Average Error: 25.9 → 25.9
Time: 46.3s
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

Original25.9
Target0.4
Herbie25.9
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.9

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

  2. Initial simplification25.9

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

  4. Applied add-sqr-sqrt25.9

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

  5. Applied associate-/r*25.9

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

  7. Applied clear-num25.9

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

  8. Final simplification25.9

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

Reproduce

herbie shell --seed 2018365 
(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: 43.5s)Debug log

start56.0ms

Algorithm
intervals

setup84.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 27.0b

localize28.0ms

Local error

Found 2 expressions with local error:

27.0b
(/ (- (* b c) (* a d)) (+ (* c c) (* d d)))
0.7b
(- (* 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))

series90.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

60.0ms
(/ (- (* b c) (* a d)) (+ (* c c) (* d d)))
30.0ms
(- (* b c) (* a d))

simplify1.4s

Counts
37 → 50
Calls

37 calls. Slowest were:

324.0ms
(- (* (* b c) (* b c)) (* (* a d) (* a d)))
297.0ms
(- (pow (* b c) 3) (pow (* a d) 3))
190.0ms
(+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d))))

prune610.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 26.6b

localize9.0ms

Local error

Found 4 expressions with local error:

29.3b
(sqrt (+ (* c c) (* d d)))
29.3b
(sqrt (+ (* c c) (* d d)))
17.6b
(/ (- (* b c) (* a d)) (sqrt (+ (* c c) (* d d))))
0.7b
(- (* b c) (* a d))

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 84
Calls

4 calls. Slowest were:

15.0ms
(/ (- (* b c) (* a d)) (sqrt (+ (* c c) (* d d))))
4.0ms
(sqrt (+ (* c c) (* d d)))
4.0ms
(sqrt (+ (* c c) (* d d)))

series207.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

104.0ms
(/ (- (* b c) (* a d)) (sqrt (+ (* c c) (* d d))))
43.0ms
(- (* b c) (* a d))
35.0ms
(sqrt (+ (* c c) (* d d)))
25.0ms
(sqrt (+ (* c c) (* d d)))

simplify2.5s

Counts
76 → 96
Calls

76 calls. Slowest were:

400.0ms
(- (pow (* b c) 3) (pow (* a d) 3))
303.0ms
(- (* (* b c) (* b c)) (* (* a d) (* a d)))
192.0ms
(/ (* (* (- (* 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)))))

prune1.3s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 24.5b

localize21.0ms

Local error

Found 4 expressions with local error:

29.3b
(sqrt (+ (* c c) (* d d)))
29.3b
(sqrt (+ (* c c) (* d d)))
17.6b
(/ (sqrt (+ (* c c) (* d d))) (- (* b c) (* a d)))
0.7b
(- (* b c) (* a d))

rewrite25.0ms

Algorithm
rewrite-expression-head
Counts
4 → 83
Calls

4 calls. Slowest were:

13.0ms
(/ (sqrt (+ (* c c) (* d d))) (- (* b c) (* a d)))
3.0ms
(sqrt (+ (* c c) (* d d)))
3.0ms
(sqrt (+ (* c c) (* d d)))

series158.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

74.0ms
(/ (sqrt (+ (* c c) (* d d))) (- (* b c) (* a d)))
28.0ms
(sqrt (+ (* c c) (* d d)))
28.0ms
(sqrt (+ (* c c) (* d d)))
28.0ms
(- (* b c) (* a d))

simplify1.7s

Counts
75 → 95
Calls

75 calls. Slowest were:

216.0ms
(/ (* (* (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))))
184.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))
141.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))

prune1.5s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 24.5b

localize11.0ms

Local error

Found 4 expressions with local error:

29.3b
(sqrt (+ (* c c) (* d d)))
29.3b
(sqrt (+ (* c c) (* d d)))
29.3b
(sqrt (+ (* c c) (* d d)))
8.9b
(/ (/ (- (* b c) (* a d)) (sqrt (sqrt (+ (* c c) (* d d))))) (sqrt (sqrt (+ (* c c) (* d d)))))

rewrite32.0ms

Algorithm
rewrite-expression-head
Counts
4 → 399
Calls

4 calls. Slowest were:

14.0ms
(/ (/ (- (* b c) (* a d)) (sqrt (sqrt (+ (* c c) (* d d))))) (sqrt (sqrt (+ (* c c) (* d d)))))
2.0ms
(sqrt (+ (* c c) (* d d)))
2.0ms
(sqrt (+ (* c c) (* d d)))

series157.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

95.0ms
(/ (/ (- (* b c) (* a d)) (sqrt (sqrt (+ (* c c) (* d d))))) (sqrt (sqrt (+ (* c c) (* d d)))))
24.0ms
(sqrt (+ (* c c) (* d d)))
19.0ms
(sqrt (+ (* c c) (* d d)))
19.0ms
(sqrt (+ (* c c) (* d d)))

simplify28.4s

Counts
673 → 411
Calls

673 calls. Slowest were:

425.0ms
(/ (* (* (/ (- (* b c) (* a d)) (sqrt (sqrt (+ (* c c) (* d d))))) (/ (- (* b c) (* a d)) (sqrt (sqrt (+ (* c c) (* d d)))))) (/ (- (* b c) (* a d)) (sqrt (sqrt (+ (* c c) (* d d)))))) (* (* (sqrt (sqrt (+ (* c c) (* d d)))) (sqrt (sqrt (+ (* c c) (* d d))))) (sqrt (sqrt (+ (* c c) (* d d))))))
403.0ms
(/ (/ (* (* (- (* b c) (* a d)) (- (* b c) (* a d))) (- (* b c) (* a d))) (* (* (sqrt (sqrt (+ (* c c) (* d d)))) (sqrt (sqrt (+ (* c c) (* d d))))) (sqrt (sqrt (+ (* c c) (* d d)))))) (* (* (sqrt (sqrt (+ (* c c) (* d d)))) (sqrt (sqrt (+ (* c c) (* d d))))) (sqrt (sqrt (+ (* c c) (* d d))))))
380.0ms
(/ (/ (- (* b c) (* a d)) (sqrt (cbrt (sqrt (+ (* c c) (* d d)))))) (cbrt (sqrt (sqrt (+ (* c c) (* d d))))))

prune5.0s

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 24.5b

regimes288.0ms

Accuracy

0% (2.6b remaining)

Error of 25.9b against oracle of 23.3b and baseline of 25.9b

bsearch2.0ms