Average Error: 43.8 → 0.2
Time: 29.6s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{\left(a \cdot -4\right) \cdot c}{2 \cdot a}}{b + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 43.8

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

  2. Simplified43.8

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

  4. Applied flip--43.8

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

  5. Applied associate-/l/43.8

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

  6. Simplified0.4

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

  8. Applied associate-/r*0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))

Details

Time bar (total: 28.7s)Debug log

sample162.0ms

Algorithm
intervals
Results
45.0ms260×body80nan
43.0ms150×body160valid
27.0ms516×pre80true
19.0ms42×body320valid
9.0ms64×body80valid

simplify66.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
66.0ms
(/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))

prune24.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 42.8b

localize34.0ms

Local error

Found 4 expressions with local error:

38.6b
(- (sqrt (fma (* a -4) c (* b b))) b)
0.2b
(/ (- (sqrt (fma (* a -4) c (* b b))) b) (* 2 a))
0.1b
(fma (* a -4) c (* b b))
0.1b
(sqrt (fma (* a -4) c (* b b)))

rewrite23.0ms

Algorithm
rewrite-expression-head
Rules
10×add-sqr-sqrt
9×add-cube-cbrt
9×*-un-lft-identity
6×sqrt-prod
6×add-exp-log
6×fma-neg
6×add-cbrt-cube
4×add-log-exp
4×log1p-expm1-u
4×pow1
4×expm1-log1p-u
3×associate-/l*
3×times-frac
2×associate-/l/
2×flip--
2×flip3--
1×div-inv
1×fma-udef
1×div-exp
1×div-sub
1×pow1/2
1×frac-2neg
1×sub-neg
1×associate-/r*
1×clear-num
1×rem-sqrt-square
1×cbrt-undiv
Counts
4 → 67
Calls
4 calls:
Slowest
11.0ms
(/ (- (sqrt (fma (* a -4) c (* b b))) b) (* 2 a))
10.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
1.0ms
(sqrt (fma (* a -4) c (* b b)))
0.0ms
(fma (* a -4) c (* b b))

series220.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
96.0ms
(/ (- (sqrt (fma (* a -4) c (* b b))) b) (* 2 a))
57.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
39.0ms
(sqrt (fma (* a -4) c (* b b)))
28.0ms
(fma (* a -4) c (* b b))

simplify1.2s

Counts
34 → 79
Calls
34 calls:
Slowest
594.0ms
(- (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (* b b))
92.0ms
(- (pow (sqrt (fma (* a -4) c (* b b))) 3) (pow b 3))
62.0ms
(/ (* (* (- (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)))
41.0ms
(- (log (- (sqrt (fma (* a -4) c (* b b))) b)) (log (* 2 a)))
37.0ms
(/ (* (cbrt (- (sqrt (fma (* a -4) c (* b b))) b)) (cbrt (- (sqrt (fma (* a -4) c (* b b))) b))) 2)

prune791.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.4b

localize10.0ms

Local error

Found 4 expressions with local error:

0.4b
(/ (* (* a -4) c) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)))
0.1b
(+ (sqrt (fma (* a -4) c (* b b))) b)
0.1b
(fma (* a -4) c (* b b))
0.1b
(* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))

rewrite38.0ms

Algorithm
rewrite-expression-head
Rules
11×add-exp-log
11×add-cbrt-cube
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×fma-def
6×pow1
4×add-log-exp
4×log1p-expm1-u
4×associate-*r/
4×expm1-log1p-u
3×flip-+
3×sqrt-prod
3×associate-*r*
3×flip3-+
2×cbrt-unprod
2×associate-/r/
2×prod-exp
2×div-exp
2×cbrt-undiv
1×associate-/l*
1×div-inv
1×fma-udef
1×*-commutative
1×pow-prod-down
1×associate-*l*
1×times-frac
1×distribute-lft-in
1×frac-2neg
1×associate-/r*
1×clear-num
1×+-commutative
1×distribute-rgt-in
Counts
4 → 70
Calls
4 calls:
Slowest
16.0ms
(* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))
13.0ms
(/ (* (* a -4) c) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)))
7.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
0.0ms
(fma (* a -4) c (* b b))

series322.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
141.0ms
(/ (* (* a -4) c) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)))
81.0ms
(* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))
65.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
35.0ms
(fma (* a -4) c (* b b))

simplify3.4s

Counts
37 → 82
Calls
37 calls:
Slowest
663.0ms
(/ (* (* (* (* 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))))
626.0ms
(- (log (* (* a -4) c)) (+ (log (* 2 a)) (log (+ (sqrt (fma (* a -4) c (* b b))) b))))
579.0ms
(/ (* (* (* (* 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))))
401.0ms
(- (log (* (* a -4) c)) (log (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))))
219.0ms
(+ (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (- (* b b) (* (sqrt (fma (* a -4) c (* b b))) b)))

prune937.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize6.0ms

Local error

Found 4 expressions with local error:

