Average Error: 28.3 → 0.3
Time: 23.2s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[-2 \cdot \frac{c}{(\left(\sqrt{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{\sqrt[3]{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_* \cdot \left((\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_* \cdot (\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*\right)}}}\right) + b)_*}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.3

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

  2. Simplified28.3

    \[\leadsto \color{blue}{\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}{2 \cdot a}}\]
  3. Using strategy rm

  4. Applied flip--28.3

    \[\leadsto \frac{\color{blue}{\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b \cdot b}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + b}}}{2 \cdot a}\]

  5. Applied associate-/l/28.3

    \[\leadsto \color{blue}{\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b \cdot b}{\left(2 \cdot a\right) \cdot \left(\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + b\right)}}\]

  6. Simplified0.5

    \[\leadsto \frac{\color{blue}{\left(a \cdot -4\right) \cdot c}}{\left(2 \cdot a\right) \cdot \left(\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + b\right)}\]
  7. Using strategy rm

  8. Applied times-frac0.3

    \[\leadsto \color{blue}{\frac{a \cdot -4}{2 \cdot a} \cdot \frac{c}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + b}}\]

  9. Simplified0.3

    \[\leadsto \color{blue}{-2} \cdot \frac{c}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + b}\]
  10. Using strategy rm

  11. Applied add-cbrt-cube0.3

    \[\leadsto -2 \cdot \frac{c}{\sqrt{\color{blue}{\sqrt[3]{\left((\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_* \cdot (\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*\right) \cdot (\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}} + b}\]
  12. Using strategy rm

  13. Applied add-sqr-sqrt0.3

    \[\leadsto -2 \cdot \frac{c}{\sqrt{\color{blue}{\sqrt{\sqrt[3]{\left((\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_* \cdot (\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*\right) \cdot (\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt[3]{\left((\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_* \cdot (\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*\right) \cdot (\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}} + b}\]

  14. Applied sqrt-prod0.4

    \[\leadsto -2 \cdot \frac{c}{\color{blue}{\sqrt{\sqrt{\sqrt[3]{\left((\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_* \cdot (\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*\right) \cdot (\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}} \cdot \sqrt{\sqrt{\sqrt[3]{\left((\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_* \cdot (\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*\right) \cdot (\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}} + b}\]

  15. Applied fma-def0.4

    \[\leadsto -2 \cdot \frac{c}{\color{blue}{(\left(\sqrt{\sqrt{\sqrt[3]{\left((\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_* \cdot (\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*\right) \cdot (\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}\right) \cdot \left(\sqrt{\sqrt{\sqrt[3]{\left((\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_* \cdot (\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*\right) \cdot (\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}\right) + b)_*}}\]

  16. Simplified0.3

    \[\leadsto -2 \cdot \frac{c}{(\color{blue}{\left(\sqrt{\sqrt{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*}}\right)} \cdot \left(\sqrt{\sqrt{\sqrt[3]{\left((\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_* \cdot (\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*\right) \cdot (\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}\right) + b)_*}\]

  17. Final simplification0.3

    \[\leadsto -2 \cdot \frac{c}{(\left(\sqrt{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{\sqrt[3]{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_* \cdot \left((\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_* \cdot (\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*\right)}}}\right) + b)_*}\]

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))

Details

Time bar (total: 21.8s)Debug log

sample206.0ms

Algorithm
intervals

simplify87.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

86.0ms
(/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))

prune33.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 28.3b

localize21.0ms

Local error

Found 4 expressions with local error:

28.2b
(- (sqrt (fma (* a -4) c (* b b))) b)
0.3b
(/ (- (sqrt (fma (* a -4) c (* b b))) b) (* 2 a))
0.2b
(fma (* a -4) c (* b b))
0.1b
(sqrt (fma (* a -4) c (* b b)))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

12.0ms
(/ (- (sqrt (fma (* a -4) c (* b b))) b) (* 2 a))
5.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
2.0ms
(sqrt (fma (* a -4) c (* b b)))

series240.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

79.0ms
(/ (- (sqrt (fma (* a -4) c (* b b))) b) (* 2 a))
75.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
46.0ms
(fma (* a -4) c (* b b))
39.0ms
(sqrt (fma (* a -4) c (* b b)))

simplify1.4s

Counts
34 → 79
Calls

34 calls. Slowest were:

761.0ms
(- (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (* b b))
119.0ms
(- (pow (sqrt (fma (* a -4) c (* b b))) 3) (pow b 3))
99.0ms
(/ (* (* (- (sqrt (fma (* a -4) c (* b b))) b) (- (sqrt (fma (* a -4) c (* b b))) b)) (- (sqrt (fma (* a -4) c (* b b))) b)) (* (* (* 2 a) (* 2 a)) (* 2 a)))

prune894.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.5b

localize12.0ms

Local error

Found 4 expressions with local error:

0.4b
(/ (* (* a -4) c) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)))
0.3b
(+ (sqrt (fma (* a -4) c (* b b))) b)
0.2b
(fma (* a -4) c (* b b))
0.2b
(* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 70
Calls

4 calls. Slowest were:

