Average Error: 32.7 → 12.9
Time: 18.6s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.02993411605426 \cdot 10^{-308}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 4.873574385931286 \cdot 10^{+105}:\\ \;\;\;\;\frac{-4 \cdot \left(c \cdot \frac{1}{2}\right)}{b + \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(c \cdot \frac{1}{2}\right)}{b + b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original32.7
Target20.4
Herbie12.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.02993411605426e-308

    1. Initial program 20.8

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

    2. Simplified20.8

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

    3. Taylor expanded around 0 20.8

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

    4. Simplified20.8

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

    if -1.02993411605426e-308 < b < 4.873574385931286e+105

    1. Initial program 31.6

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

    2. Simplified31.6

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

    4. Applied flip--31.6

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

    5. Applied associate-/l/35.7

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

    6. Simplified19.3

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

    8. Applied times-frac14.5

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

    9. Simplified8.8

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

    11. Applied associate-*r/8.6

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

    if 4.873574385931286e+105 < b

    1. Initial program 58.9

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

    2. Simplified58.9

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

    4. Applied flip--58.9

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

    5. Applied associate-/l/59.1

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

    6. Simplified32.1

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

    8. Applied times-frac31.3

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

    9. Simplified30.6

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

    11. Applied associate-*r/30.6

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

    12. Taylor expanded around 0 2.6

      \[\leadsto \frac{\left(\frac{1}{2} \cdot c\right) \cdot -4}{\color{blue}{b} + b}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification12.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.02993411605426 \cdot 10^{-308}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 4.873574385931286 \cdot 10^{+105}:\\ \;\;\;\;\frac{-4 \cdot \left(c \cdot \frac{1}{2}\right)}{b + \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(c \cdot \frac{1}{2}\right)}{b + b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

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

Details

Time bar (total: 17.2s)Debug log

sample138.0ms

Algorithm
intervals

simplify92.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

91.0ms
(/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a))

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 37.0b

localize26.0ms

Local error

Found 4 expressions with local error:

23.6b
(sqrt (fma (* a c) -4 (* b b)))
1.3b
(- (sqrt (fma (* a c) -4 (* b b))) b)
0.6b
(fma (* a c) -4 (* b b))
0.1b
(/ (- (sqrt (fma (* a c) -4 (* b b))) b) (* 2 a))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

13.0ms
(/ (- (sqrt (fma (* a c) -4 (* b b))) b) (* 2 a))
6.0ms
(- (sqrt (fma (* a c) -4 (* b b))) b)
1.0ms
(sqrt (fma (* a c) -4 (* b b)))

series241.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

75.0ms
(/ (- (sqrt (fma (* a c) -4 (* b b))) b) (* 2 a))
69.0ms
(sqrt (fma (* a c) -4 (* b b)))
55.0ms
(- (sqrt (fma (* a c) -4 (* b b))) b)
43.0ms
(fma (* a c) -4 (* b b))

simplify1.4s

Counts
34 → 79
Calls

34 calls. Slowest were:

715.0ms
(- (* (sqrt (fma (* a c) -4 (* b b))) (sqrt (fma (* a c) -4 (* b b)))) (* b b))
106.0ms
(- (pow (sqrt (fma (* a c) -4 (* b b))) 3) (pow b 3))
95.0ms
(/ (* (* (- (sqrt (fma (* a c) -4 (* b b))) b) (- (sqrt (fma (* a c) -4 (* b b))) b)) (- (sqrt (fma (* a c) -4 (* b b))) b)) (* (* (* 2 a) (* 2 a)) (* 2 a)))

prune907.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 22.1b

localize23.0ms

Local error

Found 4 expressions with local error:

23.6b
(sqrt (fma (* a c) -4 (* b b)))
14.2b
(/ (* (* a c) -4) (* (* 2 a) (+ (sqrt (fma (* a c) -4 (* b b))) b)))
1.7b
(+ (sqrt (fma (* a c) -4 (* b b))) b)
0.6b
(fma (* a c) -4 (* b b))

rewrite29.0ms

Algorithm
rewrite-expression-head
Counts
4 → 63
Calls

4 calls. Slowest were:

19.0ms
(/ (* (* a c) -4) (* (* 2 a) (+ (sqrt (fma (* a c) -4 (* b b))) b)))
7.0ms
(+ (sqrt (fma (* a c) -4 (* b b))) b)
2.0ms
(sqrt (fma (* a c) -4 (* b b)))

series287.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

124.0ms
(/ (* (* a c) -4) (* (* 2 a) (+ (sqrt (fma (* a c) -4 (* b b))) b)))
67.0ms
(sqrt (fma (* a c) -4 (* b b)))
66.0ms
(+ (sqrt (fma (* a c) -4 (* b b))) b)
29.0ms
(fma (* a c) -4 (* b b))

simplify3.0s

Counts
31 → 75
Calls

31 calls. Slowest were:

