Average Error: 33.2 → 15.4
Time: 32.2s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2.2808806815259953 \cdot 10^{-214}:\\ \;\;\;\;\left(\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.544733698813264 \cdot 10^{+98}:\\ \;\;\;\;\frac{\left(\left(a \cdot c\right) \cdot -4\right) \cdot \frac{\frac{1}{2}}{a}}{b + \sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-b}{c}}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < 2.2808806815259953e-214

    1. Initial program 20.7

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

    2. Initial simplification20.6

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

    4. Applied *-un-lft-identity20.6

      \[\leadsto \frac{\color{blue}{1 \cdot \left(\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b\right)}}{2 \cdot a}\]

    5. Applied associate-/l*20.8

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}}}\]
    6. Using strategy rm

    7. Applied div-inv20.8

      \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot a\right) \cdot \frac{1}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}}}\]

    8. Applied associate-/r*20.8

      \[\leadsto \color{blue}{\frac{\frac{1}{2 \cdot a}}{\frac{1}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}}}\]
    9. Using strategy rm

    10. Applied associate-/r/20.8

      \[\leadsto \color{blue}{\frac{\frac{1}{2 \cdot a}}{1} \cdot \left(\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b\right)}\]

    if 2.2808806815259953e-214 < b < 6.544733698813264e+98

    1. Initial program 34.4

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

    2. Initial simplification34.4

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

    4. Applied *-un-lft-identity34.4

      \[\leadsto \frac{\color{blue}{1 \cdot \left(\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b\right)}}{2 \cdot a}\]

    5. Applied associate-/l*34.4

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}}}\]
    6. Using strategy rm

    7. Applied div-inv34.4

      \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot a\right) \cdot \frac{1}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}}}\]

    8. Applied associate-/r*34.4

      \[\leadsto \color{blue}{\frac{\frac{1}{2 \cdot a}}{\frac{1}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}}}\]
    9. Using strategy rm

    10. Applied associate-/r/34.4

      \[\leadsto \color{blue}{\frac{\frac{1}{2 \cdot a}}{1} \cdot \left(\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b\right)}\]
    11. Using strategy rm

    12. Applied flip--34.5

      \[\leadsto \frac{\frac{1}{2 \cdot a}}{1} \cdot \color{blue}{\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b \cdot b}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + b}}\]

    13. Applied frac-times34.5

      \[\leadsto \color{blue}{\frac{\frac{1}{2 \cdot a} \cdot \left(\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b \cdot b\right)}{1 \cdot \left(\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + b\right)}}\]

    14. Simplified15.6

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{a} \cdot \left(-4 \cdot \left(c \cdot a\right)\right)}}{1 \cdot \left(\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + b\right)}\]

    15. Simplified15.5

      \[\leadsto \frac{\frac{\frac{1}{2}}{a} \cdot \left(-4 \cdot \left(c \cdot a\right)\right)}{\color{blue}{\sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} + b}}\]

    if 6.544733698813264e+98 < b

    1. Initial program 59.1

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

    2. Initial simplification59.1

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

    4. Applied *-un-lft-identity59.1

      \[\leadsto \frac{\color{blue}{1 \cdot \left(\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b\right)}}{2 \cdot a}\]

    5. Applied associate-/l*59.2

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}}}\]

    6. Taylor expanded around 0 3.6

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{b}{c}}}\]

    7. Simplified3.6

      \[\leadsto \frac{1}{\color{blue}{\frac{-b}{c}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification15.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2.2808806815259953 \cdot 10^{-214}:\\ \;\;\;\;\left(\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.544733698813264 \cdot 10^{+98}:\\ \;\;\;\;\frac{\left(\left(a \cdot c\right) \cdot -4\right) \cdot \frac{\frac{1}{2}}{a}}{b + \sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-b}{c}}\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 +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: 25.1s)Debug log

start178.0ms

Algorithm
intervals

setup137.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 30.9b

localize42.0ms

Local error

Found 4 expressions with local error:

21.8b
(sqrt (fma (* a -4) c (* b b)))
1.8b
(- (sqrt (fma (* a -4) c (* b b))) b)
0.4b
(/ (- (sqrt (fma (* a -4) c (* b b))) b) (* 2 a))
0.0b
(fma (* a -4) c (* b b))

rewrite41.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

28.0ms
(/ (- (sqrt (fma (* a -4) c (* b b))) b) (* 2 a))
10.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
2.0ms
(sqrt (fma (* a -4) c (* b b)))

series230.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

88.0ms
(/ (- (sqrt (fma (* a -4) c (* b b))) b) (* 2 a))
62.0ms
(sqrt (fma (* a -4) c (* b b)))
45.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
35.0ms
(fma (* a -4) c (* b b))

simplify1.4s

Counts
34 → 79
Calls

34 calls. Slowest were:

671.0ms
(- (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (* b b))
120.0ms
(- (pow (sqrt (fma (* a -4) c (* b b))) 3) (pow b 3))
88.0ms
(/ (* (* (- (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

