%

herbie shell --seed 2018354 
(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))))