Average Error: 25.7 → 25.5

Time: 19.0s

Precision: 64

Internal Precision: 128

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 2.4683185316803196 \cdot 10^{+272}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

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	25.7
Target	0.4
Herbie	25.5

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

Derivation

Split input into 2 regimes
if (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) < 2.4683185316803196e+272
1. Initial program 14.0
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Initial simplification14.0
  \[\leadsto \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
3. Using strategy rm
4. Applied add-sqr-sqrt14.0
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
5. Applied associate-/r*14.0
  \[\leadsto \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
if 2.4683185316803196e+272 < (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))
1. Initial program 60.4
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Initial simplification60.4
  \[\leadsto \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
3. Using strategy rm
4. Applied add-sqr-sqrt60.4
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
5. Applied associate-/r*60.4
  \[\leadsto \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Taylor expanded around -inf 60.1
  \[\leadsto \frac{\color{blue}{-1 \cdot b}}{\sqrt{c \cdot c + d \cdot d}}\]
7. Simplified60.1
  \[\leadsto \frac{\color{blue}{-b}}{\sqrt{c \cdot c + d \cdot d}}\]
Recombined 2 regimes into one program.
Final simplification25.5
\[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 2.4683185316803196 \cdot 10^{+272}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	25.6	25.5	23.2	2.4	3.5%

herbie shell --seed 2018354 
(FPCore (a b c d)
  :name "Complex division, imag part"

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

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

Error

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Target

Derivation

`if (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) < 2.4683185316803196e+272`

`if 2.4683185316803196e+272 < (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))`

Runtime