8 alts after pruning (8 fresh and 0 done)

Merged error: 20.8b

localize10.0ms

Local error

Found 4 expressions with local error:

21.8b
(sqrt (fma (* a -4) c (* b b)))
1.8b
(- (sqrt (fma (* a -4) c (* b b))) b)
0.3b
(/ (* 2 a) (- (sqrt (fma (* a -4) c (* b b))) b))
0.2b
(/ 1 (/ (* 2 a) (- (sqrt (fma (* a -4) c (* b b))) b)))

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 82
Calls

4 calls. Slowest were:

7.0ms
(/ (* 2 a) (- (sqrt (fma (* a -4) c (* b b))) b))
5.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
3.0ms
(/ 1 (/ (* 2 a) (- (sqrt (fma (* a -4) c (* b b))) b)))

series783.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

548.0ms
(/ (* 2 a) (- (sqrt (fma (* a -4) c (* b b))) b))
115.0ms
(/ 1 (/ (* 2 a) (- (sqrt (fma (* a -4) c (* b b))) b)))
71.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
49.0ms
(sqrt (fma (* a -4) c (* b b)))

simplify1.5s

Counts
45 → 94
Calls

45 calls. Slowest were:

644.0ms
(/ (* (* (* 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)))
121.0ms
(+ (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (+ (* b b) (* (sqrt (fma (* a -4) c (* b b))) b)))
86.0ms
(+ (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (+ (* b b) (* (sqrt (fma (* a -4) c (* b b))) b)))

prune1.4s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 13.2b

localize9.0ms

Local error

Found 4 expressions with local error:

21.8b
(sqrt (fma (* a -4) c (* b b)))
1.8b
(- (sqrt (fma (* a -4) c (* b b))) b)
0.3b
(/ 1 (- (sqrt (fma (* a -4) c (* b b))) b))
0.3b
(/ (/ 1 (* 2 a)) (/ 1 (- (sqrt (fma (* a -4) c (* b b))) b)))

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 106
Calls

4 calls. Slowest were:

5.0ms
(/ (/ 1 (* 2 a)) (/ 1 (- (sqrt (fma (* a -4) c (* b b))) b)))
5.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
3.0ms
(/ 1 (- (sqrt (fma (* a -4) c (* b b))) b))

series336.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

117.0ms
(/ 1 (- (sqrt (fma (* a -4) c (* b b))) b))
91.0ms
(/ (/ 1 (* 2 a)) (/ 1 (- (sqrt (fma (* a -4) c (* b b))) b)))
65.0ms
(sqrt (fma (* a -4) c (* b b)))
63.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)

simplify8.8s

Counts
89 → 118
Calls

89 calls. Slowest were:

659.0ms
(/ (sqrt (/ 1 (* 2 a))) (/ 1 (- (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (* b b))))
631.0ms
(/ 1 (/ 1 (- (pow (sqrt (fma (* a -4) c (* b b))) 3) (pow b 3))))
602.0ms
(/ (/ 1 (* 2 a)) (+ (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (+ (* b b) (* (sqrt (fma (* a -4) c (* b b))) b))))

prune1.9s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 13.0b

localize12.0ms

Local error

Found 4 expressions with local error:

21.8b
(sqrt (fma (* a -4) c (* b b)))
1.8b
(- (sqrt (fma (* a -4) c (* b b))) b)
0.5b
(* (/ (/ 1 (* 2 a)) 1) (- (sqrt (fma (* a -4) c (* b b))) b))
0.0b
(fma (* a -4) c (* b b))

rewrite37.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

24.0ms
(* (/ (/ 1 (* 2 a)) 1) (- (sqrt (fma (* a -4) c (* b b))) b))
10.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
2.0ms
(sqrt (fma (* a -4) c (* b b)))

series276.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

117.0ms
(* (/ (/ 1 (* 2 a)) 1) (- (sqrt (fma (* a -4) c (* b b))) b))
67.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
48.0ms
(fma (* a -4) c (* b b))
43.0ms
(sqrt (fma (* a -4) c (* b b)))

simplify5.3s

Counts
41 → 84
Calls

41 calls. Slowest were:

1.3s
(* (* (* (/ (/ 1 (* 2 a)) 1) (/ (/ 1 (* 2 a)) 1)) (/ (/ 1 (* 2 a)) 1)) (* (* (- (sqrt (fma (* a -4) c (* b b))) b) (- (sqrt (fma (* a -4) c (* b b))) b)) (- (sqrt (fma (* a -4) c (* b b))) b)))
494.0ms
(* (/ (/ 1 (* 2 a)) 1) (- b))
472.0ms
(* 1 (+ (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (+ (* b b) (* (sqrt (fma (* a -4) c (* b b))) b))))

prune1.0s

Pruning

10 alts after pruning (8 fresh and 2 done)

Merged error: 13.0b

regimes388.0ms

Accuracy

83.9% (3.1b remaining)

Error of 15.4b against oracle of 12.4b and baseline of 31.3b

bsearch234.0ms