Average Error: 26.1 → 26.1
Time: 18.1s
Precision: 64
Internal Precision: 128
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\]

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

Original26.1
Target0.5
Herbie26.1
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.1

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

  2. Initial simplification26.1

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

  4. Applied add-sqr-sqrt26.1

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

  5. Applied associate-/r*26.1

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

  6. Final simplification26.1

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

Reproduce

herbie shell --seed 2018360 
(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

start82.0ms

Algorithm
intervals

setup37.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 25.5b

localize22.0ms

Local error

Found 3 expressions with local error:

25.5b
(/ (+ (* b d) (* a c)) (+ (* c c) (* d d)))
0.5b
(+ (* b d) (* a c))
0.0b
(+ (* c c) (* d d))

rewrite29.0ms

Algorithm
rewrite-expression-head
Counts
3 → 53
Calls

3 calls. Slowest were:

19.0ms
(/ (+ (* b d) (* a c)) (+ (* c c) (* d d)))
4.0ms
(+ (* b d) (* a c))
4.0ms
(+ (* c c) (* d d))

series164.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

96.0ms
(/ (+ (* b d) (* a c)) (+ (* c c) (* d d)))
42.0ms
(+ (* b d) (* a c))
25.0ms
(+ (* c c) (* d d))

simplify1.3s

Counts
40 → 62
Calls

40 calls. Slowest were:

445.0ms
(+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d))))
257.0ms
(- (* (* b d) (* b d)) (* (* a c) (* a c)))
189.0ms
(/ (* (* (+ (* b d) (* a c)) (+ (* b d) (* a c))) (+ (* b d) (* a c))) (* (* (+ (* c c) (* d d)) (+ (* c c) (* d d))) (+ (* c c) (* d d))))

prune815.0ms

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 25.1b

localize11.0ms

Local error

Found 4 expressions with local error:

27.5b
(sqrt (+ (* c c) (* d d)))
27.5b
(sqrt (+ (* c c) (* d d)))
16.0b
(/ (+ (* b d) (* a c)) (sqrt (+ (* c c) (* d d))))
0.5b
(+ (* b d) (* a c))

rewrite33.0ms

Algorithm
rewrite-expression-head
Counts
4 → 82
Calls

4 calls. Slowest were:

19.0ms
(/ (+ (* b d) (* a c)) (sqrt (+ (* c c) (* d d))))
5.0ms
(+ (* b d) (* a c))
4.0ms
(sqrt (+ (* c c) (* d d)))

series173.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

82.0ms
(/ (+ (* b d) (* a c)) (sqrt (+ (* c c) (* d d))))
33.0ms
(sqrt (+ (* c c) (* d d)))
31.0ms
(sqrt (+ (* c c) (* d d)))
26.0ms
(+ (* b d) (* a c))

simplify1.9s

Counts
75 → 94
Calls

75 calls. Slowest were:

545.0ms
(sqrt (+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d)))))
244.0ms
(- (* (* b d) (* b d)) (* (* a c) (* a c)))
185.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))

prune1.3s

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 23.1b

localize9.0ms

Local error

Found 4 expressions with local error:

27.5b
(sqrt (+ (* c c) (* d d)))
27.5b
(sqrt (+ (* c c) (* d d)))
16.2b
(* (+ (* b d) (* a c)) (/ 1 (sqrt (+ (* c c) (* d d)))))
0.5b
(+ (* b d) (* a c))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

11.0ms
(* (+ (* b d) (* a c)) (/ 1 (sqrt (+ (* c c) (* d d)))))
7.0ms
(+ (* b d) (* a c))
2.0ms
(sqrt (+ (* c c) (* d d)))

series168.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

95.0ms
(* (+ (* b d) (* a c)) (/ 1 (sqrt (+ (* c c) (* d d)))))
31.0ms
(+ (* b d) (* a c))
23.0ms
(sqrt (+ (* c c) (* d d)))
19.0ms
(sqrt (+ (* c c) (* d d)))

simplify2.6s

Counts
45 → 78
Calls

45 calls. Slowest were:

513.0ms
(sqrt (+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d)))))
432.0ms
(* (- (* (* b d) (* b d)) (* (* a c) (* a c))) 1)
301.0ms
(* (+ (pow (* b d) 3) (pow (* a c) 3)) 1)

prune1.1s

Pruning

8 alts after pruning (5 fresh and 3 done)

Merged error: 23.1b

localize25.0ms

Local error

Found 4 expressions with local error:

27.5b
(sqrt (+ (* c c) (* d d)))
27.5b
(sqrt (+ (* c c) (* d d)))
27.5b
(sqrt (+ (* c c) (* d d)))
8.1b
(* (+ (* b d) (* a c)) (sqrt (/ 1 (sqrt (+ (* c c) (* d d))))))

rewrite31.0ms

Algorithm
rewrite-expression-head
Counts
4 → 70
Calls

4 calls. Slowest were:

14.0ms
(* (+ (* b d) (* a c)) (sqrt (/ 1 (sqrt (+ (* c c) (* d d))))))
8.0ms
(sqrt (+ (* c c) (* d d)))
4.0ms
(sqrt (+ (* c c) (* d d)))

series299.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

229.0ms
(* (+ (* b d) (* a c)) (sqrt (/ 1 (sqrt (+ (* c c) (* d d))))))
32.0ms
(sqrt (+ (* c c) (* d d)))
19.0ms
(sqrt (+ (* c c) (* d d)))
19.0ms
(sqrt (+ (* c c) (* d d)))

simplify2.8s

Counts
51 → 82
Calls

51 calls. Slowest were:

513.0ms
(sqrt (sqrt (+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d))))))
417.0ms
(* (+ (pow (* b d) 3) (pow (* a c) 3)) (sqrt (/ 1 (sqrt (+ (* c c) (* d d))))))
263.0ms
(* (- (* (* b d) (* b d)) (* (* a c) (* a c))) (sqrt 1))

prune1.3s

Pruning

8 alts after pruning (4 fresh and 4 done)

Merged error: 23.1b

regimes360.0ms

Accuracy

0% (2.6b remaining)

Error of 26.1b against oracle of 23.4b and baseline of 26.1b

bsearch3.0ms