Average Error: 33.8 → 6.5
Time: 25.5s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.170102184885046 \cdot 10^{+152}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 7.1019336895705 \cdot 10^{-310}:\\ \;\;\;\;\frac{4 \cdot c}{2 \cdot \left(\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b\right)}\\ \mathbf{elif}\;b \le 5.212862281681239 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.8
Target21.1
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -7.170102184885046e+152

    1. Initial program 62.8

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

    2. Simplified62.8

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

    3. Taylor expanded around -inf 1.4

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

    4. Simplified1.4

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

    if -7.170102184885046e+152 < b < 7.1019336895705e-310

    1. Initial program 33.7

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

    2. Simplified33.7

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

    4. Applied flip--33.8

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

    5. Applied associate-/l/37.6

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

    6. Simplified19.4

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

    8. Applied times-frac14.2

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

    9. Simplified8.6

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

    10. Simplified8.6

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

    12. Applied frac-times8.5

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

    if 7.1019336895705e-310 < b < 5.212862281681239e+99

    1. Initial program 9.0

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

    2. Simplified9.0

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

    3. Taylor expanded around -inf 9.0

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

    4. Simplified9.0

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

    if 5.212862281681239e+99 < b

    1. Initial program 45.2

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

    2. Simplified45.1

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

    4. Applied flip--61.7

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

    5. Applied associate-/l/62.0

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

    6. Simplified62.1

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

    8. Applied times-frac61.9

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

    9. Simplified61.8

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

    10. Simplified61.8

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

    11. Taylor expanded around 0 3.3

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

    12. Simplified3.3

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.170102184885046 \cdot 10^{+152}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 7.1019336895705 \cdot 10^{-310}:\\ \;\;\;\;\frac{4 \cdot c}{2 \cdot \left(\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b\right)}\\ \mathbf{elif}\;b \le 5.212862281681239 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 +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: 24.5s)Debug log

sample124.0ms

Algorithm
intervals

simplify104.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

104.0ms
(/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a))

prune27.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 34.2b

localize22.0ms

Local error

Found 4 expressions with local error:

24.5b
(sqrt (fma (* -4 a) c (* b b)))
1.7b
(- (- b) (sqrt (fma (* -4 a) c (* b b))))
0.1b
(/ (- (- b) (sqrt (fma (* -4 a) c (* b b)))) (* 2 a))
0.0b
(fma (* -4 a) c (* b b))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 100
Calls

4 calls. Slowest were:

16.0ms
(/ (- (- b) (sqrt (fma (* -4 a) c (* b b)))) (* 2 a))
5.0ms
(- (- b) (sqrt (fma (* -4 a) c (* b b))))
1.0ms
(sqrt (fma (* -4 a) c (* b b)))

series211.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

69.0ms
(/ (- (- b) (sqrt (fma (* -4 a) c (* b b)))) (* 2 a))
54.0ms
(sqrt (fma (* -4 a) c (* b b)))
54.0ms
(- (- b) (sqrt (fma (* -4 a) c (* b b))))
33.0ms
(fma (* -4 a) c (* b b))

simplify3.3s

Counts
93 → 112
Calls

93 calls. Slowest were:

271.0ms
(- (* (- b) (- b)) (* (sqrt (fma (* -4 a) c (* b b))) (sqrt (fma (* -4 a) c (* b b)))))
212.0ms
(- (pow (- b) 3) (pow (sqrt (fma (* -4 a) c (* b b))) 3))
102.0ms
(- (* 2 (/ (* a c) b)) (* 2 b))

prune1.3s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 6.3b

localize23.0ms

Local error

Found 4 expressions with local error:

24.5b
(sqrt (fma (* a -4) c (* b b)))
1.7b
(- (- b) (sqrt (fma (* a -4) c (* b b))))
0.1b
(/ (- (- b) (sqrt (fma (* a -4) c (* b b)))) (* 2 a))
0.0b
(fma (* a -4) c (* b b))

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
4 → 100
Calls

4 calls. Slowest were:

16.0ms
(/ (- (- 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)))

series222.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

74.0ms
(/ (- (- b) (sqrt (fma (* a -4) c (* b b)))) (* 2 a))
60.0ms
(sqrt (fma (* a -4) c (* b b)))
46.0ms
(fma (* a -4) c (* b b))
43.0ms
(- (- b) (sqrt (fma (* a -4) c (* b b))))

simplify3.5s

Counts
93 → 112
Calls

93 calls. Slowest were:

256.0ms
(- (* (- b) (- b)) (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))))
228.0ms
(- (pow (- b) 3) (pow (sqrt (fma (* a -4) c (* b b))) 3))
98.0ms
(/ (* (* (- (- 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

8 alts after pruning (7 fresh and 1 done)

Merged error: 6.3b

localize19.0ms

Local error

Found 4 expressions with local error:

24.5b
(sqrt (fma (* -4 a) c (* b b)))
16.8b
(/ (* (* c a) 4) (* (* 2 a) (+ (- b) (sqrt (fma (* -4 a) c (* b b))))))
1.5b
(+ (- b) (sqrt (fma (* -4 a) c (* b b))))
0.1b
(* (* 2 a) (+ (- b) (sqrt (fma (* -4 a) c (* b b)))))

rewrite25.0ms

Algorithm
rewrite-expression-head
Counts
4 → 76
Calls

4 calls. Slowest were:

10.0ms
(/ (* (* c a) 4) (* (* 2 a) (+ (- b) (sqrt (fma (* -4 a) c (* b b))))))
8.0ms
(* (* 2 a) (+ (- b) (sqrt (fma (* -4 a) c (* b b)))))
4.0ms
(+ (- b) (sqrt (fma (* -4 a) c (* b b))))

series415.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

197.0ms
(/ (* (* c a) 4) (* (* 2 a) (+ (- b) (sqrt (fma (* -4 a) c (* b b))))))
86.0ms
(* (* 2 a) (+ (- b) (sqrt (fma (* -4 a) c (* b b)))))
71.0ms
(+ (- b) (sqrt (fma (* -4 a) c (* b b))))
60.0ms
(sqrt (fma (* -4 a) c (* b b)))

simplify4.9s

Counts
41 → 88
Calls

41 calls. Slowest were:

927.0ms
(+ (* (- b) (- b)) (- (* (sqrt (fma (* -4 a) c (* b b))) (sqrt (fma (* -4 a) c (* b b)))) (* (- b) (sqrt (fma (* -4 a) c (* b b))))))
795.0ms
(/ (* (* (* (* c a) 4) (* (* c a) 4)) (* (* c a) 4)) (* (* (* (* 2 a) (+ (- b) (sqrt (fma (* -4 a) c (* b b))))) (* (* 2 a) (+ (- b) (sqrt (fma (* -4 a) c (* b b)))))) (* (* 2 a) (+ (- b) (sqrt (fma (* -4 a) c (* b b)))))))
763.0ms
(/ (* (* (* (* c a) 4) (* (* c a) 4)) (* (* c a) 4)) (* (* (* (* 2 a) (* 2 a)) (* 2 a)) (* (* (+ (- b) (sqrt (fma (* -4 a) c (* b b)))) (+ (- b) (sqrt (fma (* -4 a) c (* b b))))) (+ (- b) (sqrt (fma (* -4 a) c (* b b)))))))

prune1.1s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 6.3b

localize22.0ms

Local error

Found 4 expressions with local error:

24.5b
(sqrt (fma (* -4 a) c (* b b)))
1.5b
(- (sqrt (fma (* -4 a) c (* b b))) b)
0.2b
(* (/ c 2) (/ 4 (- (sqrt (fma (* -4 a) c (* b b))) b)))
0.1b
(/ 4 (- (sqrt (fma (* -4 a) c (* b b))) b))

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 76
Calls

4 calls. Slowest were:

10.0ms
(- (sqrt (fma (* -4 a) c (* b b))) b)
9.0ms
(* (/ c 2) (/ 4 (- (sqrt (fma (* -4 a) c (* b b))) b)))
5.0ms
(/ 4 (- (sqrt (fma (* -4 a) c (* b b))) b))

series406.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

161.0ms
(* (/ c 2) (/ 4 (- (sqrt (fma (* -4 a) c (* b b))) b)))
148.0ms
(/ 4 (- (sqrt (fma (* -4 a) c (* b b))) b))
51.0ms
(- (sqrt (fma (* -4 a) c (* b b))) b)
46.0ms
(sqrt (fma (* -4 a) c (* b b)))

simplify1.3s

Counts
41 → 88
Calls

41 calls. Slowest were:

386.0ms
(* (* (* (/ c 2) (/ c 2)) (/ c 2)) (* (* (/ 4 (- (sqrt (fma (* -4 a) c (* b b))) b)) (/ 4 (- (sqrt (fma (* -4 a) c (* b b))) b))) (/ 4 (- (sqrt (fma (* -4 a) c (* b b))) b))))
171.0ms
(+ (log (/ c 2)) (log (/ 4 (- (sqrt (fma (* -4 a) c (* b b))) b))))
72.0ms
(+ (* (sqrt (fma (* -4 a) c (* b b))) (sqrt (fma (* -4 a) c (* b b)))) (+ (* b b) (* (sqrt (fma (* -4 a) c (* b b))) b)))

prune1.1s

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 6.3b

regimes244.0ms

Accuracy

96% (1.0b remaining)

Error of 6.5b against oracle of 5.5b and baseline of 29.8b

bsearch613.0ms

end0.0ms

sample4.2s

Algorithm
intervals