Average Error: 52.6 → 0.1
Time: 21.1s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[-2 \cdot \frac{c}{b + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 52.6

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

  2. Simplified52.6

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

  4. Applied flip--52.6

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

  5. Applied associate-/l/52.6

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

  6. Simplified0.4

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

  8. Applied times-frac0.1

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019010 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))

Details

Time bar (total: 20.3s)Debug log

sample263.0ms

Algorithm
intervals

simplify104.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
104.0ms
(/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))

prune16.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 50.6b

localize25.0ms

Local error

Found 4 expressions with local error:

25.5b
(- (sqrt (fma (* a -4) c (* b b))) b)
0.1b
(fma (* a -4) c (* b b))
0.1b
(/ (- (sqrt (fma (* a -4) c (* b b))) b) (* 2 a))
0.0b
(sqrt (fma (* a -4) c (* b b)))

rewrite49.0ms

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
22.0ms
(/ (- (sqrt (fma (* a -4) c (* b b))) b) (* 2 a))
12.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
11.0ms
(sqrt (fma (* a -4) c (* b b)))
0.0ms
(fma (* a -4) c (* b b))

series305.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
121.0ms
(/ (- (sqrt (fma (* a -4) c (* b b))) b) (* 2 a))
76.0ms
(- (sqrt (fma (* a -4) c (* b b))) b)
66.0ms
(sqrt (fma (* a -4) c (* b b)))
41.0ms
(fma (* a -4) c (* b b))

simplify1.5s

Counts
34 → 79
Calls
34 calls:
Slowest
704.0ms
(- (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (* b b))
154.0ms
(- (pow (sqrt (fma (* a -4) c (* b b))) 3) (pow b 3))
82.0ms
(- (log (- (sqrt (fma (* a -4) c (* b b))) b)) (log (* 2 a)))
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)))
48.0ms
(- (pow b 2) (* 4 (* a c)))

prune796.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.4b

localize12.0ms

Local error

Found 4 expressions with local error:

0.3b
(/ (* (* a -4) c) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)))
0.1b
(fma (* a -4) c (* b b))
0.1b
(* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))
0.1b
(+ (sqrt (fma (* a -4) c (* b b))) b)

rewrite41.0ms

Algorithm
rewrite-expression-head
Rules
11×add-exp-log
11×add-cbrt-cube
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×fma-def
6×pow1
4×add-log-exp
4×log1p-expm1-u
4×associate-*r/
4×expm1-log1p-u
3×flip-+
3×sqrt-prod
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×fma-udef
1×*-commutative
1×pow-prod-down
1×associate-*l*
1×times-frac
1×distribute-lft-in
1×frac-2neg
1×associate-/r*
1×clear-num
1×+-commutative
1×distribute-rgt-in
Counts
4 → 70
Calls
4 calls:
Slowest
16.0ms
(* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))
15.0ms
(/ (* (* a -4) c) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)))
8.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
0.0ms
(fma (* a -4) c (* b b))

series358.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
144.0ms
(/ (* (* a -4) c) (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b)))
109.0ms
(* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))
68.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
37.0ms
(fma (* a -4) c (* b b))

simplify3.6s

Counts
37 → 82
Calls
37 calls:
Slowest
713.0ms
(- (log (* (* a -4) c)) (+ (log (* 2 a)) (log (+ (sqrt (fma (* a -4) c (* b b))) b))))
670.0ms
(/ (* (* (* (* 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))))
668.0ms
(/ (* (* (* (* 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))))
341.0ms
(- (log (* (* a -4) c)) (log (* (* 2 a) (+ (sqrt (fma (* a -4) c (* b b))) b))))
280.0ms
(+ (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (- (* b b) (* (sqrt (fma (* a -4) c (* b b))) b)))

prune982.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0b

localize12.0ms

Local error

Found 4 expressions with local error:

