Average Error: 25.4 → 25.3
Time: 16.1s
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

Original25.4
Target0.3
Herbie25.3
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.4

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

  3. Applied add-sqr-sqrt25.4

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

  4. Applied associate-/r*25.3

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

  5. Final simplification25.3

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

Reproduce

herbie shell --seed 2019004 
(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: 14.6s)Debug log

sample71.0ms

Algorithm
intervals

simplify10.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune12.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 30.8b

localize39.0ms

Local error

Found 3 expressions with local error:

30.8b
(/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
1.1b
(+ (* a c) (* b d))
0.0b
(+ (* c c) (* d d))

rewrite25.0ms

Algorithm
rewrite-expression-head
Counts
3 → 53
Calls

3 calls. Slowest were:

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

series143.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

89.0ms
(/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
39.0ms
(+ (* a c) (* b d))
15.0ms
(+ (* c c) (* d d))

simplify902.0ms

Counts
40 → 62
Calls

40 calls. Slowest were:

226.0ms
(- (* (* a c) (* a c)) (* (* b d) (* b d)))
153.0ms
(/ (* (* (+ (* a c) (* b d)) (+ (* a c) (* b d))) (+ (* a c) (* b d))) (* (* (+ (* c c) (* d d)) (+ (* c c) (* d d))) (+ (* c c) (* d d))))
142.0ms
(+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d))))

prune623.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 30.3b

localize10.0ms

Local error

Found 4 expressions with local error:

32.3b
(sqrt (+ (* c c) (* d d)))
32.3b
(sqrt (+ (* c c) (* d d)))
21.5b
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
1.1b
(+ (* a c) (* b d))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
4 → 82
Calls

4 calls. Slowest were:

11.0ms
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
2.0ms
(+ (* a c) (* b d))
2.0ms
(sqrt (+ (* c c) (* d d)))

series171.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

92.0ms
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
41.0ms
(+ (* a c) (* b d))
19.0ms
(sqrt (+ (* c c) (* d d)))
18.0ms
(sqrt (+ (* c c) (* d d)))

simplify1.4s

Counts
75 → 94
Calls

75 calls. Slowest were:

211.0ms
(- (* (* a c) (* a c)) (* (* b d) (* b d)))
157.0ms
(sqrt (+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d)))))
135.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))

prune1.2s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 27.9b

localize9.0ms

Local error

Found 4 expressions with local error:

32.3b
(sqrt (+ (* c c) (* d d)))
32.3b
(sqrt (+ (* c c) (* d d)))
30.8b
(/ (+ (* a c) (* b d)) (* (sqrt (+ (* c c) (* d d))) (sqrt (+ (* c c) (* d d)))))
1.1b
(+ (* a c) (* b d))

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
4 → 70
Calls

4 calls. Slowest were:

14.0ms
(/ (+ (* a c) (* b d)) (* (sqrt (+ (* c c) (* d d))) (sqrt (+ (* c c) (* d d)))))
3.0ms
(+ (* a c) (* b d))
2.0ms
(sqrt (+ (* c c) (* d d)))

series142.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

71.0ms
(/ (+ (* a c) (* b d)) (* (sqrt (+ (* c c) (* d d))) (sqrt (+ (* c c) (* d d)))))
28.0ms
(sqrt (+ (* c c) (* d d)))
24.0ms
(+ (* a c) (* b d))
19.0ms
(sqrt (+ (* c c) (* d d)))

simplify2.2s

Counts
48 → 82
Calls

48 calls. Slowest were:

260.0ms
(- (* (* a c) (* a c)) (* (* b d) (* b d)))
180.0ms
(sqrt (+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d)))))
177.0ms
(/ (* (* (+ (* a c) (* b d)) (+ (* a c) (* b d))) (+ (* a c) (* b d))) (* (* (* (sqrt (+ (* c c) (* d d))) (sqrt (+ (* c c) (* d d)))) (* (sqrt (+ (* c c) (* d d))) (sqrt (+ (* c c) (* d d))))) (* (sqrt (+ (* c c) (* d d))) (sqrt (+ (* c c) (* d d))))))

prune1.0s

Pruning

9 alts after pruning (7 fresh and 2 done)

Merged error: 27.9b

localize15.0ms

Local error

Found 4 expressions with local error:

32.3b
(sqrt (+ (* c c) (* d d)))
32.3b
(sqrt (+ (* c c) (* d d)))
32.3b
(sqrt (+ (* c c) (* d d)))
13.6b
(/ (+ (* a c) (* b d)) (sqrt (sqrt (+ (* c c) (* d d)))))

rewrite29.0ms

Algorithm
rewrite-expression-head
Counts
4 → 97
Calls

4 calls. Slowest were:

18.0ms
(/ (+ (* a c) (* b d)) (sqrt (sqrt (+ (* c c) (* d d)))))
3.0ms
(sqrt (+ (* c c) (* d d)))
3.0ms
(sqrt (+ (* c c) (* d d)))

series259.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

204.0ms
(/ (+ (* a c) (* b d)) (sqrt (sqrt (+ (* c c) (* d d)))))
19.0ms
(sqrt (+ (* c c) (* d d)))
18.0ms
(sqrt (+ (* c c) (* d d)))
18.0ms
(sqrt (+ (* c c) (* d d)))

simplify2.0s

Counts
101 → 109
Calls

101 calls. Slowest were:

249.0ms
(- (* (* a c) (* a c)) (* (* b d) (* b d)))
195.0ms
(sqrt (sqrt (+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d))))))
133.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))

prune1.6s

Pruning

10 alts after pruning (7 fresh and 3 done)

Merged error: 27.9b

regimes384.0ms

Accuracy

0% (2.5b remaining)

Error of 25.3b against oracle of 22.8b and baseline of 25.3b

bsearch2.0ms

end0.0ms

sample2.2s

Algorithm
intervals