Average Error: 30.0 → 7.5
Time: 20.9s
Precision: 64
Internal Precision: 128
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.0

    \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]

  2. Simplified7.5

    \[\leadsto \color{blue}{\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}}\]

  3. Final simplification7.5

    \[\leadsto \sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]

Reproduce

herbie shell --seed 2019016 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))

Details

Time bar (total: 20.5s)Debug log

sample27.0ms

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

simplify132.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
132.0ms
(sqrt (fabs (/ (- (* a a) (* b b)) (* a a))))

prune13.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 7.1b

localize26.0ms

Local error

Found 3 expressions with local error:

7.2b
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))
0.4b
(* (/ b a) (/ b a))
0.0b
(- 1 (* (/ b a) (/ b a)))

rewrite5.0ms

Algorithm
rewrite-expression-head
Rules
8×pow1
7×add-sqr-sqrt
6×add-cube-cbrt
6×*-un-lft-identity
5×add-exp-log
5×add-cbrt-cube
4×associate-*l*
4×associate-*r*
3×add-log-exp
3×sqrt-prod
2×div-inv
1×flip--
1×cbrt-unprod
1×pow-prod-up
1×*-commutative
1×associate-*r/
1×prod-exp
1×associate-*l/
1×pow-prod-down
1×pow1/2
1×flip3--
1×pow-plus
1×sub-neg
1×rem-sqrt-square
1×frac-times
1×pow2
Counts
3 → 48
Calls
3 calls:
Slowest
3.0ms
(* (/ b a) (/ b a))
1.0ms
(- 1 (* (/ b a) (/ b a)))
1.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

series63.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
30.0ms
(- 1 (* (/ b a) (/ b a)))
17.0ms
(* (/ b a) (/ b a))
16.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

simplify1.4s

Counts
28 → 57
Calls
28 calls:
Slowest
955.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
70.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
66.0ms
(sqrt (* (cbrt (fabs (- 1 (* (/ b a) (/ b a))))) (cbrt (fabs (- 1 (* (/ b a) (/ b a)))))))
65.0ms
(sqrt (sqrt (fabs (- 1 (* (/ b a) (/ b a))))))
62.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))

prune534.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 7.1b

localize14.0ms

Local error

Found 4 expressions with local error:

7.4b
(cbrt (- 1 (* (/ b a) (/ b a))))
7.4b
(cbrt (- 1 (* (/ b a) (/ b a))))
7.4b
(cbrt (- 1 (* (/ b a) (/ b a))))
7.2b
(sqrt (fabs (* (* (cbrt (- 1 (* (/ b a) (/ b a)))) (cbrt (- 1 (* (/ b a) (/ b a))))) (cbrt (- 1 (* (/ b a) (/ b a)))))))

rewrite3.0ms

Algorithm
rewrite-expression-head
Rules
9×add-sqr-sqrt
9×cbrt-prod
8×add-cube-cbrt
8×*-un-lft-identity
6×cbrt-div
4×add-log-exp
4×add-exp-log
4×add-cbrt-cube
4×pow1
3×flip--
3×sqrt-prod
3×pow1/3
3×flip3--
1×pow1/2
1×rem-sqrt-square
Counts
4 → 51
Calls
4 calls:
Slowest
1.0ms
(cbrt (- 1 (* (/ b a) (/ b a))))
1.0ms
(cbrt (- 1 (* (/ b a) (/ b a))))
1.0ms
(cbrt (- 1 (* (/ b a) (/ b a))))
1.0ms
(sqrt (fabs (* (* (cbrt (- 1 (* (/ b a) (/ b a)))) (cbrt (- 1 (* (/ b a) (/ b a))))) (cbrt (- 1 (* (/ b a) (/ b a)))))))

series448.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
148.0ms
(cbrt (- 1 (* (/ b a) (/ b a))))
143.0ms
(cbrt (- 1 (* (/ b a) (/ b a))))
143.0ms
(cbrt (- 1 (* (/ b a) (/ b a))))
14.0ms
(sqrt (fabs (* (* (cbrt (- 1 (* (/ b a) (/ b a)))) (cbrt (- 1 (* (/ b a) (/ b a))))) (cbrt (- 1 (* (/ b a) (/ b a)))))))

simplify3.6s

Counts
31 → 63
Calls
31 calls:
Slowest
373.0ms
(cbrt (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))))
368.0ms
(cbrt (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))))
318.0ms
(cbrt (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))))
274.0ms
(exp (* 1/3 (- (+ (* 2 (log (/ -1 a))) (log -1)) (* 2 (log (/ -1 b))))))
248.0ms
(exp (* 1/3 (- (+ (* 2 (log (/ -1 a))) (log -1)) (* 2 (log (/ -1 b))))))

prune1.2s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 7.1b

localize20.0ms

Local error

Found 4 expressions with local error:

18.5b
(/ (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))) (+ 1 (* (/ b a) (/ b a))))
7.2b
(sqrt (fabs (/ (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))) (+ 1 (* (/ b a) (/ b a))))))
0.4b
(* (/ b a) (/ b a))
0.4b
(* (/ b a) (/ b a))

rewrite25.0ms

Algorithm
rewrite-expression-head
Rules
19×add-sqr-sqrt
18×add-cube-cbrt
18×*-un-lft-identity
14×pow1
12×times-frac
10×add-exp-log
10×add-cbrt-cube
8×associate-*l*
8×associate-*r*
5×div-inv
4×difference-of-squares
4×add-log-exp
4×associate-/l*
3×sqrt-prod
3×associate-/r*
2×associate-/l/
2×cbrt-unprod
2×pow-prod-up
2×*-commutative
2×associate-/r/
2×associate-*r/
2×prod-exp
2×associate-*l/
2×pow-prod-down
2×pow-plus
2×frac-times
2×pow2
1×flip--
1×flip-+
1×div-exp
1×div-sub
1×pow1/2
1×flip3--
1×flip3-+
1×frac-2neg
1×clear-num
1×rem-sqrt-square
1×cbrt-undiv
Counts
4 → 98
Calls
4 calls:
Slowest
10.0ms
(/ (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))) (+ 1 (* (/ b a) (/ b a))))
6.0ms
(* (/ b a) (/ b a))
6.0ms
(* (/ b a) (/ b a))
1.0ms
(sqrt (fabs (/ (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))) (+ 1 (* (/ b a) (/ b a))))))

series112.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
55.0ms
(/ (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))) (+ 1 (* (/ b a) (/ b a))))
22.0ms
(* (/ b a) (/ b a))
18.0ms
(sqrt (fabs (/ (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))) (+ 1 (* (/ b a) (/ b a))))))
17.0ms
(* (/ b a) (/ b a))

simplify10.6s

Counts
83 → 110
Calls
83 calls:
Slowest
827.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
765.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
586.0ms
(- (pow (* 1 1) 3) (pow (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a))) 3))
469.0ms
(* (cbrt (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a))))) (cbrt (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a))))))
437.0ms
(- (log (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a))))) (log (+ 1 (* (/ b a) (/ b a)))))

prune1.2s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 7.1b

regimes61.0ms

Accuracy

0% (0.0b remaining)

Error of 7.5b against oracle of 7.5b and baseline of 7.5b

bsearch1.0ms

end0.0ms

sample1.1s

Algorithm
intervals
Results
8000×(pre true 80)
8000×(body real 80)