Average Error: 25.8 → 25.8

Time: 17.6s

Precision: 64

Internal Precision: 576

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

↓

\[\begin{array}{l} \mathbf{if}\;d \le 1.3100648088728728 \cdot 10^{+53}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{d \cdot d + c \cdot c}}}{\sqrt{d \cdot d + c \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\sqrt{d \cdot d + c \cdot c}}\\ \end{array}\]

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	25.8
Target	0.4
Herbie	25.8

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

Derivation

Split input into 2 regimes
if d < 1.3100648088728728e+53
1. Initial program 22.8
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Initial simplification22.8
  \[\leadsto \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
3. Using strategy rm
4. Applied add-sqr-sqrt22.8
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
5. Applied associate-/r*22.7
  \[\leadsto \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
if 1.3100648088728728e+53 < d
1. Initial program 36.3
  \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
2. Initial simplification36.3
  \[\leadsto \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
3. Using strategy rm
4. Applied add-sqr-sqrt36.3
  \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
5. Applied associate-/r*36.2
  \[\leadsto \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Taylor expanded around 0 36.9
  \[\leadsto \frac{\color{blue}{-1 \cdot a}}{\sqrt{c \cdot c + d \cdot d}}\]
7. Simplified36.9
  \[\leadsto \frac{\color{blue}{-a}}{\sqrt{c \cdot c + d \cdot d}}\]
Recombined 2 regimes into one program.
Final simplification25.8
\[\leadsto \begin{array}{l} \mathbf{if}\;d \le 1.3100648088728728 \cdot 10^{+53}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{d \cdot d + c \cdot c}}}{\sqrt{d \cdot d + c \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\sqrt{d \cdot d + c \cdot c}}\\ \end{array}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	25.7	25.8	23.0	2.7	-5.6%

herbie shell --seed 2018286 
(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 d < 1.3100648088728728e+53`

`if 1.3100648088728728e+53 < d`

Runtime