Average Error: 34.2 → 13.2
Time: 22.3s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 7.319882445655832 \cdot 10^{-308}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 3.1626152470538816 \cdot 10^{+114}:\\ \;\;\;\;-2 \cdot \frac{c}{(\left(\sqrt{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\right) + b)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b + b} \cdot -2\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < 7.319882445655832e-308

    1. Initial program 21.9

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

    2. Initial simplification21.9

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

    3. Taylor expanded around -inf 21.9

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

    4. Simplified21.9

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

    if 7.319882445655832e-308 < b < 3.1626152470538816e+114

    1. Initial program 34.0

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

    2. Initial simplification34.0

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

    4. Applied flip--34.1

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

    5. Applied associate-/l/38.3

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

    6. Simplified21.4

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

    8. Applied times-frac8.9

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

    9. Simplified8.9

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

    11. Applied add-sqr-sqrt9.1

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

    12. Applied fma-def9.0

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

    if 3.1626152470538816e+114 < b

    1. Initial program 59.4

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

    2. Initial simplification59.4

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

    4. Applied flip--59.4

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

    5. Applied associate-/l/59.5

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

    6. Simplified33.3

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

    8. Applied times-frac31.7

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

    9. Simplified31.7

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

    10. Taylor expanded around 0 1.8

      \[\leadsto -2 \cdot \frac{c}{\color{blue}{b} + b}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification13.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 7.319882445655832 \cdot 10^{-308}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 3.1626152470538816 \cdot 10^{+114}:\\ \;\;\;\;-2 \cdot \frac{c}{(\left(\sqrt{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\right) + b)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b + b} \cdot -2\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))

Details

Time bar (total: 13.5s)Debug log

start148.0ms

Algorithm
intervals

setup110.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 34.6b

localize22.0ms

Local error

Found 4 expressions with local error:

22.5b
(sqrt (fma (* a -4) c (* b b)))
0.9b
(- (sqrt (fma (* a -4) c (* b b))) b)
0.1b
(/ (- (sqrt (fma (* a -4) c (* b b))) b) (* 2 a))
0.0b
(fma (* a -4) c (* b b))

rewrite31.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

15.0ms
(/ (- (sqrt (fma (* a -4) c (* b b))) b) (* 2 a))
14.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
1.0ms
(sqrt (fma (* a -4) c (* b b)))

series223.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

77.0ms
(/ (- (sqrt (fma (* a -4) c (* b b))) b) (* 2 a))
63.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
47.0ms
(sqrt (fma (* a -4) c (* b b)))
35.0ms
(fma (* a -4) c (* b b))

simplify1.1s

Counts
34 → 79
Calls

34 calls. Slowest were:

469.0ms
(- (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (* b b))
130.0ms
(- (pow (sqrt (fma (* a -4) c (* b b))) 3) (pow b 3))
107.0ms
(/ (* (* (- (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)))

prune1.1s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 20.9b

localize21.0ms

Local error

Found 4 expressions with local error:

22.5b
(sqrt (fma (* a -4) c (* b b)))
13.3b
(/ (* (* a -4) c) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)))
0.8b
(+ (sqrt (fma (* a -4) c (* b b))) b)
0.1b
(* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))

rewrite42.0ms

Algorithm
rewrite-expression-head
Counts
4 → 74
Calls

4 calls. Slowest were:

18.0ms
(/ (* (* a -4) c) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)))
15.0ms
(* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))
7.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)

series341.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

126.0ms
(/ (* (* a -4) c) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)))
107.0ms
(* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))
58.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
50.0ms
(sqrt (fma (* a -4) c (* b b)))

simplify3.4s

Counts
41 → 86
Calls

41 calls. Slowest were:

706.0ms
(- (log (* (* a -4) c)) (+ (log (* 2 a)) (log (+ (sqrt (fma (* a -4) c (* b b))) b))))
653.0ms
(/ (* (* (* (* 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))))
634.0ms
(/ (* (* (* (* 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))))

prune1.4s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 13.2b

localize7.0ms

Local error

Found 4 expressions with local error:

22.5b
(sqrt (fma (* a -4) c (* b b)))
0.8b
(+ (sqrt (fma (* a -4) c (* b b))) b)
0.1b
(/ c (+ (sqrt (fma (* a -4) c (* b b))) b))
0.0b
(fma (* a -4) c (* b b))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 59
Calls

4 calls. Slowest were:

4.0ms
(+ (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)))

series332.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

175.0ms
(/ c (+ (sqrt (fma (* a -4) c (* b b))) b))
57.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
51.0ms
(sqrt (fma (* a -4) c (* b b)))
49.0ms
(fma (* a -4) c (* b b))

simplify554.0ms

Counts
26 → 71
Calls

26 calls. Slowest were:

257.0ms
(+ (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (- (* b b) (* (sqrt (fma (* a -4) c (* b b))) b)))
50.0ms
(- (pow b 2) (* 4 (* a c)))
34.0ms
(- (pow b 2) (* 4 (* a c)))

prune1.1s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 12.0b

localize13.0ms

Local error

Found 4 expressions with local error:

22.5b
(sqrt (fma (* a -4) c (* b b)))
22.5b
(sqrt (fma (* a -4) c (* b b)))
17.4b
(fma (sqrt (sqrt (fma (* a -4) c (* b b)))) (sqrt (sqrt (fma (* a -4) c (* b b)))) b)
0.1b
(/ c (fma (sqrt (sqrt (fma (* a -4) c (* b b)))) (sqrt (sqrt (fma (* a -4) c (* b b)))) b))

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
4 → 53
Calls

4 calls. Slowest were:

1.0ms
(sqrt (fma (* a -4) c (* b b)))
1.0ms
(sqrt (fma (* a -4) c (* b b)))
0.0ms
(/ c (fma (sqrt (sqrt (fma (* a -4) c (* b b)))) (sqrt (sqrt (fma (* a -4) c (* b b)))) b))

series709.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

319.0ms
(fma (sqrt (sqrt (fma (* a -4) c (* b b)))) (sqrt (sqrt (fma (* a -4) c (* b b)))) b)
277.0ms
(/ c (fma (sqrt (sqrt (fma (* a -4) c (* b b)))) (sqrt (sqrt (fma (* a -4) c (* b b)))) b))
57.0ms
(sqrt (fma (* a -4) c (* b b)))
57.0ms
(sqrt (fma (* a -4) c (* b b)))

simplify928.0ms

Counts
23 → 65
Calls

23 calls. Slowest were:

229.0ms
(- (/ c b) (+ (* 1/4 (/ (pow (exp (* 1/4 (- (log -4) (+ (log (/ -1 c)) (log (/ -1 a)))))) 4) (* a b))) (/ (* c (pow (exp (* 1/4 (- (log -4) (+ (log (/ -1 c)) (log (/ -1 a)))))) 2)) (pow b 2))))
219.0ms
(+ (pow (exp (* 1/4 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c)))))) 2) b)
188.0ms
(- (/ c b) (+ (/ (* (pow (exp (* 1/4 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c)))))) 2) c) (pow b 2)) (* 1/4 (/ (pow (exp (* 1/4 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c)))))) 4) (* a b)))))

prune1.1s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 12.0b

regimes315.0ms

Accuracy

96.8% (0.5b remaining)

Error of 13.2b against oracle of 12.7b and baseline of 29.8b

bsearch530.0ms