Average Error: 25.7 → 25.6
Time: 14.2s
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

Original25.7
Target0.4
Herbie25.6
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.7

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

  3. Applied add-sqr-sqrt25.7

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

  4. Applied associate-/r*25.6

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

  5. Final simplification25.6

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

Reproduce

herbie shell --seed 2019016 
(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: 12.7s)Debug log

sample35.0ms

Algorithm
intervals
Results
272×(pre true 80)
272×(body real 80)

simplify4.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
4.0ms
(/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 23.8b

localize28.0ms

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))

rewrite26.0ms

Algorithm
rewrite-expression-head
Rules
11×add-cube-cbrt
11×*-un-lft-identity
11×add-sqr-sqrt
9×times-frac
7×add-log-exp
5×add-exp-log
5×add-cbrt-cube
4×flip-+
4×flip3-+
3×associate-/l*
3×associate-/r*
3×pow1
2×associate-/l/
2×associate-/r/
2×sum-log
2×+-commutative
1×div-inv
1×div-exp
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
3 → 53
Calls
3 calls:
Slowest
16.0ms
(/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
4.0ms
(+ (* a c) (* b d))
4.0ms
(+ (* c c) (* d d))

series127.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
76.0ms
(/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
32.0ms
(+ (* a c) (* b d))
19.0ms
(+ (* c c) (* d d))

simplify827.0ms

Counts
40 → 62
Calls
40 calls:
Slowest
198.0ms
(- (* (* a c) (* a c)) (* (* b d) (* b d)))
158.0ms
(+ (* (* 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))))
87.0ms
(+ (pow (* a c) 3) (pow (* b d) 3))
43.0ms
(/ (* (cbrt (+ (* a c) (* b d))) (cbrt (+ (* a c) (* b d)))) 1)

prune592.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 23.4b

localize9.0ms

Local error

Found 4 expressions with local error:

28.4b
(sqrt (+ (* c c) (* d d)))
28.4b
(sqrt (+ (* c c) (* d d)))
13.7b
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
0.3b
(+ (* a c) (* b d))

rewrite35.0ms

Algorithm
rewrite-expression-head
Rules
23×add-sqr-sqrt
21×add-cube-cbrt
21×*-un-lft-identity
18×sqrt-prod
18×times-frac
6×add-log-exp
6×add-exp-log
6×sqrt-div
6×add-cbrt-cube
6×associate-/r*
5×flip-+
5×flip3-+
4×pow1
3×associate-/l*
2×associate-/l/
2×associate-/r/
2×pow1/2
2×rem-sqrt-square
1×div-inv
1×div-exp
1×frac-2neg
1×sum-log
1×clear-num
1×+-commutative
1×cbrt-undiv
Counts
4 → 82
Calls
4 calls:
Slowest
21.0ms
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
4.0ms
(+ (* a c) (* b d))
4.0ms
(sqrt (+ (* c c) (* d d)))
4.0ms
(sqrt (+ (* c c) (* d d)))

series191.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
91.0ms
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
38.0ms
(+ (* a c) (* b d))
31.0ms
(sqrt (+ (* c c) (* d d)))
31.0ms
(sqrt (+ (* c c) (* d d)))

simplify1.3s

Counts
75 → 94
Calls
75 calls:
Slowest
199.0ms
(- (* (* a c) (* a c)) (* (* b d) (* b d)))
194.0ms
(sqrt (+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d)))))
114.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))
108.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))
73.0ms
(+ (pow (* a c) 3) (pow (* b d) 3))

prune1.1s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 22.0b

localize11.0ms

Local error

Found 4 expressions with local error:

23.7b
(/ (+ (* c c) (* d d)) (+ (* a c) (* b d)))
0.3b
(/ 1 (/ (+ (* c c) (* d d)) (+ (* a c) (* b d))))
0.3b
(+ (* a c) (* b d))
0.0b
(+ (* c c) (* d d))

rewrite34.0ms

Algorithm
rewrite-expression-head
Rules
19×add-cube-cbrt
19×*-un-lft-identity
19×add-sqr-sqrt
18×times-frac
18×associate-/r*
9×add-exp-log
8×add-log-exp
6×add-cbrt-cube
5×flip-+
5×associate-/r/
5×flip3-+
5×pow1
3×associate-/l*
3×div-inv
2×associate-/l/
2×div-exp
2×frac-2neg
2×sum-log
2×clear-num
2×rec-exp
2×+-commutative
1×inv-pow
1×pow-flip
1×cbrt-undiv
Counts
4 → 83
Calls
4 calls:
Slowest
18.0ms
(/ (+ (* c c) (* d d)) (+ (* a c) (* b d)))
6.0ms
(/ 1 (/ (+ (* c c) (* d d)) (+ (* a c) (* b d))))
4.0ms
(+ (* a c) (* b d))
4.0ms
(+ (* c c) (* d d))

