Average Error: 33.2 → 9.6
Time: 35.4s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.9529633933266 \cdot 10^{-08}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -1.432147794020809 \cdot 10^{-104}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot 4}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) + \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}\right)}\\ \mathbf{elif}\;b \le -5.51185453096989 \cdot 10^{-127}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 8.046684077965657 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.2
Target20.4
Herbie9.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -4.9529633933266e-08 or -1.432147794020809e-104 < b < -5.51185453096989e-127

    1. Initial program 53.5

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

    2. Simplified53.5

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

    3. Taylor expanded around 0 53.5

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

    4. Simplified53.5

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

    5. Taylor expanded around -inf 7.8

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

    6. Simplified7.8

      \[\leadsto \color{blue}{\frac{-c}{b}}\]

    if -4.9529633933266e-08 < b < -1.432147794020809e-104

    1. Initial program 34.9

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

    2. Simplified34.9

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

    4. Applied flip--34.9

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

    5. Applied associate-/l/38.7

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

    6. Simplified21.5

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

    if -5.51185453096989e-127 < b < 8.046684077965657e+99

    1. Initial program 11.0

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

    2. Simplified11.0

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

    3. Taylor expanded around 0 11.0

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

    4. Simplified11.0

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

    6. Applied *-un-lft-identity11.0

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

    7. Applied *-un-lft-identity11.0

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

    8. Applied distribute-lft-out--11.0

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

    9. Applied associate-/l*11.1

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

    if 8.046684077965657e+99 < b

    1. Initial program 44.6

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

    2. Simplified44.6

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

    3. Taylor expanded around 0 44.6

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

    4. Simplified44.6

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

    5. Taylor expanded around inf 3.5

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.9529633933266 \cdot 10^{-08}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -1.432147794020809 \cdot 10^{-104}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot 4}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) + \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}\right)}\\ \mathbf{elif}\;b \le -5.51185453096989 \cdot 10^{-127}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 8.046684077965657 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

Details

Time bar (total: 34.1s)Debug log

sample174.0ms

Algorithm
intervals

simplify129.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

129.0ms
(/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a))

prune21.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 33.9b

localize44.0ms

Local error

Found 4 expressions with local error:

22.0b
(sqrt (fma (* -4 a) c (* b b)))
0.9b
(- (- b) (sqrt (fma (* -4 a) c (* b b))))
0.2b
(fma (* -4 a) c (* b b))
0.1b
(/ (- (- b) (sqrt (fma (* -4 a) c (* b b)))) (* 2 a))

rewrite36.0ms

Algorithm
rewrite-expression-head
Counts
4 → 100
Calls

4 calls. Slowest were:

22.0ms
(/ (- (- b) (sqrt (fma (* -4 a) c (* b b)))) (* 2 a))
11.0ms
(- (- b) (sqrt (fma (* -4 a) c (* b b))))
1.0ms
(sqrt (fma (* -4 a) c (* b b)))

series223.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

67.0ms
(/ (- (- b) (sqrt (fma (* -4 a) c (* b b)))) (* 2 a))
54.0ms
(- (- b) (sqrt (fma (* -4 a) c (* b b))))
51.0ms
(fma (* -4 a) c (* b b))
50.0ms
(sqrt (fma (* -4 a) c (* b b)))

simplify3.6s

Counts
93 → 112
Calls

93 calls. Slowest were:

267.0ms
(- (* (- b) (- b)) (* (sqrt (fma (* -4 a) c (* b b))) (sqrt (fma (* -4 a) c (* b b)))))
235.0ms
(- (pow (- b) 3) (pow (sqrt (fma (* -4 a) c (* b b))) 3))
95.0ms
(/ (* (* (- (- b) (sqrt (fma (* -4 a) c (* b b)))) (- (- b) (sqrt (fma (* -4 a) c (* b b))))) (- (- b) (sqrt (fma (* -4 a) c (* b b))))) (* (* (* 2 a) (* 2 a)) (* 2 a)))

prune1.3s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 5.1b

localize14.0ms

Local error

Found 4 expressions with local error:

22.0b
(sqrt (fma (* a -4) c (* b b)))
0.9b
(- (- b) (sqrt (fma (* a -4) c (* b b))))
0.2b
(fma (* a -4) c (* b b))
0.1b
(/ (- (- b) (sqrt (fma (* a -4) c (* b b)))) (* 2 a))

rewrite26.0ms

Algorithm
rewrite-expression-head
Counts
4 → 100
Calls

4 calls. Slowest were:

19.0ms
(/ (- (- b) (sqrt (fma (* a -4) c (* b b)))) (* 2 a))
5.0ms
(- (- b) (sqrt (fma (* a -4) c (* b b))))
1.0ms
(sqrt (fma (* a -4) c (* b b)))

series210.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

85.0ms
(/ (- (- b) (sqrt (fma (* a -4) c (* b b)))) (* 2 a))
52.0ms
(- (- b) (sqrt (fma (* a -4) c (* b b))))
44.0ms
(sqrt (fma (* a -4) c (* b b)))
29.0ms
(fma (* a -4) c (* b b))

simplify3.3s

Counts
93 → 112
Calls

93 calls. Slowest were:

262.0ms
(- (* (- b) (- b)) (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))))
170.0ms
(- (pow (- b) 3) (pow (sqrt (fma (* a -4) c (* b b))) 3))
81.0ms
(/ (* (* (- (- b) (sqrt (fma (* a -4) c (* b b)))) (- (- b) (sqrt (fma (* a -4) c (* b b))))) (- (- b) (sqrt (fma (* a -4) c (* b b))))) (* (* (* 2 a) (* 2 a)) (* 2 a)))

prune1.3s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 5.1b

localize6.0ms

Local error

Found 4 expressions with local error:

22.0b
(sqrt (fma (* a -4) c (* b b)))
0.9b
(- (- b) (sqrt (fma (* a -4) c (* b b))))
0.2b
(fma (* a -4) c (* b b))
0.2b
(/ 1 (/ (* 2 a) (- (- b) (sqrt (fma (* a -4) c (* b b))))))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 99
Calls

4 calls. Slowest were:

4.0ms
(- (- b) (sqrt (fma (* a -4) c (* b b))))
3.0ms
(/ 1 (/ (* 2 a) (- (- b) (sqrt (fma (* a -4) c (* b b))))))
1.0ms
(sqrt (fma (* a -4) c (* b b)))

series234.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

75.0ms
(/ 1 (/ (* 2 a) (- (- b) (sqrt (fma (* a -4) c (* b b))))))
64.0ms
(- (- b) (sqrt (fma (* a -4) c (* b b))))
58.0ms
(sqrt (fma (* a -4) c (* b b)))
36.0ms
(fma (* a -4) c (* b b))

simplify3.2s

Counts
86 → 111
Calls

86 calls. Slowest were:

444.0ms
(+ (* (- b) (- b)) (+ (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (* (- b) (sqrt (fma (* a -4) c (* b b))))))
76.0ms
(fma 1 (- b) (- (* (sqrt (fma (* a -4) c (* b b))) (sqrt 1))))
71.0ms
(- (* 2 (/ (* a c) b)) (* 2 b))

prune1.4s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 5.1b

localize9.0ms

Local error

Found 4 expressions with local error:

22.0b
(sqrt (fma (* a -4) c (* b b)))
0.9b
(- (- b) (sqrt (fma (* a -4) c (* b b))))
0.3b
(/ (/ 1 (* 2 a)) (/ 1 (- (- b) (sqrt (fma (* a -4) c (* b b))))))
0.2b
(fma (* a -4) c (* b b))

rewrite26.0ms

Algorithm
rewrite-expression-head
Counts
4 → 123
Calls

4 calls. Slowest were:

14.0ms
(/ (/ 1 (* 2 a)) (/ 1 (- (- b) (sqrt (fma (* a -4) c (* b b))))))
7.0ms
(- (- b) (sqrt (fma (* a -4) c (* b b))))
1.0ms
(sqrt (fma (* a -4) c (* b b)))

series270.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

82.0ms
(/ (/ 1 (* 2 a)) (/ 1 (- (- b) (sqrt (fma (* a -4) c (* b b))))))
74.0ms
(- (- b) (sqrt (fma (* a -4) c (* b b))))
67.0ms
(sqrt (fma (* a -4) c (* b b)))
47.0ms
(fma (* a -4) c (* b b))

simplify10.6s

Counts
134 → 135
Calls

134 calls. Slowest were:

594.0ms
(/ 1 (/ 1 (- (* (- b) (- b)) (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))))))
592.0ms
(/ (sqrt (/ 1 (* 2 a))) (+ (* (- b) (- b)) (+ (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (* (- b) (sqrt (fma (* a -4) c (* b b)))))))
575.0ms
(/ 1 (/ 1 (- (* (- b) (- b)) (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))))))

prune1.8s

Pruning

10 alts after pruning (8 fresh and 2 done)

Merged error: 5.1b

regimes404.0ms

Accuracy

85.3% (4.1b remaining)

Error of 9.6b against oracle of 5.5b and baseline of 33.2b

bsearch726.0ms

end0.0ms

sample5.0s

Algorithm
intervals