Average Error: 34.2 → 13.2
Time: 21.5s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.249183367650486 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 7.357632498724412 \cdot 10^{+115}:\\ \;\;\;\;\frac{-4 \cdot \left(c \cdot \frac{1}{2}\right)}{b + \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{b + b} \cdot \left(c \cdot \frac{1}{2}\right)\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.2
Target21.3
Herbie13.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.249183367650486e-309

    1. Initial program 21.9

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

    2. Initial simplification21.9

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

    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 -1.249183367650486e-309 < b < 7.357632498724412e+115

    1. Initial program 34.0

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

    2. Initial simplification34.0

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

    4. Applied flip--34.1

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

    5. Applied associate-/l/38.3

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

    6. Simplified21.4

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

    8. Applied times-frac16.1

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

    9. Simplified9.0

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

    11. Applied associate-*r/8.9

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

    if 7.357632498724412e+115 < b

    1. Initial program 59.5

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

    2. Initial simplification59.5

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

    4. Applied flip--59.6

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

    5. Applied associate-/l/59.7

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

    6. Simplified33.4

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

    8. Applied times-frac32.4

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

    9. Simplified31.9

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

    10. Taylor expanded around 0 1.9

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

  4. Final simplification13.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.249183367650486 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 7.357632498724412 \cdot 10^{+115}:\\ \;\;\;\;\frac{-4 \cdot \left(c \cdot \frac{1}{2}\right)}{b + \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{b + b} \cdot \left(c \cdot \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 +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: 13.4s)Debug log

start170.0ms

Algorithm
intervals

setup151.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 34.6b

localize40.0ms

Local error

Found 4 expressions with local error:

22.5b
(sqrt (fma (* a c) -4 (* b b)))
0.9b
(- (sqrt (fma (* a c) -4 (* b b))) b)
0.4b
(fma (* a c) -4 (* b b))
0.1b
(/ (- (sqrt (fma (* a c) -4 (* b b))) b) (* 2 a))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

12.0ms
(/ (- (sqrt (fma (* a c) -4 (* b b))) b) (* 2 a))
5.0ms
(- (sqrt (fma (* a c) -4 (* b b))) b)
1.0ms
(sqrt (fma (* a c) -4 (* b b)))

series282.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

101.0ms
(/ (- (sqrt (fma (* a c) -4 (* b b))) b) (* 2 a))
79.0ms
(- (sqrt (fma (* a c) -4 (* b b))) b)
70.0ms
(sqrt (fma (* a c) -4 (* b b)))
32.0ms
(fma (* a c) -4 (* b b))

simplify1.3s

Counts
34 → 79
Calls

34 calls. Slowest were:

567.0ms
(- (* (sqrt (fma (* a c) -4 (* b b))) (sqrt (fma (* a c) -4 (* b b)))) (* b b))
157.0ms
(- (pow (sqrt (fma (* a c) -4 (* b b))) 3) (pow b 3))
121.0ms
(/ (* (* (- (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)))

prune1.2s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 20.9b

localize10.0ms

Local error

Found 4 expressions with local error:

22.5b
(sqrt (fma (* a c) -4 (* b b)))
13.3b
(/ (* (* a c) -4) (* (* 2 a) (+ (sqrt (fma (* a c) -4 (* b b))) b)))
0.8b
(+ (sqrt (fma (* a c) -4 (* b b))) b)
0.4b
(fma (* a c) -4 (* b b))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 63
Calls

4 calls. Slowest were:

9.0ms
(/ (* (* a c) -4) (* (* 2 a) (+ (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)))

series219.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

100.0ms
(/ (* (* a c) -4) (* (* 2 a) (+ (sqrt (fma (* a c) -4 (* b b))) b)))
48.0ms
(+ (sqrt (fma (* a c) -4 (* b b))) b)
42.0ms
(sqrt (fma (* a c) -4 (* b b)))
29.0ms
(fma (* a c) -4 (* b b))

simplify3.0s

Counts
31 → 75
Calls

31 calls. Slowest were:

758.0ms
(/ (* (* (* (* 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))))
661.0ms
(- (log (* (* a c) -4)) (+ (log (* 2 a)) (log (+ (sqrt (fma (* a c) -4 (* b b))) b))))
565.0ms
(/ (* (* (* (* 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))))

prune1.3s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 13.2b

localize17.0ms

Local error

Found 4 expressions with local error:

22.5b
(sqrt (fma (* a c) -4 (* b b)))
0.8b
(+ (sqrt (fma (* a c) -4 (* b b))) b)
0.4b
(fma (* a c) -4 (* b b))
0.1b
(* (* 1/2 c) (/ -4 (+ (sqrt (fma (* a c) -4 (* b b))) b)))

rewrite25.0ms

Algorithm
rewrite-expression-head
Counts
4 → 63
Calls

4 calls. Slowest were:

8.0ms
(* (* 1/2 c) (/ -4 (+ (sqrt (fma (* a c) -4 (* b b))) b)))
8.0ms
(+ (sqrt (fma (* a c) -4 (* b b))) b)
6.0ms
(sqrt (fma (* a c) -4 (* b b)))

series329.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

154.0ms
(* (* 1/2 c) (/ -4 (+ (sqrt (fma (* a c) -4 (* b b))) b)))
68.0ms
(sqrt (fma (* a c) -4 (* b b)))
59.0ms
(+ (sqrt (fma (* a c) -4 (* b b))) b)
48.0ms
(fma (* a c) -4 (* b b))

simplify1.4s

Counts
31 → 75
Calls

31 calls. Slowest were:

657.0ms
(* (* (* (* 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))))
294.0ms
(+ (* (sqrt (fma (* a c) -4 (* b b))) (sqrt (fma (* a c) -4 (* b b)))) (- (* b b) (* (sqrt (fma (* a c) -4 (* b b))) b)))
103.0ms
(+ (log (* 1/2 c)) (log (/ -4 (+ (sqrt (fma (* a c) -4 (* b b))) b))))

prune1.2s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 12.0b

localize23.0ms

Local error

Found 4 expressions with local error:

22.5b
(sqrt (fma (* a c) -4 (* b b)))
22.5b
(sqrt (fma (* a c) -4 (* b b)))
17.4b
(fma (sqrt (sqrt (fma (* a c) -4 (* b b)))) (sqrt (sqrt (fma (* a c) -4 (* b b)))) b)
0.4b
(fma (* a c) -4 (* b b))

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
4 → 48
Calls

4 calls. Slowest were:

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

series461.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

328.0ms
(fma (sqrt (sqrt (fma (* a c) -4 (* b b)))) (sqrt (sqrt (fma (* a c) -4 (* b b)))) b)
50.0ms
(sqrt (fma (* a c) -4 (* b b)))
44.0ms
(sqrt (fma (* a c) -4 (* b b)))
39.0ms
(fma (* a c) -4 (* b b))

simplify494.0ms

Counts
20 → 60
Calls

20 calls. Slowest were:

196.0ms
(+ (pow (exp (* 1/4 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c)))))) 2) b)
126.0ms
(+ (pow (exp (* 1/4 (- (log -4) (+ (log (/ -1 c)) (log (/ -1 a)))))) 2) b)
35.0ms
(- (pow b 2) (* 4 (* a c)))

prune916.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 12.0b

regimes358.0ms

Accuracy

96.9% (0.5b remaining)

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

bsearch389.0ms