series215.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
104.0ms
(/ 1 (/ (+ (* c c) (* d d)) (+ (* a c) (* b d))))
68.0ms
(/ (+ (* c c) (* d d)) (+ (* a c) (* b d)))
27.0ms
(+ (* a c) (* b d))
16.0ms
(+ (* c c) (* d d))

simplify1.3s

Counts
60 → 95
Calls
60 calls:
Slowest
329.0ms
(+ (* (* a c) (* a c)) (- (* (* b d) (* b d)) (* (* a c) (* b d))))
286.0ms
(+ (* (* a c) (* a c)) (- (* (* b d) (* b d)) (* (* a c) (* b d))))
166.0ms
(- (* (* c c) (* c c)) (* (* d d) (* d d)))
143.0ms
(/ (* (* (+ (* c c) (* d d)) (+ (* c c) (* d d))) (+ (* c c) (* d d))) (* (* (+ (* a c) (* b d)) (+ (* a c) (* b d))) (+ (* a c) (* b d))))
50.0ms
(+ (pow (* c c) 3) (pow (* d d) 3))

prune1.0s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 22.0b

localize9.0ms

Local error

Found 4 expressions with local error:

28.4b
(sqrt (+ (* c c) (* d d)))
28.4b
(sqrt (+ (* c c) (* d d)))
13.7b
(/ (sqrt (+ (* c c) (* d d))) (+ (* a c) (* b d)))
0.3b
(/ 1 (* (sqrt (+ (* c c) (* d d))) (/ (sqrt (+ (* c c) (* d d))) (+ (* a c) (* b d)))))

rewrite23.0ms

Algorithm
rewrite-expression-head
Rules
23×add-sqr-sqrt
21×add-cube-cbrt
21×*-un-lft-identity
18×sqrt-prod
18×times-frac
12×add-exp-log
10×sqrt-div
7×associate-/r/
7×pow1
6×associate-/l*
6×flip-+
6×add-cbrt-cube
6×flip3-+
4×add-log-exp
4×associate-/r*
3×rec-exp
2×associate-/l/
2×pow-flip
2×div-inv
2×prod-exp
2×associate-*l/
2×div-exp
2×pow1/2
2×frac-2neg
2×clear-num
2×rem-sqrt-square
2×frac-times
1×inv-pow
1×associate-*r/
1×pow-prod-down
1×cbrt-undiv
Counts
4 → 93
Calls
4 calls:
Slowest
10.0ms
(/ 1 (* (sqrt (+ (* c c) (* d d))) (/ (sqrt (+ (* c c) (* d d))) (+ (* a c) (* b d)))))
8.0ms
(/ (sqrt (+ (* c c) (* d d))) (+ (* a c) (* b d)))
2.0ms
(sqrt (+ (* c c) (* d d)))
2.0ms
(sqrt (+ (* c c) (* d d)))

series230.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
104.0ms
(/ 1 (* (sqrt (+ (* c c) (* d d))) (/ (sqrt (+ (* c c) (* d d))) (+ (* a c) (* b d)))))
87.0ms
(/ (sqrt (+ (* c c) (* d d))) (+ (* a c) (* b d)))
20.0ms
(sqrt (+ (* c c) (* d d)))
19.0ms
(sqrt (+ (* c c) (* d d)))

simplify2.2s

Counts
82 → 105
Calls
82 calls:
Slowest
398.0ms
(+ (* (* a c) (* a c)) (- (* (* b d) (* b d)) (* (* a c) (* b d))))
187.0ms
(sqrt (+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d)))))
183.0ms
(* (sqrt (+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d))))) (+ (* a c) (* b d)))
176.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))
145.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))

prune1.5s

Pruning

10 alts after pruning (8 fresh and 2 done)

Merged error: 22.0b

regimes372.0ms

Accuracy

0% (2.6b remaining)

Error of 25.6b against oracle of 23.0b and baseline of 25.6b

bsearch3.0ms

end0.0ms

sample1.5s

Algorithm
intervals
Results
8493×(pre true 80)
8493×(body real 80)