Average Error: 26.5 → 26.5
Time: 30.7s
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

Original26.5
Target0.5
Herbie26.5
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.5

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

  3. Applied add-sqr-sqrt26.5

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

  4. Applied associate-/r*26.5

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

  5. Final simplification26.5

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

Reproduce

herbie shell --seed 2019007 
(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: 29.8s)Debug log

sample37.0ms

Algorithm
intervals

simplify9.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

9.0ms
(/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))

prune23.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 28.5b

localize36.0ms

Local error

Found 3 expressions with local error:

28.5b
(/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
0.9b
(+ (* a c) (* b d))
0.0b
(+ (* c c) (* d d))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
3 → 53
Calls

3 calls. Slowest were:

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

series133.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

71.0ms
(/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
37.0ms
(+ (* a c) (* b d))
25.0ms
(+ (* c c) (* d d))

simplify890.0ms

Counts
40 → 62
Calls

40 calls. Slowest were:

210.0ms
(- (* (* a c) (* a c)) (* (* b d) (* b d)))
146.0ms
(+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d))))
140.0ms
(/ (* (* (+ (* a c) (* b d)) (+ (* a c) (* b d))) (+ (* a c) (* b d))) (* (* (+ (* c c) (* d d)) (+ (* c c) (* d d))) (+ (* c c) (* d d))))

prune721.0ms

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 28.0b

localize15.0ms

Local error

Found 4 expressions with local error:

32.9b
(sqrt (+ (* c c) (* d d)))
32.9b
(sqrt (+ (* c c) (* d d)))
19.4b
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
0.9b
(+ (* a c) (* b d))

rewrite29.0ms

Algorithm
rewrite-expression-head
Counts
4 → 82
Calls

4 calls. Slowest were:

17.0ms
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
4.0ms
(+ (* a c) (* b d))
3.0ms
(sqrt (+ (* c c) (* d d)))

series141.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

67.0ms
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
28.0ms
(sqrt (+ (* c c) (* d d)))
26.0ms
(+ (* a c) (* b d))
21.0ms
(sqrt (+ (* c c) (* d d)))

simplify1.4s

Counts
75 → 94
Calls

75 calls. Slowest were:

259.0ms
(- (* (* a c) (* a c)) (* (* b d) (* b d)))
194.0ms
(sqrt (+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d)))))
142.0ms
(+ (pow (* a c) 3) (pow (* b d) 3))

prune1.1s

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 26.6b

localize13.0ms

Local error

Found 4 expressions with local error:

32.9b
(sqrt (+ (* c c) (* d d)))
32.9b
(sqrt (+ (* c c) (* d d)))
32.9b
(sqrt (+ (* c c) (* d d)))
10.0b
(/ (/ (+ (* a c) (* b d)) (sqrt (sqrt (+ (* c c) (* d d))))) (sqrt (sqrt (+ (* c c) (* d d)))))

rewrite44.0ms

Algorithm
rewrite-expression-head
Counts
4 → 398
Calls

4 calls. Slowest were:

17.0ms
(/ (/ (+ (* a c) (* b 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)))

series187.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

113.0ms
(/ (/ (+ (* a c) (* b d)) (sqrt (sqrt (+ (* c c) (* d d))))) (sqrt (sqrt (+ (* c c) (* d d)))))
32.0ms
(sqrt (+ (* c c) (* d d)))
23.0ms
(sqrt (+ (* c c) (* d d)))
18.0ms
(sqrt (+ (* c c) (* d d)))

simplify17.0s

Counts
672 → 410
Calls

672 calls. Slowest were:

535.0ms
(/ (sqrt (sqrt (+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d)))))) (sqrt (sqrt (sqrt (+ (* c c) (* d d))))))
276.0ms
(/ (/ (* (cbrt (+ (* a c) (* b d))) (cbrt (+ (* a c) (* b d)))) (sqrt (sqrt 1))) (* (cbrt (sqrt (sqrt (+ (* c c) (* d d))))) (cbrt (sqrt (sqrt (+ (* c c) (* d d)))))))
239.0ms
(/ (* (* (/ (+ (* a c) (* b d)) (sqrt (sqrt (+ (* c c) (* d d))))) (/ (+ (* a c) (* b d)) (sqrt (sqrt (+ (* c c) (* d d)))))) (/ (+ (* a c) (* b 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))))))

prune4.9s

Pruning

10 alts after pruning (8 fresh and 2 done)

Merged error: 26.6b

localize20.0ms

Local error

Found 4 expressions with local error:

34.0b
(cbrt (+ (* c c) (* d d)))
34.0b
(cbrt (+ (* c c) (* d d)))
32.9b
(sqrt (+ (* c c) (* d d)))
32.9b
(sqrt (+ (* c c) (* d d)))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 54
Calls

4 calls. Slowest were:

2.0ms
(sqrt (+ (* c c) (* d d)))
2.0ms
(sqrt (+ (* c c) (* d d)))
1.0ms
(cbrt (+ (* c c) (* d d)))

series234.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

94.0ms
(cbrt (+ (* c c) (* d d)))
80.0ms
(cbrt (+ (* c c) (* d d)))
31.0ms
(sqrt (+ (* c c) (* d d)))
29.0ms
(sqrt (+ (* c c) (* d d)))

simplify598.0ms

Counts
34 → 66
Calls

34 calls. Slowest were:

92.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))
90.0ms
(cbrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))
90.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))

prune855.0ms

Pruning

10 alts after pruning (7 fresh and 3 done)

Merged error: 26.6b

regimes335.0ms

Accuracy

0% (2.7b remaining)

Error of 26.5b against oracle of 23.8b and baseline of 26.5b

bsearch2.0ms

end0.0ms

sample1.0s

Algorithm
intervals