Average Error: 25.5 → 26.0

Time: 22.2s

Precision: 64

Internal Precision: 128

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{b \cdot d + a \cdot c}{c \cdot c + d \cdot d} \le 1.2550484536608203 \cdot 10^{+222}:\\ \;\;\;\;\frac{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Error

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

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In
Out

Enter valid numbers for all inputs

Target

Original	25.5
Target	0.4
Herbie	26.0

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

Derivation

Split input into 2 regimes
if (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) < 1.2550484536608203e+222
1. Initial program 14.1
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Initial simplification14.1
  \[\leadsto \frac{b \cdot d + a \cdot c}{c \cdot c + d \cdot d}\]
3. Using strategy rm
4. Applied add-sqr-sqrt14.2
  \[\leadsto \frac{b \cdot d + a \cdot c}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
5. Applied associate-/r*14.1
  \[\leadsto \color{blue}{\frac{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
if 1.2550484536608203e+222 < (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
1. Initial program 57.3
  \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
2. Initial simplification57.3
  \[\leadsto \frac{b \cdot d + a \cdot c}{c \cdot c + d \cdot d}\]
3. Using strategy rm
4. Applied add-sqr-sqrt57.3
  \[\leadsto \frac{b \cdot d + a \cdot c}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
5. Applied associate-/r*57.3
  \[\leadsto \color{blue}{\frac{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Taylor expanded around -inf 59.6
  \[\leadsto \frac{\color{blue}{-1 \cdot b}}{\sqrt{c \cdot c + d \cdot d}}\]
7. Simplified59.6
  \[\leadsto \frac{\color{blue}{-b}}{\sqrt{c \cdot c + d \cdot d}}\]
Recombined 2 regimes into one program.
Final simplification26.0
\[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot d + a \cdot c}{c \cdot c + d \cdot d} \le 1.2550484536608203 \cdot 10^{+222}:\\ \;\;\;\;\frac{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Runtime

herbie shell --seed 2018346 
(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))))

Error

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Target

Derivation

`if (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) < 1.2550484536608203e+222`

`if 1.2550484536608203e+222 < (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))`

Runtime