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

rewrite9.0ms

Algorithm
rewrite-expression-head
Rules
9×add-sqr-sqrt
8×add-cube-cbrt
8×*-un-lft-identity
6×sqrt-prod
6×fma-def
4×add-log-exp
4×log1p-expm1-u
4×add-exp-log
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
3×associate-/r*
2×flip-+
2×associate-/r/
2×flip3-+
1×div-inv
1×fma-udef
1×pow1/2
1×frac-2neg
1×clear-num
1×rem-sqrt-square
1×+-commutative
Counts
4 → 59
Calls
4 calls:
Slowest
4.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
2.0ms
(/ c (+ (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))

series309.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
159.0ms
(/ c (+ (sqrt (fma (* a -4) c (* b b))) b))
63.0ms
(+ (sqrt (fma (* a -4) c (* b b))) b)
53.0ms
(sqrt (fma (* a -4) c (* b b)))
34.0ms
(fma (* a -4) c (* b b))

simplify560.0ms

Counts
26 → 71
Calls
26 calls:
Slowest
203.0ms
(+ (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (- (* b b) (* (sqrt (fma (* a -4) c (* b b))) b)))
49.0ms
(- (pow b 2) (* 4 (* a c)))
36.0ms
(- (pow b 2) (* 4 (* a c)))
30.0ms
(- (pow b 2) (* 4 (* a c)))
28.0ms
(cbrt (+ (sqrt (fma (* a -4) c (* b b))) b))

prune737.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0b

localize19.0ms

Local error

Found 4 expressions with local error:

0.3b
(/ (* (* a -4) c) (+ (* (* 2 a) (sqrt (fma (* a -4) c (* b b)))) (* (* 2 a) b)))
0.1b
(fma (* a -4) c (* b b))
0.1b
(* (* 2 a) (sqrt (fma (* a -4) c (* b b))))
0.1b
(+ (* (* 2 a) (sqrt (fma (* a -4) c (* b b)))) (* (* 2 a) b))

rewrite41.0ms

Algorithm
rewrite-expression-head
Rules
8×add-cube-cbrt
8×add-exp-log
8×add-cbrt-cube
8×*-un-lft-identity
8×add-sqr-sqrt
6×add-log-exp
6×associate-*r*
6×pow1
4×log1p-expm1-u
4×times-frac
4×associate-/r*
4×expm1-log1p-u
3×distribute-lft-out
3×sqrt-prod
2×flip-+
2×associate-/r/
2×flip3-+
1×associate-/l*
1×div-inv
1×fma-udef
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×associate-*l*
1×div-exp
1×frac-2neg
1×fma-def
1×sum-log
1×clear-num
1×+-commutative
1×cbrt-undiv
Counts
4 → 70
Calls
4 calls:
Slowest
19.0ms
(/ (* (* a -4) c) (+ (* (* 2 a) (sqrt (fma (* a -4) c (* b b)))) (* (* 2 a) b)))
11.0ms
(+ (* (* 2 a) (sqrt (fma (* a -4) c (* b b)))) (* (* 2 a) b))
9.0ms
(* (* 2 a) (sqrt (fma (* a -4) c (* b b))))
0.0ms
(fma (* a -4) c (* b b))

series364.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
175.0ms
(/ (* (* a -4) c) (+ (* (* 2 a) (sqrt (fma (* a -4) c (* b b)))) (* (* 2 a) b)))
98.0ms
(+ (* (* 2 a) (sqrt (fma (* a -4) c (* b b)))) (* (* 2 a) b))
61.0ms
(* (* 2 a) (sqrt (fma (* a -4) c (* b b))))
31.0ms
(fma (* a -4) c (* b b))

simplify2.7s

Counts
42 → 82
Calls
42 calls:
Slowest
692.0ms
(* (* (* (* 2 a) (* 2 a)) (* 2 a)) (* (* (sqrt (fma (* a -4) c (* b b))) (sqrt (fma (* a -4) c (* b b)))) (sqrt (fma (* a -4) c (* b b)))))
411.0ms
(/ (* (* (* (* a -4) c) (* (* a -4) c)) (* (* a -4) c)) (* (* (+ (* (* 2 a) (sqrt (fma (* a -4) c (* b b)))) (* (* 2 a) b)) (+ (* (* 2 a) (sqrt (fma (* a -4) c (* b b)))) (* (* 2 a) b))) (+ (* (* 2 a) (sqrt (fma (* a -4) c (* b b)))) (* (* 2 a) b))))
189.0ms
(* (exp (* (* 2 a) (sqrt (fma (* a -4) c (* b b))))) (exp (* (* 2 a) b)))
184.0ms
(- (log (* (* a -4) c)) (log (+ (* (* 2 a) (sqrt (fma (* a -4) c (* b b)))) (* (* 2 a) b))))
143.0ms
(- (* (* 2 a) (sqrt (fma (* a -4) c (* b b)))) (* (* 2 a) b))

prune888.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0b

regimes252.0ms

Accuracy

0% (0.1b remaining)

Error of 0.1b against oracle of 0.0b and baseline of 0.1b

bsearch3.0ms

end0.0ms

sample6.4s

Algorithm
intervals