Average Error: 19.4 → 13.7
Time: 20.8s
Precision: 64
Internal Precision: 128
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 7.79324369207276 \cdot 10^{+95}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{-(\left(\left|\sqrt[3]{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right|\right) \cdot \left(\sqrt{\sqrt[3]{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\right) + b)_*}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 7.79324369207276e+95

    1. Initial program 15.3

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

    2. Simplified15.3

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

    4. Applied add-cube-cbrt15.4

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

    5. Applied sqrt-prod15.4

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

    6. Applied *-un-lft-identity15.4

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

    7. Applied prod-diff15.5

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

    8. Simplified15.5

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

    9. Simplified15.4

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

    if 7.79324369207276e+95 < b

    1. Initial program 42.9

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

    2. Simplified42.8

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

    3. Taylor expanded around 0 4.0

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

  4. Final simplification13.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 7.79324369207276 \cdot 10^{+95}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{-(\left(\left|\sqrt[3]{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right|\right) \cdot \left(\sqrt{\sqrt[3]{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\right) + b)_*}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))

Details

Time bar (total: 19.0s)Debug log

sample119.0ms

Algorithm
intervals

simplify98.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

98.0ms
(if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c))))))

prune13.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 19.9b

localize159.0ms

Local error

Found 4 expressions with local error:

24.3b
(sqrt (fma c (* a -4) (* b b)))
24.3b
(sqrt (fma c (* a -4) (* b b)))
1.1b
(- (sqrt (fma c (* a -4) (* b b))) b)
1.1b
(- (- b) (sqrt (fma c (* a -4) (* b b))))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 93
Calls

4 calls. Slowest were:

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

series245.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

70.0ms
(sqrt (fma c (* a -4) (* b b)))
69.0ms
(- (- b) (sqrt (fma c (* a -4) (* b b))))
53.0ms
(sqrt (fma c (* a -4) (* b b)))
53.0ms
(- (sqrt (fma c (* a -4) (* b b))) b)

simplify2.3s

Counts
81 → 105
Calls

81 calls. Slowest were:

76.0ms
(fma 1 (- b) (- (* (sqrt (fma c (* a -4) (* b b))) 1)))
71.0ms
(fma 1 (- b) (- (* (sqrt (cbrt (fma c (* a -4) (* b b)))) (sqrt (* (cbrt (fma c (* a -4) (* b b))) (cbrt (fma c (* a -4) (* b b))))))))
65.0ms
(fma (* (cbrt (- b)) (cbrt (- b))) (cbrt (- b)) (- (* (sqrt (fma c (* a -4) (* b b))) (sqrt 1))))

prune1.5s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 13.9b

localize159.0ms

Local error

Found 4 expressions with local error:

25.1b
(cbrt (fma (* a -4) c (* b b)))
25.1b
(cbrt (fma (* a -4) c (* b b)))
24.3b
(sqrt (fma c (* a -4) (* b b)))
22.7b
(fma (fabs (cbrt (fma (* a -4) c (* b b)))) (sqrt (cbrt (fma (* a -4) c (* b b)))) b)

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 50
Calls

4 calls. Slowest were:

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

series809.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

313.0ms
(fma (fabs (cbrt (fma (* a -4) c (* b b)))) (sqrt (cbrt (fma (* a -4) c (* b b)))) b)
233.0ms
(cbrt (fma (* a -4) c (* b b)))
210.0ms
(cbrt (fma (* a -4) c (* b b)))
51.0ms
(sqrt (fma c (* a -4) (* b b)))

simplify1.5s

Counts
22 → 62
Calls

22 calls. Slowest were:

205.0ms
(exp (* 1/3 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c))))))
186.0ms
(- (pow b 2/3) (* 4/3 (* (* a c) (pow (/ 1 (pow b 4)) 1/3))))
179.0ms
(- (+ b (* (fabs (pow (fma (* -4 a) c (pow b 2)) 1/3)) (pow b 1/3))) (* 2/3 (* (* a (* (fabs (pow (fma (* -4 a) c (pow b 2)) 1/3)) c)) (pow (/ 1 (pow b 5)) 1/3))))

prune1.5s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 13.9b

localize121.0ms

Local error

Found 4 expressions with local error:

24.3b
(sqrt (fma c (* a -4) (* b b)))
24.3b
(sqrt (fma c (* a -4) (* b b)))
24.3b
(sqrt (fma c (* a -4) (* b b)))
1.1b
(- (sqrt (fma c (* a -4) (* b b))) b)

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 61
Calls

4 calls. Slowest were:

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

series239.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

72.0ms
(sqrt (fma c (* a -4) (* b b)))
67.0ms
(sqrt (fma c (* a -4) (* b b)))
55.0ms
(- (sqrt (fma c (* a -4) (* b b))) b)
45.0ms
(sqrt (fma c (* a -4) (* b b)))

simplify176.0ms

Counts
29 → 73
Calls

29 calls. Slowest were:

21.0ms
(sqrt (* (cbrt (fma c (* a -4) (* b b))) (cbrt (fma c (* a -4) (* b b)))))
19.0ms
(sqrt (sqrt (fma c (* a -4) (* b b))))
17.0ms
(sqrt (fma c (* a -4) (* b b)))

prune2.0s

Pruning

10 alts after pruning (8 fresh and 2 done)

Merged error: 13.9b

localize135.0ms

Local error

Found 4 expressions with local error:

25.1b
(cbrt (fma c (* a -4) (* b b)))
25.1b
(cbrt (fma (* c -4) a (* b b)))
24.3b
(sqrt (fma c (* a -4) (* b b)))
5.5b
(- (* (fabs (cbrt (fma (* c -4) a (* b b)))) (sqrt (cbrt (fma c (* a -4) (* b b))))) b)

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
4 → 54
Calls

4 calls. Slowest were:

14.0ms
(- (* (fabs (cbrt (fma (* c -4) a (* b b)))) (sqrt (cbrt (fma c (* a -4) (* b b))))) b)
2.0ms
(sqrt (fma c (* a -4) (* b b)))
1.0ms
(cbrt (fma c (* a -4) (* b b)))

series808.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

314.0ms
(- (* (fabs (cbrt (fma (* c -4) a (* b b)))) (sqrt (cbrt (fma c (* a -4) (* b b))))) b)
235.0ms
(cbrt (fma c (* a -4) (* b b)))
214.0ms
(cbrt (fma (* c -4) a (* b b)))
44.0ms
(sqrt (fma c (* a -4) (* b b)))

simplify1.7s

Counts
23 → 66
Calls

23 calls. Slowest were:

224.0ms
(exp (* 1/3 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c))))))
215.0ms
(- (pow b 2/3) (* 4/3 (* (* a c) (pow (/ 1 (pow b 4)) 1/3))))
201.0ms
(- (pow b 2/3) (* 4/3 (* (* a c) (pow (/ 1 (pow b 4)) 1/3))))

prune1.5s

Pruning

11 alts after pruning (8 fresh and 3 done)

Merged error: 13.9b

regimes327.0ms

Accuracy

89.1% (0.7b remaining)

Error of 13.7b against oracle of 13.0b and baseline of 19.5b

bsearch236.0ms

end0.0ms

sample3.4s

Algorithm
intervals