9.0ms
(/ (* (* a -4) c) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)))
8.0ms
(* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))
4.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)

series329.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

120.0ms
(/ (* (* a -4) c) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)))
98.0ms
(* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))
79.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
33.0ms
(fma (* a -4) c (* b b))

simplify3.5s

Counts
37 → 82
Calls

37 calls. Slowest were:

790.0ms
(- (log (* (* a -4) c)) (+ (log (* 2 a)) (log (+ (sqrt (fma (* a -4) c (* b b))) b))))
676.0ms
(/ (* (* (* (* a -4) c) (* (* a -4) c)) (* (* a -4) c)) (* (* (* (* 2 a) (* 2 a)) (* 2 a)) (* (* (+ (sqrt (fma (* a -4) c (* b b))) b) (+ (sqrt (fma (* a -4) c (* b b))) b)) (+ (sqrt (fma (* a -4) c (* b b))) b))))
584.0ms
(/ (* (* (* (* a -4) c) (* (* a -4) c)) (* (* a -4) c)) (* (* (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))))

prune1.2s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

0.3b
(+ (sqrt (fma (* a -4) c (* b b))) b)
0.2b
(fma (* a -4) c (* b b))
0.2b
(/ c (+ (sqrt (fma (* a -4) c (* b b))) b))
0.1b
(sqrt (fma (* a -4) c (* b b)))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 59
Calls

4 calls. Slowest were:

7.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
2.0ms
(/ c (+ (sqrt (fma (* a -4) c (* b b))) b))
1.0ms
(sqrt (fma (* a -4) c (* b b)))

series297.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

133.0ms
(/ c (+ (sqrt (fma (* a -4) c (* b b))) b))
67.0ms
(sqrt (fma (* a -4) c (* b b)))
64.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
33.0ms
(fma (* a -4) c (* b b))

simplify651.0ms

Counts
26 → 71
Calls

26 calls. Slowest were:

245.0ms
(+ (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (- (* b b) (* (sqrt (fma (* a -4) c (* b b))) b)))
49.0ms
(- (pow b 2) (* 4 (* a c)))
47.0ms
(- (pow b 2) (* 4 (* a c)))

prune768.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0b

localize17.0ms

Local error

Found 4 expressions with local error:

0.5b
(cbrt (* (* (fma (* a -4) c (* b b)) (fma (* a -4) c (* b b))) (fma (* a -4) c (* b b))))
0.5b
(* (fma (* a -4) c (* b b)) (fma (* a -4) c (* b b)))
0.4b
(* (* (fma (* a -4) c (* b b)) (fma (* a -4) c (* b b))) (fma (* a -4) c (* b b)))
0.3b
(+ (sqrt (cbrt (* (* (fma (* a -4) c (* b b)) (fma (* a -4) c (* b b))) (fma (* a -4) c (* b b))))) b)

rewrite41.0ms

Algorithm
rewrite-expression-head
Counts
4 → 84
Calls

4 calls. Slowest were:

15.0ms
(* (* (fma (* a -4) c (* b b)) (fma (* a -4) c (* b b))) (fma (* a -4) c (* b b)))
11.0ms
(cbrt (* (* (fma (* a -4) c (* b b)) (fma (* a -4) c (* b b))) (fma (* a -4) c (* b b))))
8.0ms
(* (fma (* a -4) c (* b b)) (fma (* a -4) c (* b b)))

series244.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

102.0ms
(* (* (fma (* a -4) c (* b b)) (fma (* a -4) c (* b b))) (fma (* a -4) c (* b b)))
60.0ms
(* (fma (* a -4) c (* b b)) (fma (* a -4) c (* b b)))
49.0ms
(+ (sqrt (cbrt (* (* (fma (* a -4) c (* b b)) (fma (* a -4) c (* b b))) (fma (* a -4) c (* b b))))) b)
34.0ms
(cbrt (* (* (fma (* a -4) c (* b b)) (fma (* a -4) c (* b b))) (fma (* a -4) c (* b b))))

simplify6.1s

Counts
55 → 96
Calls

55 calls. Slowest were:

1.1s
(* (* (* (* (fma (* a -4) c (* b b)) (fma (* a -4) c (* b b))) (fma (* a -4) c (* b b))) (* (* (fma (* a -4) c (* b b)) (fma (* a -4) c (* b b))) (fma (* a -4) c (* b b)))) (* (* (fma (* a -4) c (* b b)) (fma (* a -4) c (* b b))) (fma (* a -4) c (* b b))))
719.0ms
(* (* (fma (* a -4) c (* b b)) (fma (* a -4) c (* b b))) (* b b))
497.0ms
(* (cbrt (sqrt (cbrt (* (* (fma (* a -4) c (* b b)) (fma (* a -4) c (* b b))) (fma (* a -4) c (* b b)))))) (cbrt (sqrt (cbrt (* (* (fma (* a -4) c (* b b)) (fma (* a -4) c (* b b))) (fma (* a -4) c (* b b)))))))

prune1.0s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0b

regimes197.0ms

Accuracy

0% (0.3b remaining)

Error of 0.3b against oracle of 0.0b and baseline of 0.3b

bsearch4.0ms

end0.0ms

sample4.4s

Algorithm
intervals