Average Error: 33.4 → 7.4
Time: 20.5s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.1317624771383882 \cdot 10^{+58}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.080192248888389 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{2 \cdot a} - \frac{b}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.3660877782595927 \cdot 10^{+23}:\\ \;\;\;\;\left(c \cdot -2\right) \cdot \frac{1}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if b < -2.1317624771383882e+58

    1. Initial program 37.4

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

    2. Simplified37.4

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

    3. Taylor expanded around -inf 5.4

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

    if -2.1317624771383882e+58 < b < 1.080192248888389e-171

    1. Initial program 11.7

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

    2. Simplified11.7

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

    4. Applied div-sub11.7

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

    if 1.080192248888389e-171 < b < 1.3660877782595927e+23

    1. Initial program 32.3

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

    2. Simplified32.3

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

    4. Applied flip--32.4

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

    5. Applied associate-/l/37.4

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

    6. Simplified20.4

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

    8. Applied associate-/r*14.6

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

    9. Taylor expanded around inf 5.8

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

    11. Applied div-inv5.9

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

    if 1.3660877782595927e+23 < b

    1. Initial program 55.5

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

    2. Simplified55.5

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

    3. Taylor expanded around inf 4.4

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

    4. Simplified4.4

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

  4. Final simplification7.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.1317624771383882 \cdot 10^{+58}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.080192248888389 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{2 \cdot a} - \frac{b}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.3660877782595927 \cdot 10^{+23}:\\ \;\;\;\;\left(c \cdot -2\right) \cdot \frac{1}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019022 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))

Details

Time bar (total: 19.7s)Debug log

sample154.0ms

Algorithm
intervals
Results
58.0ms40×body2560valid
28.0ms185×body80valid
20.0ms27×body1280valid
14.0ms97×body80nan
9.0ms15×body640valid
4.0ms7×body320valid
3.0ms3×body5120valid
2.0ms6×body160valid

simplify80.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
80.0ms
(/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))

prune19.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 32.5b

localize50.0ms

Local error

Found 4 expressions with local error:

21.7b
(sqrt (- (* b b) (* (* c a) 4)))
2.0b
(- (sqrt (- (* b b) (* (* c a) 4))) b)
0.4b
(- (* b b) (* (* c a) 4))
0.1b
(/ (- (sqrt (- (* b b) (* (* c a) 4))) b) (* 2 a))

rewrite65.0ms

Algorithm
rewrite-expression-head
Rules
8×add-sqr-sqrt
7×add-cube-cbrt
7×*-un-lft-identity
6×add-log-exp
6×add-exp-log
6×add-cbrt-cube
4×flip--
4×flip3--
4×pow1
3×associate-/l*
3×sqrt-prod
3×times-frac
2×associate-/l/
2×sqrt-div
2×sub-neg
1×div-inv
1×div-exp
1×div-sub
1×diff-log
1×pow1/2
1×frac-2neg
1×associate-/r*
1×clear-num
1×rem-sqrt-square
1×cbrt-undiv
Counts
4 → 59
Calls
4 calls:
Slowest
37.0ms
(/ (- (sqrt (- (* b b) (* (* c a) 4))) b) (* 2 a))
19.0ms
(- (sqrt (- (* b b) (* (* c a) 4))) b)
4.0ms
(- (* b b) (* (* c a) 4))
4.0ms
(sqrt (- (* b b) (* (* c a) 4)))

series239.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
86.0ms
(/ (- (sqrt (- (* b b) (* (* c a) 4))) b) (* 2 a))
70.0ms
(sqrt (- (* b b) (* (* c a) 4)))
54.0ms
(- (sqrt (- (* b b) (* (* c a) 4))) b)
30.0ms
(- (* b b) (* (* c a) 4))

simplify2.0s

Counts
32 → 71
Calls
32 calls:
Slowest
321.0ms
(sqrt (- (pow (* b b) 3) (pow (* (* c a) 4) 3)))
295.0ms
(- (* (sqrt (- (* b b) (* (* c a) 4))) (sqrt (- (* b b) (* (* c a) 4)))) (* b b))
169.0ms
(- (pow (sqrt (- (* b b) (* (* c a) 4))) 3) (pow b 3))
134.0ms
(- b (* 2 (/ (* a c) b)))
112.0ms
(sqrt (- (* (* b b) (* b b)) (* (* (* c a) 4) (* (* c a) 4))))

prune821.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 5.5b

localize25.0ms

Local error

Found 4 expressions with local error:

21.7b
(sqrt (- (* b b) (* (* c a) 4)))
13.5b
(/ (* (* c -4) a) (* (* 2 a) (+ (sqrt (- (* b b) (* (* c a) 4))) b)))
1.3b
(+ (sqrt (- (* b b) (* (* c a) 4))) b)
0.4b
(- (* b b) (* (* c a) 4))

rewrite47.0ms

Algorithm
rewrite-expression-head
Rules
9×add-exp-log
9×add-cbrt-cube
6×add-log-exp
6×add-sqr-sqrt
5×add-cube-cbrt
5×*-un-lft-identity
4×pow1
3×sqrt-prod
2×flip--
2×flip-+
2×associate-/r/
2×associate-*r/
2×div-exp
2×sqrt-div
2×flip3--
2×flip3-+
2×cbrt-undiv
1×associate-/l*
1×div-inv
1×cbrt-unprod
1×prod-exp
1×diff-log
1×pow1/2
1×times-frac
1×frac-2neg
1×sub-neg
1×associate-/r*
1×clear-num
1×rem-sqrt-square
1×+-commutative
Counts
4 → 55
Calls
4 calls:
Slowest
24.0ms
(/ (* (* c -4) a) (* (* 2 a) (+ (sqrt (- (* b b) (* (* c a) 4))) b)))
14.0ms
(+ (sqrt (- (* b b) (* (* c a) 4))) b)
4.0ms
(sqrt (- (* b b) (* (* c a) 4)))
4.0ms
(- (* b b) (* (* c a) 4))

series284.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
143.0ms
(/ (* (* c -4) a) (* (* 2 a) (+ (sqrt (- (* b b) (* (* c a) 4))) b)))
61.0ms
(sqrt (- (* b b) (* (* c a) 4)))
43.0ms
(+ (sqrt (- (* b b) (* (* c a) 4))) b)
36.0ms
(- (* b b) (* (* c a) 4))

simplify3.4s

Counts
29 → 67
Calls
29 calls:
Slowest
857.0ms
(/ (* (* (* (* c -4) a) (* (* c -4) a)) (* (* c -4) a)) (* (* (* (* 2 a) (* 2 a)) (* 2 a)) (* (* (+ (sqrt (- (* b b) (* (* c a) 4))) b) (+ (sqrt (- (* b b) (* (* c a) 4))) b)) (+ (sqrt (- (* b b) (* (* c a) 4))) b))))
833.0ms
(/ (* (* (* (* c -4) a) (* (* c -4) a)) (* (* c -4) a)) (* (* (* (* 2 a) (+ (sqrt (- (* b b) (* (* c a) 4))) b)) (* (* 2 a) (+ (sqrt (- (* b b) (* (* c a) 4))) b))) (* (* 2 a) (+ (sqrt (- (* b b) (* (* c a) 4))) b))))
310.0ms
(sqrt (- (pow (* b b) 3) (pow (* (* c a) 4) 3)))
267.0ms
(+ (* (sqrt (- (* b b) (* (* c a) 4))) (sqrt (- (* b b) (* (* c a) 4)))) (- (* b b) (* (sqrt (- (* b b) (* (* c a) 4))) b)))
152.0ms
(- (log (* (* c -4) a)) (+ (log (* 2 a)) (log (+ (sqrt (- (* b b) (* (* c a) 4))) b))))

prune769.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 5.5b

localize6.0ms

Local error

Found 4 expressions with local error:

21.7b
(sqrt (- (* b b) (* (* c a) 4)))
12.1b
(/ (* (* c -4) a) (* 2 a))
1.3b
(+ (sqrt (- (* b b) (* (* c a) 4))) b)
0.4b
(- (* b b) (* (* c a) 4))

rewrite23.0ms

Algorithm
rewrite-expression-head
Rules
6×add-log-exp
6×add-exp-log
6×add-cbrt-cube
6×add-sqr-sqrt
5×add-cube-cbrt
5×*-un-lft-identity
4×pow1
3×sqrt-prod
2×flip--
2×sqrt-div
2×flip3--
1×associate-/l*
1×div-inv
1×flip-+
1×div-exp
1×diff-log
1×pow1/2
1×times-frac
1×flip3-+
1×frac-2neg
1×sub-neg
1×associate-/r*
1×clear-num
1×rem-sqrt-square
1×+-commutative
1×cbrt-undiv
Counts
4 → 51
Calls
4 calls:
Slowest
11.0ms
(+ (sqrt (- (* b b) (* (* c a) 4))) b)
5.0ms
(/ (* (* c -4) a) (* 2 a))
4.0ms
(- (* b b) (* (* c a) 4))
2.0ms
(sqrt (- (* b b) (* (* c a) 4)))

