Average Error: 33.4 → 13.4
Time: 19.0s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 4.891503048541261 \cdot 10^{-173}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}{2 \cdot a} - \frac{b}{2 \cdot a}\\ \mathbf{elif}\;b \le 7.50780044314549 \cdot 10^{+35}:\\ \;\;\;\;\left(c \cdot -2\right) \cdot \frac{1}{b + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{b + b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < 4.891503048541261e-173

    1. Initial program 21.0

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

    2. Simplified20.9

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

    4. Applied div-sub20.9

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

    if 4.891503048541261e-173 < b < 7.50780044314549e+35

    1. Initial program 33.2

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

    2. Simplified33.2

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

    4. Applied flip--33.3

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

    5. Applied associate-/l/38.1

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

    6. Simplified19.9

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

    8. Applied times-frac5.8

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

    9. Simplified5.8

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

    11. Applied div-inv5.9

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

    12. Applied associate-*r*5.9

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

    if 7.50780044314549e+35 < b

    1. Initial program 55.8

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

    2. Simplified55.8

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

    4. Applied flip--55.9

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

    5. Applied associate-/l/56.5

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

    6. Simplified28.1

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

    8. Applied times-frac24.4

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

    9. Simplified24.4

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

    10. Taylor expanded around 0 4.1

      \[\leadsto -2 \cdot \frac{c}{\color{blue}{b} + b}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification13.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 4.891503048541261 \cdot 10^{-173}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}{2 \cdot a} - \frac{b}{2 \cdot a}\\ \mathbf{elif}\;b \le 7.50780044314549 \cdot 10^{+35}:\\ \;\;\;\;\left(c \cdot -2\right) \cdot \frac{1}{b + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{b + b}\\ \end{array}\]

Reproduce

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

Details

Time bar (total: 17.8s)Debug log

sample96.0ms

Algorithm
intervals
Results
22.0ms40×body2560valid
17.0ms185×body80valid
14.0ms6×body160valid
13.0ms27×body1280valid
8.0ms97×body80nan
6.0ms15×body640valid
2.0ms3×body5120valid
2.0ms7×body320valid

simplify56.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
56.0ms
(/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))

prune13.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 32.5b

localize22.0ms

Local error

Found 4 expressions with local error:

21.7b
(sqrt (fma (* a -4) c (* b b)))
2.0b
(- (sqrt (fma (* a -4) c (* b b))) b)
0.1b
(/ (- (sqrt (fma (* a -4) c (* b b))) b) (* 2 a))
0.0b
(fma (* a -4) c (* b b))

rewrite18.0ms

Algorithm
rewrite-expression-head
Rules
10×add-sqr-sqrt
9×add-cube-cbrt
9×*-un-lft-identity
6×sqrt-prod
6×add-exp-log
6×fma-neg
6×add-cbrt-cube
4×add-log-exp
4×log1p-expm1-u
4×pow1
4×expm1-log1p-u
3×associate-/l*
3×times-frac
2×associate-/l/
2×flip--
2×flip3--
1×div-inv
1×fma-udef
1×div-exp
1×div-sub
1×pow1/2
1×frac-2neg
1×sub-neg
1×associate-/r*
1×clear-num
1×rem-sqrt-square
1×cbrt-undiv
Counts
4 → 67
Calls
4 calls:
Slowest
11.0ms
(/ (- (sqrt (fma (* a -4) c (* b b))) b) (* 2 a))
5.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
1.0ms
(sqrt (fma (* a -4) c (* b b)))
0.0ms
(fma (* a -4) c (* b b))

series227.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
79.0ms
(/ (- (sqrt (fma (* a -4) c (* b b))) b) (* 2 a))
54.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
47.0ms
(fma (* a -4) c (* b b))
46.0ms
(sqrt (fma (* a -4) c (* b b)))

simplify1.3s

Counts
34 → 79
Calls
34 calls:
Slowest
612.0ms
(- (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (* b b))
109.0ms
(- (pow (sqrt (fma (* a -4) c (* b b))) 3) (pow b 3))
63.0ms
(/ (* (* (- (sqrt (fma (* a -4) c (* b b))) b) (- (sqrt (fma (* a -4) c (* b b))) b)) (- (sqrt (fma (* a -4) c (* b b))) b)) (* (* (* 2 a) (* 2 a)) (* 2 a)))
58.0ms
(- (log (- (sqrt (fma (* a -4) c (* b b))) b)) (log (* 2 a)))
36.0ms
(/ (* (cbrt (- (sqrt (fma (* a -4) c (* b b))) b)) (cbrt (- (sqrt (fma (* a -4) c (* b b))) b))) 2)

prune943.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 17.8b

localize22.0ms

Local error

Found 4 expressions with local error:

