Average Error: 25.5 → 25.5
Time: 15.8s
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

Original25.5
Target0.4
Herbie25.5
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.5

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

  3. Applied add-sqr-sqrt25.6

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

  4. Applied associate-/r*25.5

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

  5. Final simplification25.5

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

Reproduce

herbie shell --seed 2018362 
(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: 12.2s)Debug log

start61.0ms

Algorithm
intervals

setup14.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 23.8b

localize19.0ms

Local error

Found 3 expressions with local error:

23.8b
(/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
0.3b
(+ (* a c) (* b d))
0.0b
(+ (* c c) (* d d))

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
3 → 53
Calls

3 calls. Slowest were:

14.0ms
(+ (* a c) (* b d))
9.0ms
(/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
4.0ms
(+ (* c c) (* d d))

series130.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

85.0ms
(/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
30.0ms
(+ (* a c) (* b d))
15.0ms
(+ (* c c) (* d d))

simplify891.0ms

Counts
40 → 62
Calls

40 calls. Slowest were:

220.0ms
(- (* (* a c) (* a c)) (* (* b d) (* b d)))
160.0ms
(+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d))))
138.0ms
(/ (* (* (+ (* a c) (* b d)) (+ (* a c) (* b d))) (+ (* a c) (* b d))) (* (* (+ (* c c) (* d d)) (+ (* c c) (* d d))) (+ (* c c) (* d d))))

prune707.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 23.4b

localize18.0ms

Local error

Found 4 expressions with local error:

28.4b
(sqrt (+ (* c c) (* d d)))
28.4b
(sqrt (+ (* c c) (* d d)))
16.0b
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
0.3b
(+ (* a c) (* b d))

rewrite33.0ms

Algorithm
rewrite-expression-head
Counts
4 → 82
Calls

4 calls. Slowest were:

20.0ms
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
4.0ms
(+ (* a c) (* b d))
4.0ms
(sqrt (+ (* c c) (* d d)))

series185.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

91.0ms
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
32.0ms
(sqrt (+ (* c c) (* d d)))
31.0ms
(sqrt (+ (* c c) (* d d)))
30.0ms
(+ (* a c) (* b d))

simplify1.5s

Counts
75 → 94
Calls

75 calls. Slowest were:

285.0ms
(- (* (* a c) (* a c)) (* (* b d) (* b d)))
179.0ms
(sqrt (+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d)))))
152.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))

prune1.3s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 21.0b

localize14.0ms

Local error

Found 4 expressions with local error:

28.4b
(sqrt (+ (* c c) (* d d)))
28.4b
(sqrt (+ (* c c) (* d d)))
16.0b
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
0.3b
(+ (* a c) (* b d))

rewrite33.0ms

Algorithm
rewrite-expression-head
Counts
4 → 82
Calls

4 calls. Slowest were:

20.0ms
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
4.0ms
(sqrt (+ (* c c) (* d d)))
4.0ms
(+ (* a c) (* b d))

series171.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

78.0ms
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
44.0ms
(+ (* a c) (* b d))
29.0ms
(sqrt (+ (* c c) (* d d)))
19.0ms
(sqrt (+ (* c c) (* d d)))

simplify1.5s

Counts
75 → 94
Calls

75 calls. Slowest were:

249.0ms
(- (* (* a c) (* a c)) (* (* b d) (* b d)))
189.0ms
(sqrt (+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d)))))
171.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))

prune1.7s

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 21.0b

localize6.0ms

Local error

Found 4 expressions with local error:

23.3b
(* (+ (* a c) (* b d)) (/ 1 (+ (* c c) (* d d))))
0.6b
(/ 1 (+ (* c c) (* d d)))
0.3b
(+ (* a c) (* b d))
0.0b
(+ (* c c) (* d d))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

6.0ms
(* (+ (* a c) (* b d)) (/ 1 (+ (* c c) (* d d))))
2.0ms
(+ (* c c) (* d d))
2.0ms
(+ (* a c) (* b d))

series156.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

82.0ms
(* (+ (* a c) (* b d)) (/ 1 (+ (* c c) (* d d))))
30.0ms
(/ 1 (+ (* c c) (* d d)))
29.0ms
(+ (* a c) (* b d))
15.0ms
(+ (* c c) (* d d))

simplify2.4s

Counts
40 → 79
Calls

40 calls. Slowest were:

433.0ms
(* (- (* (* a c) (* a c)) (* (* b d) (* b d))) 1)
384.0ms
(* (+ (* (* a c) (* a c)) (- (* (* b d) (* b d)) (* (* a c) (* b d)))) (+ (* c c) (* d d)))
326.0ms
(* (+ (pow (* a c) 3) (pow (* b d) 3)) 1)

prune867.0ms

Pruning

11 alts after pruning (8 fresh and 3 done)

Merged error: 21.0b

regimes461.0ms

Accuracy

0% (2.7b remaining)

Error of 25.5b against oracle of 22.8b and baseline of 25.5b

bsearch3.0ms