series147.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
54.0ms
(+ (sqrt (- (* b b) (* (* c a) 4))) b)
54.0ms
(sqrt (- (* b b) (* (* c a) 4)))
34.0ms
(- (* b b) (* (* c a) 4))
5.0ms
(/ (* (* c -4) a) (* 2 a))

simplify2.2s

Counts
24 → 63
Calls
24 calls:
Slowest
613.0ms
(- (log (* (* c -4) a)) (log (* 2 a)))
444.0ms
(/ (* (* (* (* c -4) a) (* (* c -4) a)) (* (* c -4) a)) (* (* (* 2 a) (* 2 a)) (* 2 a)))
333.0ms
(sqrt (- (pow (* b b) 3) (pow (* (* c a) 4) 3)))
141.0ms
(sqrt (- (* (* b b) (* b b)) (* (* (* c a) 4) (* (* c a) 4))))
126.0ms
(- b (* 2 (/ (* a c) b)))

prune869.0ms

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 5.5b

localize12.0ms

Local error

Found 4 expressions with local error:

21.7b
(sqrt (- (* b b) (* (* c a) 4)))
1.3b
(+ (sqrt (- (* b b) (* (* c a) 4))) b)
0.4b
(- (* b b) (* (* c a) 4))
0.1b
(/ (* -2 c) (+ (sqrt (- (* b b) (* (* c a) 4))) b))

rewrite47.0ms

Algorithm
rewrite-expression-head
Rules
8×add-sqr-sqrt
7×add-cube-cbrt
7×*-un-lft-identity
6×add-log-exp
6×add-exp-log
6×add-cbrt-cube
4×pow1
3×sqrt-prod
3×times-frac
3×associate-/r*
2×flip--
2×flip-+
2×associate-/r/
2×sqrt-div
2×flip3--
2×flip3-+
1×associate-/l*
1×div-inv
1×div-exp
1×diff-log
1×pow1/2
1×frac-2neg
1×sub-neg
1×clear-num
1×rem-sqrt-square
1×+-commutative
1×cbrt-undiv
Counts
4 → 57
Calls
4 calls:
Slowest
24.0ms
(/ (* -2 c) (+ (sqrt (- (* b b) (* (* c a) 4))) b))
15.0ms
(+ (sqrt (- (* b b) (* (* c a) 4))) b)
4.0ms
(- (* b b) (* (* c a) 4))
4.0ms
(sqrt (- (* b b) (* (* c a) 4)))

series269.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
130.0ms
(/ (* -2 c) (+ (sqrt (- (* b b) (* (* c a) 4))) b))
56.0ms
(+ (sqrt (- (* b b) (* (* c a) 4))) b)
56.0ms
(sqrt (- (* b b) (* (* c a) 4)))
27.0ms
(- (* b b) (* (* c a) 4))

simplify1.7s

Counts
32 → 69
Calls
32 calls:
Slowest
300.0ms
(sqrt (- (pow (* b b) 3) (pow (* (* c a) 4) 3)))
254.0ms
(+ (* (sqrt (- (* b b) (* (* c a) 4))) (sqrt (- (* b b) (* (* c a) 4)))) (- (* b b) (* (sqrt (- (* b b) (* (* c a) 4))) b)))
135.0ms
(- b (* 2 (/ (* a c) b)))
121.0ms
(sqrt (- (* (* b b) (* b b)) (* (* (* c a) 4) (* (* c a) 4))))
107.0ms
(- (* 2 b) (* 2 (/ (* a c) b)))

prune699.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 5.5b

regimes279.0ms

Accuracy

92.1% (1.9b remaining)

Error of 7.4b against oracle of 5.5b and baseline of 29.4b

bsearch395.0ms

end0.0ms

sample5.1s

Algorithm
intervals
Results
1.9s1063×body2560valid
963.0ms5847×body80valid
781.0ms924×body1280valid
508.0ms2979×body80nan
341.0ms435×body640valid
143.0ms102×body5120valid
133.0ms222×body320valid
42.0ms145×body160valid