Average Error: 26.0 → 26.0
Time: 14.8s
Precision: 64
Internal Precision: 128
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\]

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

Original26.0
Target0.4
Herbie26.0
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.0

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

  3. Applied add-sqr-sqrt26.0

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

  4. Applied associate-/r*26.0

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

  5. Final simplification26.0

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

Reproduce

herbie shell --seed 2019008 
(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: 14.0s)Debug log

sample68.0ms

Algorithm
intervals

simplify36.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

36.0ms
(/ (- (* b c) (* a d)) (+ (* c c) (* d d)))

prune7.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 26.9b

localize50.0ms

Local error

Found 3 expressions with local error:

26.9b
(/ (- (* b c) (* a d)) (+ (* c c) (* d d)))
0.7b
(- (* b c) (* a d))
0.0b
(+ (* c c) (* d d))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
3 → 55
Calls

3 calls. Slowest were:

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

series140.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

63.0ms
(/ (- (* b c) (* a d)) (+ (* c c) (* d d)))
52.0ms
(- (* b c) (* a d))
24.0ms
(+ (* c c) (* d d))

simplify1.3s

Counts
41 → 64
Calls

41 calls. Slowest were:

298.0ms
(- (pow (* b c) 3) (pow (* a d) 3))
218.0ms
(- (* (* b c) (* b c)) (* (* a d) (* a d)))
149.0ms
(+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d))))

prune653.0ms

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 26.4b

localize17.0ms

Local error

Found 4 expressions with local error:

30.1b
(sqrt (+ (* c c) (* d d)))
30.1b
(sqrt (+ (* c c) (* d d)))
16.4b
(/ (- (* b c) (* a d)) (sqrt (+ (* c c) (* d d))))
0.7b
(- (* b c) (* a d))

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
4 → 84
Calls

4 calls. Slowest were:

14.0ms
(/ (- (* b c) (* a d)) (sqrt (+ (* c c) (* d d))))
4.0ms
(- (* b c) (* a d))
4.0ms
(sqrt (+ (* c c) (* d d)))

series219.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

104.0ms
(/ (- (* b c) (* a d)) (sqrt (+ (* c c) (* d d))))
47.0ms
(- (* b c) (* a d))
34.0ms
(sqrt (+ (* c c) (* d d)))
34.0ms
(sqrt (+ (* c c) (* d d)))

simplify2.2s

Counts
76 → 96
Calls

76 calls. Slowest were:

291.0ms
(- (pow (* b c) 3) (pow (* a d) 3))
261.0ms
(- (* (* b c) (* b c)) (* (* a d) (* a d)))
223.0ms
(/ (* (* (- (* 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)))))

prune1.2s

Pruning

10 alts after pruning (8 fresh and 2 done)

Merged error: 23.7b

localize16.0ms

Local error

Found 4 expressions with local error:

30.1b
(sqrt (+ (* c c) (* d d)))
30.1b
(sqrt (+ (* c c) (* d d)))
16.5b
(/ (sqrt (+ (* c c) (* d d))) (- (* b c) (* a d)))
0.7b
(- (* b c) (* a d))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 83
Calls

4 calls. Slowest were:

11.0ms
(/ (sqrt (+ (* c c) (* d d))) (- (* b c) (* a d)))
3.0ms
(sqrt (+ (* c c) (* d d)))
3.0ms
(sqrt (+ (* c c) (* d d)))

series130.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

52.0ms
(/ (sqrt (+ (* c c) (* d d))) (- (* b c) (* a d)))
38.0ms
(- (* b c) (* a d))
21.0ms
(sqrt (+ (* c c) (* d d)))
19.0ms
(sqrt (+ (* c c) (* d d)))

simplify1.3s

Counts
75 → 95
Calls

75 calls. Slowest were:

171.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))
158.0ms
(/ (* (* (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))))
115.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))

prune1.0s

Pruning

10 alts after pruning (8 fresh and 2 done)

Merged error: 23.7b

localize14.0ms

Local error

Found 4 expressions with local error:

30.1b
(sqrt (+ (* c c) (* d d)))
30.1b
(sqrt (+ (* c c) (* d d)))
30.1b
(sqrt (+ (* c c) (* d d)))
8.3b
(/ (- (* b c) (* a d)) (sqrt (sqrt (+ (* c c) (* d d)))))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 98
Calls

4 calls. Slowest were:

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

series246.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

187.0ms
(/ (- (* b c) (* a d)) (sqrt (sqrt (+ (* c c) (* d d)))))
21.0ms
(sqrt (+ (* c c) (* d d)))
19.0ms
(sqrt (+ (* c c) (* d d)))
18.0ms
(sqrt (+ (* c c) (* d d)))

simplify2.3s

Counts
102 → 110
Calls

102 calls. Slowest were:

210.0ms
(- (pow (* b c) 3) (pow (* a d) 3))
163.0ms
(sqrt (sqrt (+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d))))))
162.0ms
(- (* (* b c) (* b c)) (* (* a d) (* a d)))

prune1.1s

Pruning

10 alts after pruning (7 fresh and 3 done)

Merged error: 23.7b

regimes330.0ms

Accuracy

0% (2.7b remaining)

Error of 26.0b against oracle of 23.3b and baseline of 26.0b

bsearch2.0ms

end0.0ms

sample1.4s

Algorithm
intervals