21.7b
(sqrt (fma (* a -4) c (* b b)))
13.5b
(/ (* (* a -4) c) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)))
1.3b
(+ (sqrt (fma (* a -4) c (* b b))) b)
0.1b
(* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))

rewrite52.0ms

Algorithm
rewrite-expression-head
Rules
11×add-exp-log
11×add-cbrt-cube
9×add-sqr-sqrt
8×add-cube-cbrt
8×*-un-lft-identity
6×sqrt-prod
6×fma-def
6×pow1
4×add-log-exp
4×log1p-expm1-u
4×associate-*r/
4×expm1-log1p-u
3×flip-+
3×associate-*r*
3×flip3-+
2×cbrt-unprod
2×associate-/r/
2×prod-exp
2×div-exp
2×cbrt-undiv
1×associate-/l*
1×div-inv
1×*-commutative
1×pow-prod-down
1×associate-*l*
1×pow1/2
1×times-frac
1×distribute-lft-in
1×frac-2neg
1×associate-/r*
1×clear-num
1×rem-sqrt-square
1×+-commutative
1×distribute-rgt-in
Counts
4 → 74
Calls
4 calls:
Slowest
25.0ms
(/ (* (* a -4) c) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)))
16.0ms
(* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))
7.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
2.0ms
(sqrt (fma (* a -4) c (* b b)))

series378.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
133.0ms
(/ (* (* a -4) c) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)))
108.0ms
(* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))
69.0ms
(sqrt (fma (* a -4) c (* b b)))
68.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)

simplify3.3s

Counts
41 → 86
Calls
41 calls:
Slowest
738.0ms
(/ (* (* (* (* a -4) c) (* (* a -4) c)) (* (* a -4) c)) (* (* (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))))
567.0ms
(- (log (* (* a -4) c)) (+ (log (* 2 a)) (log (+ (sqrt (fma (* a -4) c (* b b))) b))))
554.0ms
(/ (* (* (* (* a -4) c) (* (* a -4) c)) (* (* a -4) c)) (* (* (* (* 2 a) (* 2 a)) (* 2 a)) (* (* (+ (sqrt (fma (* a -4) c (* b b))) b) (+ (sqrt (fma (* a -4) c (* b b))) b)) (+ (sqrt (fma (* a -4) c (* b b))) b))))
316.0ms
(- (log (* (* a -4) c)) (log (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))))
242.0ms
(+ (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (- (* b b) (* (sqrt (fma (* a -4) c (* b b))) b)))

prune1.1s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 12.3b

localize10.0ms

Local error

Found 4 expressions with local error:

21.7b
(sqrt (fma (* a -4) c (* b b)))
1.3b
(+ (sqrt (fma (* a -4) c (* b b))) b)
0.1b
(/ c (+ (sqrt (fma (* a -4) c (* b b))) b))
0.0b
(* -2 (/ c (+ (sqrt (fma (* a -4) c (* b b))) b)))

rewrite18.0ms

Algorithm
rewrite-expression-head
Rules
10×add-sqr-sqrt
9×add-cube-cbrt
9×*-un-lft-identity
6×sqrt-prod
6×associate-*r*
6×fma-def
4×add-log-exp
4×log1p-expm1-u
4×associate-/r/
4×add-exp-log
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
3×flip-+
3×flip3-+
3×associate-/r*
2×div-inv
1×*-commutative
1×associate-*r/
1×pow1/2
1×frac-2neg
1×clear-num
1×rem-sqrt-square
1×+-commutative
Counts
4 → 66
Calls
4 calls:
Slowest
7.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
5.0ms
(/ c (+ (sqrt (fma (* a -4) c (* b b))) b))
3.0ms
(* -2 (/ c (+ (sqrt (fma (* a -4) c (* b b))) b)))
2.0ms
(sqrt (fma (* a -4) c (* b b)))