0.1b
(+ (sqrt (fma (* a -4) c (* b b))) b)
0.1b
(fma (* a -4) c (* b b))
0.1b
(/ (/ (* (* a -4) c) (* 2 a)) (+ (sqrt (fma (* a -4) c (* b b))) b))
0.1b
(/ (* (* a -4) c) (* 2 a))

rewrite34.0ms

Algorithm
rewrite-expression-head
Rules
20×times-frac
16×add-cube-cbrt
16×*-un-lft-identity
16×add-sqr-sqrt
11×add-exp-log
11×add-cbrt-cube
6×associate-/l*
6×div-inv
6×fma-def
4×add-log-exp
4×log1p-expm1-u
4×div-exp
4×associate-/r*
4×cbrt-undiv
4×pow1
4×expm1-log1p-u
3×sqrt-prod
2×flip-+
2×associate-/r/
2×flip3-+
2×frac-2neg
2×clear-num
1×associate-/l/
1×fma-udef
1×+-commutative
Counts
4 → 87
Calls
4 calls:
Slowest
19.0ms
(/ (/ (* (* a -4) c) (* 2 a)) (+ (sqrt (fma (* a -4) c (* b b))) b))
10.0ms
(/ (* (* a -4) c) (* 2 a))
4.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
0.0ms
(fma (* a -4) c (* b b))

series226.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
122.0ms
(/ (/ (* (* a -4) c) (* 2 a)) (+ (sqrt (fma (* a -4) c (* b b))) b))
59.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
35.0ms
(fma (* a -4) c (* b b))
11.0ms
(/ (* (* a -4) c) (* 2 a))

simplify11.3s

Counts
66 → 99
Calls
66 calls:
Slowest
622.0ms
(/ (/ (* (* (* (* 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)))
611.0ms
(+ (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (- (* b b) (* (sqrt (fma (* a -4) c (* b b))) b)))
571.0ms
(/ (* (* (/ (* (* a -4) c) (* 2 a)) (/ (* (* a -4) c) (* 2 a))) (/ (* (* a -4) c) (* 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)))
505.0ms
(- (log (* (* a -4) c)) (log (* 2 a)))
473.0ms
(sqrt (/ (* (* a -4) c) (* 2 a)))

prune1.0s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0b

localize12.0ms

Local error

Found 4 expressions with local error:

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

rewrite39.0ms

Algorithm
rewrite-expression-head
Rules
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
6×add-exp-log
6×add-cbrt-cube
6×fma-def
4×add-log-exp
4×div-inv
4×log1p-expm1-u
4×times-frac
4×associate-/r*
4×pow1
4×expm1-log1p-u
3×sqrt-prod
2×associate-/l/
2×associate-/l*
2×flip-+
2×associate-*r/
2×flip3-+
2×frac-2neg
2×clear-num
1×fma-udef
1×associate-/r/
1×div-exp
1×+-commutative
1×cbrt-undiv
Counts
4 → 67
Calls
4 calls:
Slowest
18.0ms
(/ (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)) c)
15.0ms
(/ (* a -4) (/ (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)) c))
6.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
0.0ms
(fma (* a -4) c (* b b))

series324.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
115.0ms
(/ (* a -4) (/ (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)) c))
113.0ms
(/ (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)) c)
56.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
40.0ms
(fma (* a -4) c (* b b))

simplify2.6s

Counts
34 → 79
Calls
34 calls:
Slowest
296.0ms
(/ -4 (/ (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)) c))
276.0ms
(- (log (* a -4)) (log (/ (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)) c)))
219.0ms
(/ (* (* (* a -4) (* a -4)) (* a -4)) (* (* (/ (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)) c) (/ (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)) c)) (/ (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)) c)))
208.0ms
(/ a (* (cbrt (/ (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)) c)) (cbrt (/ (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)) c))))
180.0ms
(/ -4 (cbrt (/ (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)) c)))

prune804.0ms

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0b

regimes189.0ms

Accuracy

0% (0.2b remaining)

Error of 0.2b against oracle of 0.0b and baseline of 0.2b

bsearch2.0ms

end0.0ms

sample4.9s

Algorithm
intervals
Results
1.4s8020×body80nan
1.3s4851×body160valid
983.0ms16020×pre80true
609.0ms1350×body320valid
218.0ms1799×body80valid