Average Error: 19.2 → 12.9
Time: 18.8s
Precision: 64
Internal Precision: 128
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2.1097448153148673 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 2.1097448153148673e+99

    1. Initial program 16.0

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

    2. Simplified16.0

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}}\]

    3. Taylor expanded around inf 16.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}{2 \cdot a}\\ \end{array}\]

    4. Simplified16.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt16.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\sqrt{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\]

    if 2.1097448153148673e+99 < b

    1. Initial program 29.3

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

    2. Simplified29.3

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}}\]

    3. Taylor expanded around inf 29.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}{2 \cdot a}\\ \end{array}\]

    4. Simplified29.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\]

    5. Taylor expanded around 0 2.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification12.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2.1097448153148673 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))

Details

Time bar (total: 17.4s)Debug log

sample120.0ms

Algorithm
intervals

simplify136.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

136.0ms
(if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))

prune15.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 19.7b

localize129.0ms

Local error

Found 4 expressions with local error:

22.0b
(sqrt (fma -4 (* c a) (* b b)))
22.0b
(sqrt (fma -4 (* c a) (* b b)))
0.9b
(- (- b) (sqrt (fma -4 (* c a) (* b b))))
0.8b
(fma -4 (* c a) (* b b))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
4 → 84
Calls

4 calls. Slowest were:

9.0ms
(- (- b) (sqrt (fma -4 (* c a) (* b b))))
4.0ms
(sqrt (fma -4 (* c a) (* b b)))
2.0ms
(sqrt (fma -4 (* c a) (* b b)))

series226.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

91.0ms
(sqrt (fma -4 (* c a) (* b b)))
62.0ms
(- (- b) (sqrt (fma -4 (* c a) (* b b))))
40.0ms
(sqrt (fma -4 (* c a) (* b b)))
33.0ms
(fma -4 (* c a) (* b b))

simplify2.5s

Counts
76 → 96
Calls

76 calls. Slowest were:

77.0ms
(fma 1 (- b) (- (* (sqrt (fma -4 (* c a) (* b b))) 1)))
74.0ms
(fma 1 (- b) (- (* (sqrt (fma -4 (* c a) (* b b))) (sqrt 1))))
69.0ms
(fma 1 (- b) (- (* (sqrt (cbrt (fma -4 (* c a) (* b b)))) (sqrt (* (cbrt (fma -4 (* c a) (* b b))) (cbrt (fma -4 (* c a) (* b b))))))))

prune1.6s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 13.0b

localize106.0ms

Local error

Found 4 expressions with local error:

22.0b
(sqrt (fma -4 (* c a) (* b b)))
22.0b
(sqrt (fma (* a -4) c (* b b)))
0.9b
(- (- b) (sqrt (fma -4 (* c a) (* b b))))
0.8b
(fma -4 (* c a) (* b b))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 84
Calls

4 calls. Slowest were:

9.0ms
(- (- b) (sqrt (fma -4 (* c a) (* b b))))
2.0ms
(sqrt (fma -4 (* c a) (* b b)))
2.0ms
(sqrt (fma (* a -4) c (* b b)))

series224.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

72.0ms
(- (- b) (sqrt (fma -4 (* c a) (* b b))))
66.0ms
(sqrt (fma (* a -4) c (* b b)))
54.0ms
(sqrt (fma -4 (* c a) (* b b)))
32.0ms
(fma -4 (* c a) (* b b))

simplify2.2s

Counts
76 → 96
Calls

76 calls. Slowest were:

77.0ms
(fma 1 (- b) (- (* (sqrt (fma -4 (* c a) (* b b))) (sqrt 1))))
72.0ms
(- (* 2 (/ (* a c) b)) (* 2 b))
63.0ms
(/ (exp (- b)) (exp (sqrt (fma -4 (* c a) (* b b)))))

prune1.8s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 13.0b

localize103.0ms

Local error

Found 4 expressions with local error:

22.0b
(sqrt (fma (* a -4) c (* b b)))
22.0b
(sqrt (fma -4 (* c a) (* b b)))
22.0b
(sqrt (fma -4 (* c a) (* b b)))
10.6b
(- (- b) (* (sqrt (sqrt (fma -4 (* c a) (* b b)))) (sqrt (sqrt (fma -4 (* c a) (* b b))))))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

4.0ms
(- (- b) (* (sqrt (sqrt (fma -4 (* c a) (* b b)))) (sqrt (sqrt (fma -4 (* c a) (* b b))))))
1.0ms
(sqrt (fma (* a -4) c (* b b)))
1.0ms
(sqrt (fma -4 (* c a) (* b b)))

series187.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

52.0ms
(sqrt (fma -4 (* c a) (* b b)))
50.0ms
(- (- b) (* (sqrt (sqrt (fma -4 (* c a) (* b b)))) (sqrt (sqrt (fma -4 (* c a) (* b b))))))
45.0ms
(sqrt (fma (* a -4) c (* b b)))
41.0ms
(sqrt (fma -4 (* c a) (* b b)))

simplify523.0ms

Counts
37 → 78
Calls

37 calls. Slowest were:

81.0ms
(- (* 2 (/ (* a c) b)) (* 2 b))
53.0ms
(* -2 (/ (* a c) b))
47.0ms
(fma (sqrt (- b)) (sqrt (- b)) (- (* (sqrt (sqrt (fma -4 (* c a) (* b b)))) (sqrt (sqrt (fma -4 (* c a) (* b b)))))))

prune1.7s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 13.0b

localize150.0ms

Local error

Found 4 expressions with local error:

22.0b
(sqrt (fma (* a -4) c (* b b)))
22.0b
(sqrt (fma (* a -4) c (* b b)))
22.0b
(sqrt (fma -4 (* c a) (* b b)))
10.5b
(- (* (sqrt (sqrt (fma (* a -4) c (* b b)))) (sqrt (sqrt (fma (* a -4) c (* b b))))) b)

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

24.0ms
(- (* (sqrt (sqrt (fma (* a -4) c (* b b)))) (sqrt (sqrt (fma (* a -4) c (* b b))))) b)
2.0ms
(sqrt (fma (* a -4) c (* b b)))
2.0ms
(sqrt (fma -4 (* c a) (* b b)))

series240.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

67.0ms
(sqrt (fma (* a -4) c (* b b)))
66.0ms
(sqrt (fma (* a -4) c (* b b)))
55.0ms
(- (* (sqrt (sqrt (fma (* a -4) c (* b b)))) (sqrt (sqrt (fma (* a -4) c (* b b))))) b)
52.0ms
(sqrt (fma -4 (* c a) (* b b)))

simplify130.0ms

Counts
25 → 68
Calls

25 calls. Slowest were:

17.0ms
(sqrt (fma (* a -4) c (* b b)))
15.0ms
(sqrt (sqrt (fma (* a -4) c (* b b))))
13.0ms
(sqrt (sqrt (fma (* a -4) c (* b b))))

prune1.4s

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 13.0b

regimes313.0ms

Accuracy

91.6% (0.6b remaining)

Error of 12.9b against oracle of 12.3b and baseline of 19.2b

bsearch198.0ms

end0.0ms

sample3.3s

Algorithm
intervals