series416.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
166.0ms
(/ c (+ (sqrt (fma (* a -4) c (* b b))) b))
146.0ms
(* -2 (/ c (+ (sqrt (fma (* a -4) c (* b b))) b)))
62.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
42.0ms
(sqrt (fma (* a -4) c (* b b)))

simplify707.0ms

Counts
33 → 78
Calls
33 calls:
Slowest
232.0ms
(+ (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (- (* b b) (* (sqrt (fma (* a -4) c (* b b))) b)))
192.0ms
(+ (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (- (* b b) (* (sqrt (fma (* a -4) c (* b b))) b)))
30.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
26.0ms
(sqrt (+ (sqrt (fma (* a -4) c (* b b))) b))
22.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)

prune1.1s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 10.3b

localize10.0ms

Local error

Found 4 expressions with local error:

21.7b
(sqrt (fma (* a -4) c (* b b)))
1.3b
(+ (sqrt (fma (* a -4) c (* b b))) b)
0.2b
(* (* -2 c) (/ 1 (+ (sqrt (fma (* a -4) c (* b b))) b)))
0.1b
(/ 1 (+ (sqrt (fma (* a -4) c (* b b))) b))

rewrite18.0ms

Algorithm
rewrite-expression-head
Rules
10×add-sqr-sqrt
9×add-cube-cbrt
9×add-exp-log
9×*-un-lft-identity
7×pow1
6×sqrt-prod
6×associate-*r*
6×add-cbrt-cube
6×fma-def
4×add-log-exp
4×log1p-expm1-u
4×associate-/r/
4×expm1-log1p-u
3×flip-+
3×flip3-+
3×associate-/r*
2×div-inv
2×prod-exp
2×rec-exp
1×inv-pow
1×pow-flip
1×cbrt-unprod
1×*-commutative
1×un-div-inv
1×associate-*r/
1×pow-prod-down
1×associate-*l*
1×pow1/2
1×frac-2neg
1×clear-num
1×rem-sqrt-square
1×+-commutative
Counts
4 → 75
Calls
4 calls:
Slowest
6.0ms
(/ 1 (+ (sqrt (fma (* a -4) c (* b b))) b))
6.0ms
(* (* -2 c) (/ 1 (+ (sqrt (fma (* a -4) c (* b b))) b)))
4.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
1.0ms
(sqrt (fma (* a -4) c (* b b)))

series346.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
116.0ms
(/ 1 (+ (sqrt (fma (* a -4) c (* b b))) b))
109.0ms
(* (* -2 c) (/ 1 (+ (sqrt (fma (* a -4) c (* b b))) b)))
69.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
51.0ms
(sqrt (fma (* a -4) c (* b b)))

simplify1.7s

Counts
39 → 87
Calls
39 calls:
Slowest
597.0ms
(* (* (* (* -2 c) (* -2 c)) (* -2 c)) (* (* (/ 1 (+ (sqrt (fma (* a -4) c (* b b))) b)) (/ 1 (+ (sqrt (fma (* a -4) c (* b b))) b))) (/ 1 (+ (sqrt (fma (* a -4) c (* b b))) b))))
246.0ms
(+ (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (- (* b b) (* (sqrt (fma (* a -4) c (* b b))) b)))
197.0ms
(+ (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (- (* b b) (* (sqrt (fma (* a -4) c (* b b))) b)))
114.0ms
(+ (log (* -2 c)) (log (/ 1 (+ (sqrt (fma (* a -4) c (* b b))) b))))
104.0ms
(* (* -2 c) (/ 1 (+ (sqrt (fma (* a -4) c (* b b))) b)))

prune894.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 10.3b

regimes339.0ms

Accuracy

93.6% (1.1b remaining)

Error of 13.4b against oracle of 12.3b and baseline of 29.4b

bsearch294.0ms

end0.0ms

sample4.5s

Algorithm
intervals
Results
1.4s1063×body2560valid
991.0ms5847×body80valid
755.0ms924×body1280valid
501.0ms2979×body80nan
324.0ms435×body640valid
136.0ms102×body5120valid
88.0ms222×body320valid
38.0ms145×body160valid