Average Error: 33.8 → 6.5
Time: 25.2s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.170102184885046 \cdot 10^{+152}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 7.1019336895705 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\frac{1}{2} \cdot c\right) \cdot 4}{\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b}\\ \mathbf{elif}\;b \le 5.212862281681239 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

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. Taylor expanded around -inf 33.7

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

    4. Simplified33.7

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

    6. Applied flip--33.8

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

    7. Applied associate-/l/37.6

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

    8. Simplified19.4

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

    10. Applied times-frac14.2

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

    11. Simplified8.6

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

    12. Simplified8.6

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

    14. Applied associate-*r/8.5

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

    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. Taylor expanded around -inf 45.2

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

    4. Simplified45.1

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

    6. Applied flip--61.7

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

    7. Applied associate-/l/62.0

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

    8. Simplified62.1

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

    9. Taylor expanded around 0 3.3

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

    10. Simplified3.3

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.170102184885046 \cdot 10^{+152}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 7.1019336895705 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\frac{1}{2} \cdot c\right) \cdot 4}{\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b}\\ \mathbf{elif}\;b \le 5.212862281681239 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :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: 24.2s)Debug log

sample154.0ms

Algorithm
intervals

simplify110.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

109.0ms
(/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a))

prune13.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 34.2b

localize27.0ms

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))

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

series267.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

83.0ms
(/ (- (- b) (sqrt (fma (* -4 a) c (* b b)))) (* 2 a))
74.0ms
(- (- b) (sqrt (fma (* -4 a) c (* b b))))
69.0ms
(sqrt (fma (* -4 a) c (* b b)))
40.0ms
(fma (* -4 a) c (* b b))

simplify3.4s

Counts
93 → 112
Calls

93 calls. Slowest were:

252.0ms
(- (* (- b) (- b)) (* (sqrt (fma (* -4 a) c (* b b))) (sqrt (fma (* -4 a) c (* b b)))))
207.0ms
(- (pow (- b) 3) (pow (sqrt (fma (* -4 a) c (* b b))) 3))
102.0ms
(- (* 2 (/ (* a c) b)) (* 2 b))

prune1.4s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 6.3b

localize29.0ms

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))

rewrite37.0ms

Algorithm
rewrite-expression-head
Counts
4 → 100
Calls

4 calls. Slowest were:

23.0ms
(/ (- (- b) (sqrt (fma (* a -4) c (* b b)))) (* 2 a))
10.0ms
(- (- b) (sqrt (fma (* a -4) c (* b b))))
2.0ms
(sqrt (fma (* a -4) c (* b b)))

series244.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

79.0ms
(/ (- (- b) (sqrt (fma (* a -4) c (* b b)))) (* 2 a))
72.0ms
(sqrt (fma (* a -4) c (* b b)))
55.0ms
(- (- b) (sqrt (fma (* a -4) c (* b b))))
38.0ms
(fma (* a -4) c (* b b))

simplify3.3s

Counts
93 → 112
Calls

93 calls. Slowest were:

294.0ms
(- (* (- b) (- b)) (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))))
224.0ms
(- (pow (- b) 3) (pow (sqrt (fma (* a -4) c (* b b))) 3))
77.0ms
(fma 1 (- b) (- (* (sqrt (cbrt (fma (* a -4) c (* b b)))) (sqrt (* (cbrt (fma (* a -4) c (* b b))) (cbrt (fma (* a -4) c (* b b))))))))

prune1.2s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 6.3b

localize28.0ms

Local error

Found 4 expressions with local error:

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

rewrite51.0ms

Algorithm
rewrite-expression-head
Counts
4 → 76
Calls

4 calls. Slowest were:

21.0ms
(* (* 2 a) (+ (- b) (sqrt (fma (* a -4) c (* b b)))))
19.0ms
(/ (* (* a c) 4) (* (* 2 a) (+ (- b) (sqrt (fma (* a -4) c (* b b))))))
8.0ms
(+ (- b) (sqrt (fma (* a -4) c (* b b))))

series345.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

160.0ms
(/ (* (* a c) 4) (* (* 2 a) (+ (- b) (sqrt (fma (* a -4) c (* b b))))))
66.0ms
(sqrt (fma (* a -4) c (* b b)))
64.0ms
(+ (- b) (sqrt (fma (* a -4) c (* b b))))
55.0ms
(* (* 2 a) (+ (- b) (sqrt (fma (* a -4) c (* b b)))))

simplify4.0s

Counts
41 → 88
Calls

41 calls. Slowest were:

690.0ms
(/ (* (* (* (* a c) 4) (* (* a c) 4)) (* (* a c) 4)) (* (* (* (* 2 a) (* 2 a)) (* 2 a)) (* (* (+ (- b) (sqrt (fma (* a -4) c (* b b)))) (+ (- b) (sqrt (fma (* a -4) c (* b b))))) (+ (- b) (sqrt (fma (* a -4) c (* b b)))))))
619.0ms
(/ (* (* (* (* a c) 4) (* (* a c) 4)) (* (* a c) 4)) (* (* (* (* 2 a) (+ (- b) (sqrt (fma (* a -4) c (* b b))))) (* (* 2 a) (+ (- b) (sqrt (fma (* a -4) c (* b b)))))) (* (* 2 a) (+ (- b) (sqrt (fma (* a -4) c (* b b)))))))
576.0ms
(- (log (* (* a c) 4)) (+ (log (* 2 a)) (log (+ (- b) (sqrt (fma (* a -4) c (* b b)))))))

prune1.1s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 6.3b

localize11.0ms

Local error

Found 4 expressions with local error:

24.5b
(sqrt (fma (* a -4) c (* b b)))
1.5b
(- (sqrt (fma (* a -4) c (* b b))) b)
0.2b
(* (* 1/2 c) (/ 4 (- (sqrt (fma (* a -4) c (* b b))) b)))
0.1b
(/ 4 (- (sqrt (fma (* a -4) c (* b b))) b))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 71
Calls

4 calls. Slowest were:

5.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
5.0ms
(* (* 1/2 c) (/ 4 (- (sqrt (fma (* a -4) c (* b b))) b)))
3.0ms
(/ 4 (- (sqrt (fma (* a -4) c (* b b))) b))

series424.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

180.0ms
(* (* 1/2 c) (/ 4 (- (sqrt (fma (* a -4) c (* b b))) b)))
131.0ms
(/ 4 (- (sqrt (fma (* a -4) c (* b b))) b))
69.0ms
(sqrt (fma (* a -4) c (* b b)))
43.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)

simplify1.4s

Counts
36 → 83
Calls

36 calls. Slowest were:

582.0ms
(* (* (* (* 1/2 c) (* 1/2 c)) (* 1/2 c)) (* (* (/ 4 (- (sqrt (fma (* a -4) c (* b b))) b)) (/ 4 (- (sqrt (fma (* a -4) c (* b b))) b))) (/ 4 (- (sqrt (fma (* a -4) c (* b b))) b))))
91.0ms
(+ (log (* 1/2 c)) (log (/ 4 (- (sqrt (fma (* a -4) c (* b b))) b))))
87.0ms
(+ (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (+ (* b b) (* (sqrt (fma (* a -4) c (* b b))) b)))

prune1.3s

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 6.3b

regimes230.0ms

Accuracy

96% (1.0b remaining)

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

bsearch654.0ms

end0.0ms

sample4.4s

Algorithm
intervals