Average Error: 33.5 → 13.3
Time: 21.3s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.012283951834148 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.7747799476926007 \cdot 10^{+107}:\\ \;\;\;\;\left(c \cdot \frac{1}{2}\right) \cdot \frac{-4}{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{b + b} \cdot \left(c \cdot \frac{1}{2}\right)\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.5
Target20.8
Herbie13.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -7.012283951834148e-219

    1. Initial program 22.5

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

    2. Initial simplification22.5

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

    3. Taylor expanded around inf 22.5

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

    4. Simplified22.4

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

    if -7.012283951834148e-219 < b < 1.7747799476926007e+107

    1. Initial program 29.5

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

    2. Initial simplification29.5

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

    4. Applied flip--29.7

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

    5. Applied associate-/l/34.5

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

    6. Simplified21.6

      \[\leadsto \frac{\color{blue}{\left(a \cdot c\right) \cdot -4}}{\left(2 \cdot a\right) \cdot \left(\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} + b\right)}\]
    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. Simplified10.1

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

    if 1.7747799476926007e+107 < b

    1. Initial program 59.1

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

    2. Initial simplification59.1

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

    4. Applied flip--59.2

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

    5. Applied associate-/l/59.3

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

    6. Simplified33.0

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

    8. Applied times-frac31.7

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

    9. Simplified31.1

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

    10. Taylor expanded around 0 2.7

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

  4. Final simplification13.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.012283951834148 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.7747799476926007 \cdot 10^{+107}:\\ \;\;\;\;\left(c \cdot \frac{1}{2}\right) \cdot \frac{-4}{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{b + b} \cdot \left(c \cdot \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2018365 +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

start194.0ms

Algorithm
intervals

setup110.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 35.7b

localize34.0ms

Local error

Found 4 expressions with local error:

25.1b
(sqrt (fma (* a c) -4 (* b b)))
1.2b
(- (sqrt (fma (* a c) -4 (* b b))) b)
0.8b
(fma (* a c) -4 (* b b))
0.1b
(/ (- (sqrt (fma (* a c) -4 (* b b))) b) (* 2 a))

rewrite32.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

19.0ms
(/ (- (sqrt (fma (* a c) -4 (* b b))) b) (* 2 a))
10.0ms
(- (sqrt (fma (* a c) -4 (* b b))) b)
2.0ms
(sqrt (fma (* a c) -4 (* b b)))

series252.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

90.0ms
(/ (- (sqrt (fma (* a c) -4 (* b b))) b) (* 2 a))
74.0ms
(sqrt (fma (* a c) -4 (* b b)))
49.0ms
(- (sqrt (fma (* a c) -4 (* b b))) b)
39.0ms
(fma (* a c) -4 (* b b))

simplify1.6s

Counts
34 → 79
Calls

34 calls. Slowest were:

729.0ms
(- (* (sqrt (fma (* a c) -4 (* b b))) (sqrt (fma (* a c) -4 (* b b)))) (* b b))
163.0ms
(- (pow (sqrt (fma (* a c) -4 (* b b))) 3) (pow b 3))
85.0ms
(- (log (- (sqrt (fma (* a c) -4 (* b b))) b)) (log (* 2 a)))

prune1.3s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 22.7b

localize23.0ms

Local error

Found 4 expressions with local error:

25.1b
(sqrt (fma (* a c) -4 (* b b)))
17.5b
(/ (* (* a c) -4) (* (* 2 a) (+ (sqrt (fma (* a c) -4 (* b b))) b)))
1.2b
(+ (sqrt (fma (* a c) -4 (* b b))) b)
0.8b
(fma (* a c) -4 (* b b))

rewrite15.0ms

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)))

series315.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

141.0ms
(/ (* (* a c) -4) (* (* 2 a) (+ (sqrt (fma (* a c) -4 (* b b))) b)))
82.0ms
(+ (sqrt (fma (* a c) -4 (* b b))) b)
56.0ms
(sqrt (fma (* a c) -4 (* b b)))
35.0ms
(fma (* a c) -4 (* b b))

simplify3.1s

Counts
31 → 75
Calls

31 calls. Slowest were:

715.0ms
(/ (* (* (* (* 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))))
702.0ms
(/ (* (* (* (* 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))))
686.0ms
(- (log (* (* a c) -4)) (+ (log (* 2 a)) (log (+ (sqrt (fma (* a c) -4 (* b b))) b))))

prune1.2s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 16.3b

localize20.0ms

Local error

Found 4 expressions with local error:

25.1b
(sqrt (fma (* a c) -4 (* b b)))
1.2b
(+ (sqrt (fma (* a c) -4 (* b b))) b)
0.8b
(fma (* a c) -4 (* b b))
0.3b
(/ -4 (+ (sqrt (fma (* a c) -4 (* b b))) b))

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 59
Calls

4 calls. Slowest were:

8.0ms
(+ (sqrt (fma (* a c) -4 (* b b))) b)
5.0ms
(/ -4 (+ (sqrt (fma (* a c) -4 (* b b))) b))
2.0ms
(sqrt (fma (* a c) -4 (* b b)))

series286.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

147.0ms
(/ -4 (+ (sqrt (fma (* a c) -4 (* b b))) b))
55.0ms
(+ (sqrt (fma (* a c) -4 (* b b))) b)
45.0ms
(sqrt (fma (* a c) -4 (* b b)))
39.0ms
(fma (* a c) -4 (* b b))

simplify660.0ms

Counts
26 → 71
Calls

26 calls. Slowest were:

293.0ms
(+ (* (sqrt (fma (* a c) -4 (* b b))) (sqrt (fma (* a c) -4 (* b b)))) (- (* b b) (* (sqrt (fma (* a c) -4 (* b b))) b)))
32.0ms
(- (pow b 2) (* 4 (* a c)))
30.0ms
(- (pow b 2) (* 4 (* a c)))

prune1.5s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 13.1b

localize12.0ms

Local error

Found 4 expressions with local error:

25.1b
(sqrt (fma (* a c) -4 (* b b)))
25.1b
(sqrt (fma (* a c) -4 (* b b)))
1.2b
(+ (sqrt (fma (* a c) -4 (* b b))) b)
1.2b
(+ (sqrt (fma (* a c) -4 (* b b))) b)

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 64
Calls

4 calls. Slowest were:

4.0ms
(+ (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)))

series272.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

78.0ms
(+ (sqrt (fma (* a c) -4 (* b b))) b)
69.0ms
(sqrt (fma (* a c) -4 (* b b)))
63.0ms
(sqrt (fma (* a c) -4 (* b b)))
61.0ms
(+ (sqrt (fma (* a c) -4 (* b b))) b)

simplify374.0ms

Counts
30 → 76
Calls

30 calls. Slowest were:

27.0ms
(sqrt (* (cbrt (fma (* a c) -4 (* b b))) (cbrt (fma (* a c) -4 (* b b)))))
27.0ms
(sqrt (* (cbrt (fma (* a c) -4 (* b b))) (cbrt (fma (* a c) -4 (* b b)))))
27.0ms
(* (cbrt (sqrt (fma (* a c) -4 (* b b)))) (cbrt (sqrt (fma (* a c) -4 (* b b)))))

prune1.3s

Pruning

10 alts after pruning (8 fresh and 2 done)

Merged error: 13.1b

regimes362.0ms

Accuracy

95.6% (0.7b remaining)

Error of 13.3b against oracle of 12.5b and baseline of 29.1b

bsearch420.0ms