Average Error: 25.4 → 25.4
Time: 14.0s
Precision: 64
Internal Precision: 128
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{c \cdot b - d \cdot a}}}{\sqrt{c \cdot c + d \cdot d}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original25.4
Target0.4
Herbie25.4
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.4

    \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt25.4

    \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

  4. Applied associate-/r*25.3

    \[\leadsto \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
  5. Using strategy rm

  6. Applied clear-num25.4

    \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{b \cdot c - a \cdot d}}}}{\sqrt{c \cdot c + d \cdot d}}\]

  7. Final simplification25.4

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

Reproduce

herbie shell --seed 2019002 
(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))))

Details

Time bar (total: 12.4s)Debug log

sample73.0ms

Algorithm
intervals

simplify37.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

37.0ms
(/ (- (* b c) (* a d)) (+ (* c c) (* d d)))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 25.6b

localize26.0ms

Local error

Found 2 expressions with local error:

25.6b
(/ (- (* b c) (* a d)) (+ (* c c) (* d d)))
0.3b
(- (* b c) (* a d))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
2 → 44
Calls

2 calls. Slowest were:

7.0ms
(/ (- (* b c) (* a d)) (+ (* c c) (* d d)))
2.0ms
(- (* b c) (* a d))

series110.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

62.0ms
(/ (- (* b c) (* a d)) (+ (* c c) (* d d)))
49.0ms
(- (* b c) (* a d))

simplify1.2s

Counts
37 → 50
Calls

37 calls. Slowest were:

265.0ms
(- (pow (* b c) 3) (pow (* a d) 3))
236.0ms
(- (* (* b c) (* b c)) (* (* a d) (* a d)))
122.0ms
(+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d))))

prune619.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 25.1b

localize8.0ms

Local error

Found 4 expressions with local error:

28.8b
(sqrt (+ (* c c) (* d d)))
28.8b
(sqrt (+ (* c c) (* d d)))
16.5b
(/ (- (* b c) (* a d)) (sqrt (+ (* c c) (* d d))))
0.3b
(- (* b c) (* a d))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 84
Calls

4 calls. Slowest were:

7.0ms
(/ (- (* b c) (* a d)) (sqrt (+ (* c c) (* d d))))
2.0ms
(sqrt (+ (* c c) (* d d)))
2.0ms
(- (* b c) (* a d))

series234.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

126.0ms
(/ (- (* b c) (* a d)) (sqrt (+ (* c c) (* d d))))
45.0ms
(- (* b c) (* a d))
33.0ms
(sqrt (+ (* c c) (* d d)))
31.0ms
(sqrt (+ (* c c) (* d d)))

simplify2.2s

Counts
76 → 96
Calls

76 calls. Slowest were:

277.0ms
(- (pow (* b c) 3) (pow (* a d) 3))
211.0ms
(/ (* (* (- (* b c) (* a d)) (- (* b c) (* a d))) (- (* b c) (* a d))) (* (* (sqrt (+ (* c c) (* d d))) (sqrt (+ (* c c) (* d d)))) (sqrt (+ (* c c) (* d d)))))
193.0ms
(- (* (* b c) (* b c)) (* (* a d) (* a d)))

prune1.3s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 22.9b

localize9.0ms

Local error

Found 4 expressions with local error:

28.8b
(sqrt (+ (* c c) (* d d)))
28.8b
(sqrt (+ (* c c) (* d d)))
16.5b
(/ (sqrt (+ (* c c) (* d d))) (- (* b c) (* a d)))
0.3b
(- (* b c) (* a d))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 83
Calls

4 calls. Slowest were:

13.0ms
(/ (sqrt (+ (* c c) (* d d))) (- (* b c) (* a d)))
4.0ms
(- (* b c) (* a d))
2.0ms
(sqrt (+ (* c c) (* d d)))

series131.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

57.0ms
(/ (sqrt (+ (* c c) (* d d))) (- (* b c) (* a d)))
27.0ms
(- (* b c) (* a d))
25.0ms
(sqrt (+ (* c c) (* d d)))
21.0ms
(sqrt (+ (* c c) (* d d)))

simplify1.3s

Counts
75 → 95
Calls

75 calls. Slowest were:

167.0ms
(/ (* (* (sqrt (+ (* c c) (* d d))) (sqrt (+ (* c c) (* d d)))) (sqrt (+ (* c c) (* d d)))) (* (* (- (* b c) (* a d)) (- (* b c) (* a d))) (- (* b c) (* a d))))
136.0ms
(+ (* (* b c) (* b c)) (+ (* (* a d) (* a d)) (* (* b c) (* a d))))
118.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))

prune1.4s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 22.9b

localize12.0ms

Local error

Found 3 expressions with local error:

25.2b
(/ (+ (* c c) (* d d)) (- (* b c) (* a d)))
0.4b
(/ 1 (/ (+ (* c c) (* d d)) (- (* b c) (* a d))))
0.3b
(- (* b c) (* a d))

rewrite32.0ms

Algorithm
rewrite-expression-head
Counts
3 → 73
Calls

3 calls. Slowest were:

21.0ms
(/ (+ (* c c) (* d d)) (- (* b c) (* a d)))
6.0ms
(/ 1 (/ (+ (* c c) (* d d)) (- (* b c) (* a d))))
4.0ms
(- (* b c) (* a d))

series236.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

119.0ms
(/ 1 (/ (+ (* c c) (* d d)) (- (* b c) (* a d))))
67.0ms
(/ (+ (* c c) (* d d)) (- (* b c) (* a d)))
50.0ms
(- (* b c) (* a d))

simplify958.0ms

Counts
56 → 82
Calls

56 calls. Slowest were:

110.0ms
(- (* (* c c) (* c c)) (* (* d d) (* d d)))
103.0ms
(+ (* (* b c) (* b c)) (+ (* (* a d) (* a d)) (* (* b c) (* a d))))
100.0ms
(+ (* (* b c) (* b c)) (+ (* (* a d) (* a d)) (* (* b c) (* a d))))

prune818.0ms

Pruning

13 alts after pruning (11 fresh and 2 done)

Merged error: 22.9b

regimes359.0ms

Accuracy

0% (2.5b remaining)

Error of 25.4b against oracle of 22.9b and baseline of 25.4b

bsearch3.0ms

end0.0ms

sample1.4s

Algorithm
intervals