Average Error: 19.7 → 13.3
Time: 22.2s
Precision: 64
Internal Precision: 128
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 5.966382005329617 \cdot 10^{+105}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt[3]{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}} \cdot \left|\sqrt[3]{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot -4}{\left(2 \cdot a\right) \cdot \left(b + \sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 5.966382005329617e+105

    1. Initial program 16.3

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

    2. Initial simplification16.3

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

    4. Applied add-cube-cbrt16.5

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

    5. Applied sqrt-prod16.5

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

    6. Applied add-cube-cbrt16.6

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

    7. Applied prod-diff16.6

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

    8. Simplified16.5

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

    9. Simplified16.5

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

    if 5.966382005329617e+105 < b

    1. Initial program 31.0

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

    2. Initial simplification31.0

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

    3. Taylor expanded around 0 2.5

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

    4. Taylor expanded around inf 2.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\]

    5. Simplified2.5

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

    7. Applied flip--2.5

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

    8. Applied associate-/l/2.5

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

    9. Simplified2.5

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

  4. Final simplification13.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 5.966382005329617 \cdot 10^{+105}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt[3]{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}} \cdot \left|\sqrt[3]{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot -4}{\left(2 \cdot a\right) \cdot \left(b + \sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2018365 +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: 15.1s)Debug log

start120.0ms

Algorithm
intervals

setup173.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 20.7b

localize190.0ms

Local error

Found 4 expressions with local error:

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

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 93
Calls

4 calls. Slowest were:

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

series241.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

69.0ms
(- (- b) (sqrt (fma -4 (* c a) (* b b))))
68.0ms
(sqrt (fma -4 (* c a) (* b b)))
53.0ms
(- (sqrt (fma -4 (* c a) (* b b))) b)
50.0ms
(sqrt (fma -4 (* c a) (* b b)))

simplify2.3s

Counts
81 → 105
Calls

81 calls. Slowest were:

78.0ms
(- (* 2 (/ (* a c) b)) (* 2 b))
67.0ms
(fma (- (sqrt (fma -4 (* c a) (* b b)))) (sqrt 1) (* (sqrt (fma -4 (* c a) (* b b))) (sqrt 1)))
66.0ms
(fma (* (cbrt (- b)) (cbrt (- b))) (cbrt (- b)) (- (* (sqrt (fma -4 (* c a) (* b b))) (sqrt 1))))

prune2.1s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 13.1b

localize143.0ms

Local error

Found 4 expressions with local error:

25.1b
(sqrt (fma -4 (* c a) (* b b)))
1.2b
(- (sqrt (fma -4 (* c a) (* b b))) b)
0.8b
(fma -4 (* c a) (* b b))
0.2b
(/ (* 2 c) (- (- b) b))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

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

series146.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

51.0ms
(sqrt (fma -4 (* c a) (* b b)))
46.0ms
(- (sqrt (fma -4 (* c a) (* b b))) b)
40.0ms
(fma -4 (* c a) (* b b))
9.0ms
(/ (* 2 c) (- (- b) b))

simplify1.4s

Counts
34 → 78
Calls

34 calls. Slowest were:

678.0ms
(/ (* (* (* 2 c) (* 2 c)) (* 2 c)) (* (* (- (- b) b) (- (- b) b)) (- (- b) b)))
406.0ms
(+ (* (- b) (- b)) (+ (* b b) (* (- b) b)))
50.0ms
(- (pow b 2) (* 4 (* a c)))

prune1.5s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 12.8b

localize116.0ms

Local error

Found 4 expressions with local error:

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

rewrite40.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

28.0ms
(/ (- (sqrt (fma -4 (* c a) (* b b))) b) (* 2 a))
9.0ms
(- (sqrt (fma -4 (* c a) (* b b))) b)
2.0ms
(sqrt (fma -4 (* c a) (* b b)))

series238.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

75.0ms
(- (sqrt (fma -4 (* c a) (* b b))) b)
68.0ms
(/ (- (sqrt (fma -4 (* c a) (* b b))) b) (* 2 a))
48.0ms
(sqrt (fma -4 (* c a) (* b b)))
46.0ms
(fma -4 (* c a) (* b b))

simplify1.4s

Counts
34 → 79
Calls

34 calls. Slowest were:

789.0ms
(- (* (sqrt (fma -4 (* c a) (* b b))) (sqrt (fma -4 (* c a) (* b b)))) (* b b))
140.0ms
(- (pow (sqrt (fma -4 (* c a) (* b b))) 3) (pow b 3))
107.0ms
(/ (* (* (- (sqrt (fma -4 (* c a) (* b b))) b) (- (sqrt (fma -4 (* c a) (* b b))) b)) (- (sqrt (fma -4 (* c a) (* b b))) b)) (* (* (* 2 a) (* 2 a)) (* 2 a)))

prune1.5s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 12.8b

localize139.0ms

Local error

Found 4 expressions with local error:

25.1b
(sqrt (fma -4 (* c a) (* b b)))
1.2b
(- (sqrt (fma -4 (* c a) (* b b))) b)
0.8b
(fma -4 (* c a) (* b b))
0.2b
(/ 1 (/ (* 2 a) (- (sqrt (fma -4 (* c a) (* b b))) b)))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

6.0ms
(- (sqrt (fma -4 (* c a) (* b b))) b)
4.0ms
(/ 1 (/ (* 2 a) (- (sqrt (fma -4 (* c a) (* b b))) b)))
2.0ms
(sqrt (fma -4 (* c a) (* b b)))

series222.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

76.0ms
(/ 1 (/ (* 2 a) (- (sqrt (fma -4 (* c a) (* b b))) b)))
55.0ms
(- (sqrt (fma -4 (* c a) (* b b))) b)
49.0ms
(sqrt (fma -4 (* c a) (* b b)))
41.0ms
(fma -4 (* c a) (* b b))

simplify539.0ms

Counts
32 → 81
Calls

32 calls. Slowest were:

110.0ms
(+ (* (sqrt (fma -4 (* c a) (* b b))) (sqrt (fma -4 (* c a) (* b b)))) (+ (* b b) (* (sqrt (fma -4 (* c a) (* b b))) b)))
50.0ms
(- (pow b 2) (* 4 (* a c)))
45.0ms
(- (pow b 2) (* 4 (* a c)))

prune1.4s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 12.8b

regimes921.0ms

Accuracy

91.7% (0.7b remaining)

Error of 13.3b against oracle of 12.6b and baseline of 21.4b

bsearch240.0ms