Average Error: 26.3 → 26.2
Time: 13.4s
Precision: 64
Internal Precision: 128
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\]

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

Original26.3
Target0.4
Herbie26.2
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.3

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

  3. Applied add-sqr-sqrt26.3

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

  4. Applied associate-/r*26.2

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

  5. Final simplification26.2

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

Reproduce

herbie shell --seed 2019005 
(FPCore (a b c d)
  :name "Complex division, real part"

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

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

Details

Time bar (total: 12.1s)Debug log

sample77.0ms

Algorithm
intervals

simplify10.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

10.0ms
(/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))

prune12.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 23.6b

localize24.0ms

Local error

Found 3 expressions with local error:

23.6b
(/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
0.3b
(+ (* a c) (* b d))
0.0b
(+ (* c c) (* d d))

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
3 → 53
Calls

3 calls. Slowest were:

10.0ms
(/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
3.0ms
(+ (* c c) (* d d))
2.0ms
(+ (* a c) (* b d))

series117.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

66.0ms
(/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
36.0ms
(+ (* a c) (* b d))
15.0ms
(+ (* c c) (* d d))

simplify966.0ms

Counts
40 → 62
Calls

40 calls. Slowest were:

255.0ms
(- (* (* a c) (* a c)) (* (* b d) (* b d)))
166.0ms
(+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d))))
161.0ms
(/ (* (* (+ (* a c) (* b d)) (+ (* a c) (* b d))) (+ (* a c) (* b d))) (* (* (+ (* c c) (* d d)) (+ (* c c) (* d d))) (+ (* c c) (* d d))))

prune746.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 23.2b

localize17.0ms

Local error

Found 4 expressions with local error:

28.6b
(sqrt (+ (* c c) (* d d)))
28.6b
(sqrt (+ (* c c) (* d d)))
14.4b
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
0.3b
(+ (* a c) (* b d))

rewrite34.0ms

Algorithm
rewrite-expression-head
Counts
4 → 82
Calls

4 calls. Slowest were:

21.0ms
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
4.0ms
(+ (* a c) (* b d))
4.0ms
(sqrt (+ (* c c) (* d d)))

series217.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

114.0ms
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
40.0ms
(+ (* a c) (* b d))
32.0ms
(sqrt (+ (* c c) (* d d)))
30.0ms
(sqrt (+ (* c c) (* d d)))

simplify1.4s

Counts
75 → 94
Calls

75 calls. Slowest were:

242.0ms
(- (* (* a c) (* a c)) (* (* b d) (* b d)))
170.0ms
(sqrt (+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d)))))
121.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))

prune1.1s

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 20.5b

localize14.0ms

Local error

Found 4 expressions with local error:

28.6b
(sqrt (+ (* c c) (* d d)))
28.6b
(sqrt (+ (* c c) (* d d)))
14.4b
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
0.3b
(+ (* a c) (* b d))

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
4 → 82
Calls

4 calls. Slowest were:

16.0ms
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
4.0ms
(sqrt (+ (* c c) (* d d)))
4.0ms
(sqrt (+ (* c c) (* d d)))

series173.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

89.0ms
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
39.0ms
(+ (* a c) (* b d))
25.0ms
(sqrt (+ (* c c) (* d d)))
19.0ms
(sqrt (+ (* c c) (* d d)))

simplify1.8s

Counts
75 → 94
Calls

75 calls. Slowest were:

512.0ms
(- (* (* a c) (* a c)) (* (* b d) (* b d)))
169.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))
140.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))

prune1.3s

Pruning

9 alts after pruning (6 fresh and 3 done)

Merged error: 20.5b

localize37.0ms

Local error

Found 4 expressions with local error:

28.6b
(sqrt (+ (* c c) (* d d)))
28.6b
(sqrt (+ (* c c) (* d d)))
28.6b
(sqrt (+ (* c c) (* d d)))
28.6b
(sqrt (+ (* c c) (* d d)))

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

4.0ms
(sqrt (+ (* c c) (* d d)))
4.0ms
(sqrt (+ (* c c) (* d d)))
4.0ms
(sqrt (+ (* c c) (* d d)))

series126.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

36.0ms
(sqrt (+ (* c c) (* d d)))
31.0ms
(sqrt (+ (* c c) (* d d)))
30.0ms
(sqrt (+ (* c c) (* d d)))
28.0ms
(sqrt (+ (* c c) (* d d)))

simplify740.0ms

Counts
36 → 68
Calls

36 calls. Slowest were:

143.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))
135.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))
132.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))

prune956.0ms

Pruning

9 alts after pruning (5 fresh and 4 done)

Merged error: 20.5b

regimes396.0ms

Accuracy

0% (2.8b remaining)

Error of 26.2b against oracle of 23.4b and baseline of 26.2b

bsearch2.0ms

end0.0ms

sample1.7s

Algorithm
intervals