Average Error: 33.4 → 13.1
Time: 18.8s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.139372088040273 \cdot 10^{-292}:\\ \;\;\;\;\frac{(\left(\sqrt{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.4569203446844513 \cdot 10^{+101}:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -5.139372088040273e-292

    1. Initial program 21.0

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

    2. Simplified21.0

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

    4. Applied add-sqr-sqrt21.2

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

    5. Applied fma-neg21.1

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

    if -5.139372088040273e-292 < b < 1.4569203446844513e+101

    1. Initial program 32.1

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

    2. Simplified32.1

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

    4. Applied div-inv32.1

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

    6. Applied flip--32.2

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

    7. Applied associate-*l/32.2

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

    8. Simplified16.3

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

    9. Taylor expanded around inf 9.3

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

    if 1.4569203446844513e+101 < b

    1. Initial program 58.9

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

    2. Simplified58.9

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

    4. Applied div-inv58.9

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

    6. Applied flip--59.0

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

    7. Applied associate-*l/59.0

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

    8. Simplified30.1

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

    9. Taylor expanded around inf 29.2

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

    10. Taylor expanded around 0 3.0

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

  4. Final simplification13.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.139372088040273 \cdot 10^{-292}:\\ \;\;\;\;\frac{(\left(\sqrt{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.4569203446844513 \cdot 10^{+101}:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 +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: 17.6s)Debug log

sample92.0ms

Algorithm
intervals

simplify59.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

59.0ms
(/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))

prune13.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 34.5b

localize22.0ms

Local error

Found 4 expressions with local error:

23.3b
(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))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

15.0ms
(/ (- (sqrt (fma (* a -4) c (* b b))) b) (* 2 a))
5.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
1.0ms
(sqrt (fma (* a -4) c (* b b)))

series231.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

92.0ms
(/ (- (sqrt (fma (* a -4) c (* b b))) b) (* 2 a))
57.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
48.0ms
(sqrt (fma (* a -4) c (* b b)))
33.0ms
(fma (* a -4) c (* b b))

simplify1.3s

Counts
34 → 79
Calls

34 calls. Slowest were:

579.0ms
(- (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (* b b))
148.0ms
(- (pow (sqrt (fma (* a -4) c (* b b))) 3) (pow b 3))
85.0ms
(/ (* (* (- (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

11 alts after pruning (11 fresh and 0 done)

Merged error: 22.1b

localize12.0ms

Local error

Found 4 expressions with local error:

23.3b
(sqrt (fma (* a -4) c (* b b)))
0.9b
(- (sqrt (fma (* a -4) c (* b b))) b)
0.2b
(* (- (sqrt (fma (* a -4) c (* b b))) b) (/ 1 (* 2 a)))
0.0b
(fma (* a -4) c (* b b))

rewrite43.0ms

Algorithm
rewrite-expression-head
Counts
4 → 70
Calls

4 calls. Slowest were:

25.0ms
(* (- (sqrt (fma (* a -4) c (* b b))) b) (/ 1 (* 2 a)))
14.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
2.0ms
(sqrt (fma (* a -4) c (* b b)))

series281.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

99.0ms
(* (- (sqrt (fma (* a -4) c (* b b))) b) (/ 1 (* 2 a)))
68.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
66.0ms
(sqrt (fma (* a -4) c (* b b)))
48.0ms
(fma (* a -4) c (* b b))

simplify2.1s

Counts
39 → 82
Calls

39 calls. Slowest were:

320.0ms
(* (- (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (* b b)) 1)
283.0ms
(* (- (pow (sqrt (fma (* a -4) c (* b b))) 3) (pow b 3)) 1)
239.0ms
(* (- (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (* b b)) (/ 1 (* 2 a)))

prune995.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 21.4b

localize17.0ms

Local error

Found 4 expressions with local error:

23.3b
(sqrt (fma (* a -4) c (* b b)))
15.2b
(* (* a (* c -4)) (/ 1/2 a))
0.8b
(+ (sqrt (fma (* a -4) c (* b b))) b)
0.1b
(/ (* (* a (* c -4)) (/ 1/2 a)) (+ (sqrt (fma (* a -4) c (* b b))) b))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 77
Calls

4 calls. Slowest were:

11.0ms
(/ (* (* a (* c -4)) (/ 1/2 a)) (+ (sqrt (fma (* a -4) c (* b b))) b))
5.0ms
(* (* a (* c -4)) (/ 1/2 a))
4.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)

series282.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

126.0ms
(/ (* (* a (* c -4)) (/ 1/2 a)) (+ (sqrt (fma (* a -4) c (* b b))) b))
79.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
66.0ms
(sqrt (fma (* a -4) c (* b b)))
10.0ms
(* (* a (* c -4)) (/ 1/2 a))

simplify2.3s

Counts
45 → 89
Calls

45 calls. Slowest were:

263.0ms
(/ (* (* (* (* a (* c -4)) (/ 1/2 a)) (* (* a (* c -4)) (/ 1/2 a))) (* (* a (* c -4)) (/ 1/2 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)))
248.0ms
(/ (* (* (* (* a (* c -4)) (* a (* c -4))) (* a (* c -4))) (* (* (/ 1/2 a) (/ 1/2 a)) (/ 1/2 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)))
226.0ms
(+ (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (- (* b b) (* (sqrt (fma (* a -4) c (* b b))) b)))

prune1.5s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 16.4b

localize16.0ms

Local error

Found 4 expressions with local error:

23.3b
(sqrt (fma (* a -4) c (* b b)))
0.8b
(+ (sqrt (fma (* a -4) c (* b b))) b)
0.1b
(/ (* -2 c) (+ (sqrt (fma (* a -4) c (* b b))) b))
0.0b
(fma (* a -4) c (* b b))

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

12.0ms
(/ (* -2 c) (+ (sqrt (fma (* a -4) c (* b b))) b))
8.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
2.0ms
(sqrt (fma (* a -4) c (* b b)))

series301.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

145.0ms
(/ (* -2 c) (+ (sqrt (fma (* a -4) c (* b b))) b))
78.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
47.0ms
(sqrt (fma (* a -4) c (* b b)))
31.0ms
(fma (* a -4) c (* b b))

simplify813.0ms

Counts
34 → 77
Calls

34 calls. Slowest were:

253.0ms
(+ (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (- (* b b) (* (sqrt (fma (* a -4) c (* b b))) b)))
105.0ms
(/ (* (* (* -2 c) (* -2 c)) (* -2 c)) (* (* (+ (sqrt (fma (* a -4) c (* b b))) b) (+ (sqrt (fma (* a -4) c (* b b))) b)) (+ (sqrt (fma (* a -4) c (* b b))) b)))
52.0ms
(- (log (* -2 c)) (log (+ (sqrt (fma (* a -4) c (* b b))) b)))

prune910.0ms

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 15.2b

regimes452.0ms

Accuracy

95.8% (0.7b remaining)

Error of 13.1b against oracle of 12.4b and baseline of 28.9b

bsearch458.0ms

end0.0ms

sample4.3s

Algorithm
intervals