Average Error: 30.0 → 7.5
Time: 22.5s
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 +o rules:numerics
(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: 22.0s)Debug log

sample30.0ms

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

simplify179.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
179.0ms
(sqrt (fabs (/ (- (* a a) (* b b)) (* a a))))

prune14.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 7.1b

localize22.0ms

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)))

rewrite6.0ms

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×log1p-expm1-u
3×sqrt-prod
3×expm1-log1p-u
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 → 54
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)))))

series46.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
20.0ms
(- 1 (* (/ b a) (/ b a)))
13.0ms
(* (/ b a) (/ b a))
12.0ms
(sqrt (fabs (- 1 (* (/ b a) (/ b a)))))

simplify1.2s

Counts
28 → 63
Calls
28 calls:
Slowest
805.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
61.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
52.0ms
(sqrt (sqrt (fabs (- 1 (* (/ b a) (/ b a))))))
48.0ms
(sqrt (fabs (- 1 (/ (pow b 2) (pow a 2)))))
44.0ms
(sqrt (* (cbrt (fabs (- 1 (* (/ b a) (/ b a))))) (cbrt (fabs (- 1 (* (/ b a) (/ b a)))))))

prune658.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 7.1b

localize32.0ms

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)))))))

rewrite7.0ms

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×log1p-expm1-u
4×add-exp-log
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
3×flip--
3×sqrt-prod
3×pow1/3
3×flip3--
1×pow1/2
1×rem-sqrt-square
Counts
4 → 59
Calls
4 calls:
Slowest
2.0ms
(cbrt (- 1 (* (/ b a) (/ b a))))
2.0ms
(cbrt (- 1 (* (/ b a) (/ b a))))
2.0ms
(cbrt (- 1 (* (/ b a) (/ b a))))
2.0ms
(sqrt (fabs (* (* (cbrt (- 1 (* (/ b a) (/ b a)))) (cbrt (- 1 (* (/ b a) (/ b a))))) (cbrt (- 1 (* (/ b a) (/ b a)))))))

series430.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
159.0ms
(cbrt (- 1 (* (/ b a) (/ b a))))
134.0ms
(cbrt (- 1 (* (/ b a) (/ b a))))
118.0ms
(cbrt (- 1 (* (/ b a) (/ b a))))
19.0ms
(sqrt (fabs (* (* (cbrt (- 1 (* (/ b a) (/ b a)))) (cbrt (- 1 (* (/ b a) (/ b a))))) (cbrt (- 1 (* (/ b a) (/ b a)))))))

simplify4.0s

Counts
31 → 71
Calls
31 calls:
Slowest
408.0ms
(cbrt (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))))
366.0ms
(cbrt (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))))
332.0ms
(exp (* 1/3 (- (+ (* 2 (log (/ 1 a))) (log -1)) (* 2 (log (/ 1 b))))))
308.0ms
(exp (* 1/3 (- (+ (* 2 (log (/ 1 a))) (log -1)) (* 2 (log (/ 1 b))))))
297.0ms
(cbrt (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))))

prune887.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 7.1b

localize38.0ms

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))

rewrite28.0ms

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*
4×log1p-expm1-u
4×expm1-log1p-u
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 → 106
Calls
4 calls:
Slowest
15.0ms
(/ (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))) (+ 1 (* (/ b a) (/ b a))))
7.0ms
(* (/ b a) (/ b a))
3.0ms
(* (/ b a) (/ b a))
1.0ms
(sqrt (fabs (/ (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))) (+ 1 (* (/ b a) (/ b a))))))

series90.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
42.0ms
(/ (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))) (+ 1 (* (/ b a) (/ b a))))
19.0ms
(sqrt (fabs (/ (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))) (+ 1 (* (/ b a) (/ b a))))))
15.0ms
(* (/ b a) (/ b a))
14.0ms
(* (/ b a) (/ b a))

simplify11.3s

Counts
83 → 118
Calls
83 calls:
Slowest
911.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
814.0ms
(* (* (* (/ b a) (/ b a)) (/ b a)) (* (* (/ b a) (/ b a)) (/ b a)))
682.0ms
(sqrt (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))))
473.0ms
(/ (* (cbrt (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a))))) (cbrt (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a)))))) (* (cbrt (+ 1 (* (/ b a) (/ b a)))) (cbrt (+ 1 (* (/ b a) (/ b a))))))
460.0ms
(/ (sqrt (- (* 1 1) (* (* (/ b a) (/ b a)) (* (/ b a) (/ b a))))) (sqrt (+ 1 (* (/ b a) (/ b a)))))

prune1.5s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 7.1b

regimes103.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample1.4s

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