Average Error: 26.1 → 26.0
Time: 11.2s
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

Original26.1
Target0.4
Herbie26.0
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.1

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

  3. Applied add-sqr-sqrt26.1

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

  4. Applied associate-/r*26.0

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

  5. Final simplification26.0

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

Reproduce

herbie shell --seed 2019022 
(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: 10.4s)Debug log

sample33.0ms

Algorithm
intervals
Results
23.0ms270×body80valid

simplify4.0ms

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

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 26.8b

localize24.0ms

Local error

Found 3 expressions with local error:

26.8b
(/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
0.7b
(+ (* a c) (* b d))
0.0b
(+ (* c c) (* d d))

rewrite29.0ms

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
13.0ms
(+ (* c c) (* d d))
13.0ms
(/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
3.0ms
(+ (* a c) (* b d))

series119.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
67.0ms
(/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
33.0ms
(+ (* a c) (* b d))
19.0ms
(+ (* c c) (* d d))

simplify750.0ms

Counts
40 → 62
Calls
40 calls:
Slowest
200.0ms
(- (* (* a c) (* a c)) (* (* b d) (* b d)))
148.0ms
(+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d))))
129.0ms
(/ (* (* (+ (* a c) (* b d)) (+ (* a c) (* b d))) (+ (* a c) (* b d))) (* (* (+ (* c c) (* d d)) (+ (* c c) (* d d))) (+ (* c c) (* d d))))
74.0ms
(+ (pow (* a c) 3) (pow (* b d) 3))
26.0ms
(/ (* (cbrt (+ (* a c) (* b d))) (cbrt (+ (* a c) (* b d)))) (* (cbrt (+ (* c c) (* d d))) (cbrt (+ (* c c) (* d d)))))

prune533.0ms

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 26.4b

localize8.0ms

Local error

Found 4 expressions with local error:

29.7b
(sqrt (+ (* c c) (* d d)))
29.7b
(sqrt (+ (* c c) (* d d)))
16.2b
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
0.7b
(+ (* a c) (* b d))

rewrite24.0ms

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
13.0ms
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
4.0ms
(sqrt (+ (* c c) (* d d)))
3.0ms
(+ (* a c) (* b d))
2.0ms
(sqrt (+ (* c c) (* d d)))

series162.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
80.0ms
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
33.0ms
(+ (* a c) (* b d))
26.0ms
(sqrt (+ (* c c) (* d d)))
23.0ms
(sqrt (+ (* c c) (* d d)))

simplify1.1s

Counts
75 → 94
Calls
75 calls:
Slowest
183.0ms
(- (* (* a c) (* a c)) (* (* b d) (* b d)))
125.0ms
(sqrt (+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d)))))
108.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))
91.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))
78.0ms
(/ (* (* (+ (* 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)))))

prune872.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 23.6b

localize16.0ms

Local error

Found 4 expressions with local error:

29.7b
(sqrt (+ (* c c) (* d d)))
29.7b
(sqrt (+ (* c c) (* d d)))
16.3b
(* (+ (* a c) (* b d)) (/ 1 (sqrt (+ (* c c) (* d d)))))
0.7b
(+ (* a c) (* b d))

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
10×add-sqr-sqrt
8×add-cube-cbrt
8×add-exp-log
8×*-un-lft-identity
6×add-log-exp
6×flip-+
6×sqrt-prod
6×associate-*r*
6×sqrt-div
6×add-cbrt-cube
6×flip3-+
6×pow1
3×associate-*l*
2×associate-/r/
2×prod-exp
2×associate-*l/
2×pow1/2
2×rem-sqrt-square
2×frac-times
1×div-inv
1×cbrt-unprod
1×*-commutative
1×un-div-inv
1×associate-*r/
1×pow-prod-down
1×sum-log
1×rec-exp
1×+-commutative
Counts
4 → 66
Calls
4 calls:
Slowest
6.0ms
(* (+ (* a c) (* b d)) (/ 1 (sqrt (+ (* c c) (* d d)))))
3.0ms
(sqrt (+ (* c c) (* d d)))
3.0ms
(sqrt (+ (* c c) (* d d)))
2.0ms
(+ (* a c) (* b d))

series149.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
75.0ms
(* (+ (* a c) (* b d)) (/ 1 (sqrt (+ (* c c) (* d d)))))
27.0ms
(+ (* a c) (* b d))
26.0ms
(sqrt (+ (* c c) (* d d)))
21.0ms
(sqrt (+ (* c c) (* d d)))

simplify1.8s

Counts
45 → 78
Calls
45 calls:
Slowest
333.0ms
(* (- (* (* a c) (* a c)) (* (* b d) (* b d))) 1)
274.0ms
(* (+ (pow (* a c) 3) (pow (* b d) 3)) 1)
156.0ms
(* (+ (* (* a c) (* a c)) (- (* (* b d) (* b d)) (* (* a c) (* b d)))) (sqrt (+ (* c c) (* d d))))
133.0ms
(sqrt (+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d)))))
126.0ms
(* (- (* (* a c) (* a c)) (* (* b d) (* b d))) (/ 1 (sqrt (+ (* c c) (* d d)))))

prune804.0ms

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 23.6b

localize4.0ms

Local error

Found 4 expressions with local error:

29.7b
(sqrt (+ (* c c) (* d d)))
29.7b
(sqrt (+ (* c c) (* d d)))
16.2b
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
0.7b
(+ (* a c) (* b d))

rewrite15.0ms

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
9.0ms
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
2.0ms
(+ (* a c) (* b d))
2.0ms
(sqrt (+ (* c c) (* d d)))
2.0ms
(sqrt (+ (* c c) (* d d)))

series149.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
81.0ms
(/ (+ (* a c) (* b d)) (sqrt (+ (* c c) (* d d))))
29.0ms
(+ (* a c) (* b d))
21.0ms
(sqrt (+ (* c c) (* d d)))
19.0ms
(sqrt (+ (* c c) (* d d)))

simplify1.2s

Counts
75 → 94
Calls
75 calls:
Slowest
213.0ms
(- (* (* a c) (* a c)) (* (* b d) (* b d)))
143.0ms
(sqrt (+ (* (* c c) (* c c)) (- (* (* d d) (* d d)) (* (* c c) (* d d)))))
117.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))
98.0ms
(sqrt (- (* (* c c) (* c c)) (* (* d d) (* d d))))
89.0ms
(/ (* (* (+ (* 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)))))

prune924.0ms

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 23.6b

regimes239.0ms

Accuracy

0% (2.6b remaining)

Error of 26.0b against oracle of 23.4b and baseline of 26.0b

bsearch2.0ms

end0.0ms

sample1.4s

Algorithm
intervals
Results
1.3s8447×body80valid