Average Error: 19.5 → 13.6
Time: 17.6s
Precision: 64
Internal Precision: 128
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 3.995303483597858 \cdot 10^{+95}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{-(\left(\left|\sqrt[3]{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*}\right|\right) \cdot \left(\sqrt{\sqrt[3]{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*}}\right) + b)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 3.995303483597858e+95

    1. Initial program 16.7

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

    2. Simplified16.7

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

    4. Applied add-cube-cbrt16.9

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

    5. Applied sqrt-prod16.9

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

    6. Applied *-un-lft-identity16.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{1 \cdot \left(-b\right)} - \sqrt{\sqrt[3]{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt[3]{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\]

    7. Applied prod-diff16.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{(1 \cdot \left(-b\right) + \left(-\sqrt{\sqrt[3]{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt[3]{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right))_* + (\left(-\sqrt{\sqrt[3]{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt[3]{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right) + \left(\sqrt{\sqrt[3]{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt[3]{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\]

    8. Simplified16.9

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

    9. Simplified16.8

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

    if 3.995303483597858e+95 < 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 0 2.7

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

  4. Final simplification13.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 3.995303483597858 \cdot 10^{+95}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{-(\left(\left|\sqrt[3]{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*}\right|\right) \cdot \left(\sqrt{\sqrt[3]{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*}}\right) + b)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 +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: 16.3s)Debug log

sample151.0ms

Algorithm
intervals

simplify141.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

141.0ms
(if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))

prune29.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 20.7b

localize141.0ms

Local error

Found 4 expressions with local error:

23.2b
(sqrt (fma -4 (* c a) (* b b)))
23.2b
(sqrt (fma -4 (* c a) (* b b)))
1.1b
(- (sqrt (fma -4 (* c a) (* b b))) b)
1.1b
(- (- b) (sqrt (fma -4 (* c a) (* b b))))

rewrite29.0ms

Algorithm
rewrite-expression-head
Counts
4 → 93
Calls

4 calls. Slowest were:

10.0ms
(- (sqrt (fma -4 (* c a) (* b b))) b)
9.0ms
(- (- b) (sqrt (fma -4 (* c a) (* b b))))
4.0ms
(sqrt (fma -4 (* c a) (* b b)))

series274.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

82.0ms
(sqrt (fma -4 (* c a) (* b b)))
73.0ms
(- (- b) (sqrt (fma -4 (* c a) (* b b))))
72.0ms
(- (sqrt (fma -4 (* c a) (* b b))) b)
47.0ms
(sqrt (fma -4 (* c a) (* b b)))

simplify2.2s

Counts
81 → 105
Calls

81 calls. Slowest were:

92.0ms
(- (* 2 (/ (* a c) b)) (* 2 b))
75.0ms
(fma 1 (- b) (- (* (sqrt (fma -4 (* c a) (* b b))) (sqrt 1))))
67.0ms
(fma (sqrt (- b)) (sqrt (- b)) (- (* (sqrt (sqrt (fma -4 (* c a) (* b b)))) (sqrt (sqrt (fma -4 (* c a) (* b b)))))))

prune1.8s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 13.7b

localize109.0ms

Local error

Found 4 expressions with local error:

23.9b
(cbrt (fma (* c a) -4 (* b b)))
23.9b
(cbrt (fma (* c a) -4 (* b b)))
23.2b
(sqrt (fma -4 (* c a) (* b b)))
17.1b
(fma (fabs (cbrt (fma (* c a) -4 (* b b)))) (sqrt (cbrt (fma (* c a) -4 (* b b)))) b)

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 50
Calls

4 calls. Slowest were:

2.0ms
(sqrt (fma -4 (* c a) (* b b)))
1.0ms
(cbrt (fma (* c a) -4 (* b b)))
1.0ms
(cbrt (fma (* c a) -4 (* b b)))

series777.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

337.0ms
(fma (fabs (cbrt (fma (* c a) -4 (* b b)))) (sqrt (cbrt (fma (* c a) -4 (* b b)))) b)
224.0ms
(cbrt (fma (* c a) -4 (* b b)))
168.0ms
(cbrt (fma (* c a) -4 (* b b)))
48.0ms
(sqrt (fma -4 (* c a) (* b b)))

simplify1.5s

Counts
22 → 62
Calls

22 calls. Slowest were:

206.0ms
(- (+ (* (fabs (pow (fma (* a c) -4 (pow b 2)) 1/3)) (pow b 1/3)) b) (* 2/3 (* (* (fabs (pow (fma (* a c) -4 (pow b 2)) 1/3)) (* a c)) (pow (/ 1 (pow b 5)) 1/3))))
199.0ms
(+ (* (exp (* 1/6 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c)))))) (fabs (pow (fma (* a c) -4 (pow b 2)) 1/3))) b)
183.0ms
(- (pow b 2/3) (* 4/3 (* (* a c) (pow (/ 1 (pow b 4)) 1/3))))

prune1.5s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 13.7b

localize119.0ms

Local error

Found 4 expressions with local error:

23.2b
(sqrt (fma -4 (* c a) (* b b)))
23.2b
(sqrt (fma -4 (* c a) (* b b)))
23.2b
(sqrt (fma -4 (* c a) (* b b)))
1.1b
(- (sqrt (fma -4 (* c a) (* b b))) b)

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 61
Calls

4 calls. Slowest were:

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

series221.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

73.0ms
(- (sqrt (fma -4 (* c a) (* b b))) b)
63.0ms
(sqrt (fma -4 (* c a) (* b b)))
45.0ms
(sqrt (fma -4 (* c a) (* b b)))
39.0ms
(sqrt (fma -4 (* c a) (* b b)))

simplify194.0ms

Counts
29 → 73
Calls

29 calls. Slowest were:

20.0ms
(* (cbrt (sqrt (fma -4 (* c a) (* b b)))) (cbrt (sqrt (fma -4 (* c a) (* b b)))))
20.0ms
(sqrt (* (cbrt (fma -4 (* c a) (* b b))) (cbrt (fma -4 (* c a) (* b b)))))
19.0ms
(sqrt (sqrt (fma -4 (* c a) (* b b))))

prune1.4s

Pruning

9 alts after pruning (7 fresh and 2 done)

Merged error: 13.7b

localize137.0ms

Local error

Found 4 expressions with local error:

23.2b
(sqrt (fma -4 (* c a) (* b b)))
23.2b
(sqrt (fma -4 (* c a) (* b b)))
23.2b
(sqrt (fma -4 (* c a) (* b b)))
23.2b
(sqrt (fma -4 (* c a) (* b b)))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

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

series275.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

71.0ms
(sqrt (fma -4 (* c a) (* b b)))
69.0ms
(sqrt (fma -4 (* c a) (* b b)))
69.0ms
(sqrt (fma -4 (* c a) (* b b)))
65.0ms
(sqrt (fma -4 (* c a) (* b b)))

simplify221.0ms

Counts
28 → 68
Calls

28 calls. Slowest were:

23.0ms
(sqrt (* (cbrt (fma -4 (* c a) (* b b))) (cbrt (fma -4 (* c a) (* b b)))))
21.0ms
(sqrt (sqrt (fma -4 (* c a) (* b b))))
21.0ms
(sqrt (* (cbrt (fma -4 (* c a) (* b b))) (cbrt (fma -4 (* c a) (* b b)))))

prune1.3s

Pruning

9 alts after pruning (6 fresh and 3 done)

Merged error: 13.7b

regimes317.0ms

Accuracy

89.3% (0.7b remaining)

Error of 13.6b against oracle of 12.9b and baseline of 19.7b

bsearch260.0ms

end0.0ms

sample3.3s

Algorithm
intervals