713.0ms
(/ (* (* (* (* a c) -4) (* (* a c) -4)) (* (* a c) -4)) (* (* (* (* 2 a) (+ (sqrt (fma (* a c) -4 (* b b))) b)) (* (* 2 a) (+ (sqrt (fma (* a c) -4 (* b b))) b))) (* (* 2 a) (+ (sqrt (fma (* a c) -4 (* b b))) b))))
674.0ms
(/ (* (* (* (* a c) -4) (* (* a c) -4)) (* (* a c) -4)) (* (* (* (* 2 a) (* 2 a)) (* 2 a)) (* (* (+ (sqrt (fma (* a c) -4 (* b b))) b) (+ (sqrt (fma (* a c) -4 (* b b))) b)) (+ (sqrt (fma (* a c) -4 (* b b))) b))))
570.0ms
(- (log (* (* a c) -4)) (+ (log (* 2 a)) (log (+ (sqrt (fma (* a c) -4 (* b b))) b))))

prune888.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 13.3b

localize19.0ms

Local error

Found 4 expressions with local error:

23.6b
(sqrt (fma (* a c) -4 (* b b)))
1.7b
(+ (sqrt (fma (* a c) -4 (* b b))) b)
0.6b
(fma (* a c) -4 (* b b))
0.2b
(* (* 1/2 c) (/ -4 (+ (sqrt (fma (* a c) -4 (* b b))) b)))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 63
Calls

4 calls. Slowest were:

7.0ms
(+ (sqrt (fma (* a c) -4 (* b b))) b)
4.0ms
(* (* 1/2 c) (/ -4 (+ (sqrt (fma (* a c) -4 (* b b))) b)))
2.0ms
(sqrt (fma (* a c) -4 (* b b)))

series271.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

147.0ms
(* (* 1/2 c) (/ -4 (+ (sqrt (fma (* a c) -4 (* b b))) b)))
55.0ms
(+ (sqrt (fma (* a c) -4 (* b b))) b)
40.0ms
(sqrt (fma (* a c) -4 (* b b)))
28.0ms
(fma (* a c) -4 (* b b))

simplify1.3s

Counts
31 → 75
Calls

31 calls. Slowest were:

566.0ms
(* (* (* (* 1/2 c) (* 1/2 c)) (* 1/2 c)) (* (* (/ -4 (+ (sqrt (fma (* a c) -4 (* b b))) b)) (/ -4 (+ (sqrt (fma (* a c) -4 (* b b))) b))) (/ -4 (+ (sqrt (fma (* a c) -4 (* b b))) b))))
230.0ms
(+ (* (sqrt (fma (* a c) -4 (* b b))) (sqrt (fma (* a c) -4 (* b b)))) (- (* b b) (* (sqrt (fma (* a c) -4 (* b b))) b)))
120.0ms
(+ (log (* 1/2 c)) (log (/ -4 (+ (sqrt (fma (* a c) -4 (* b b))) b))))

prune888.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 11.4b

localize6.0ms

Local error

Found 4 expressions with local error:

23.6b
(sqrt (fma (* a c) -4 (* b b)))
1.7b
(+ (sqrt (fma (* a c) -4 (* b b))) b)
0.6b
(fma (* a c) -4 (* b b))
0.0b
(/ (* (* 1/2 c) -4) (+ (sqrt (fma (* a c) -4 (* b b))) b))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

9.0ms
(/ (* (* 1/2 c) -4) (+ (sqrt (fma (* a c) -4 (* b b))) b))
4.0ms
(+ (sqrt (fma (* a c) -4 (* b b))) b)
1.0ms
(sqrt (fma (* a c) -4 (* b b)))

series310.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

171.0ms
(/ (* (* 1/2 c) -4) (+ (sqrt (fma (* a c) -4 (* b b))) b))
62.0ms
(+ (sqrt (fma (* a c) -4 (* b b))) b)
42.0ms
(sqrt (fma (* a c) -4 (* b b)))
36.0ms
(fma (* a c) -4 (* b b))

simplify1.1s

Counts
35 → 77
Calls

35 calls. Slowest were:

265.0ms
(+ (* (sqrt (fma (* a c) -4 (* b b))) (sqrt (fma (* a c) -4 (* b b)))) (- (* b b) (* (sqrt (fma (* a c) -4 (* b b))) b)))
132.0ms
(/ (* (* (* (* 1/2 c) -4) (* (* 1/2 c) -4)) (* (* 1/2 c) -4)) (* (* (+ (sqrt (fma (* a c) -4 (* b b))) b) (+ (sqrt (fma (* a c) -4 (* b b))) b)) (+ (sqrt (fma (* a c) -4 (* b b))) b)))
100.0ms
(- (log (* (* 1/2 c) -4)) (log (+ (sqrt (fma (* a c) -4 (* b b))) b)))

prune911.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 11.4b

regimes315.0ms

Accuracy

96.6% (0.6b remaining)

Error of 12.9b against oracle of 12.3b and baseline of 29.3b

bsearch335.0ms

end0.0ms

sample4.6s

Algorithm
intervals