Average Error: 19.6 → 6.9
Time: 30.3s
Precision: 64
Internal Precision: 128
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.8021396801207425 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left|b\right|}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 3.2165368891128914 \cdot 10^{+63}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\left|\sqrt[3]{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left|\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right|}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - (\left(a \cdot \frac{c}{b}\right) \cdot -2 + b)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -4.8021396801207425e+125

    1. Initial program 50.9

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt50.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

    4. Applied rem-sqrt-square50.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left|\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right|}{2 \cdot a}\\ \end{array}\]

    5. Simplified50.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left|\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right|}{2 \cdot a}\\ \end{array}\]

    6. Taylor expanded around 0 3.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left|b\right|}{2 \cdot a}\\ \end{array}\]

    if -4.8021396801207425e+125 < b < 3.2165368891128914e+63

    1. Initial program 9.1

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt9.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

    4. Applied rem-sqrt-square9.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left|\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right|}{2 \cdot a}\\ \end{array}\]

    5. Simplified9.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left|\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right|}{2 \cdot a}\\ \end{array}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt9.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left|\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right|}{2 \cdot a}\\ \end{array}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt9.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{\color{blue}{\left(\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left|\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right|}{2 \cdot a}\\ \end{array}\]

    10. Applied sqrt-prod9.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left|\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right|}{2 \cdot a}\\ \end{array}\]

    11. Simplified9.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\left|\sqrt[3]{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}\right|} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left|\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right|}{2 \cdot a}\\ \end{array}\]

    if 3.2165368891128914e+63 < b

    1. Initial program 26.7

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Taylor expanded around inf 6.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    3. Simplified3.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{(\left(\frac{c}{b} \cdot a\right) \cdot -2 + b)_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt3.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - (\left(\frac{c}{b} \cdot a\right) \cdot -2 + b)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.8021396801207425 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left|b\right|}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 3.2165368891128914 \cdot 10^{+63}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\left|\sqrt[3]{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left|\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right|}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - (\left(a \cdot \frac{c}{b}\right) \cdot -2 + b)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))

Details

Time bar (total: 28.6s)Debug log

sample155.0ms

Algorithm
intervals
Results
378×(pre true 80)
285×(body real 80)
93×(body nan 80)

simplify96.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
96.0ms
(if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))

prune27.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 19.9b

localize128.0ms

Local error

Found 4 expressions with local error:

23.2b
(sqrt (- (* b b) (* (* 4 a) c)))
23.2b
(sqrt (- (* b b) (* (* 4 a) c)))
2.2b
(+ (- b) (sqrt (- (* b b) (* (* 4 a) c))))
1.2b
(- (- b) (sqrt (- (* b b) (* (* 4 a) c))))

rewrite26.0ms

Algorithm
rewrite-expression-head
Rules
28×add-sqr-sqrt
26×*-un-lft-identity
24×prod-diff
22×add-cube-cbrt
19×sqrt-prod
8×add-log-exp
8×neg-mul-1
4×log1p-expm1-u
4×add-exp-log
4×sqrt-div
4×fma-neg
4×add-cbrt-cube
4×fma-def
4×pow1
4×expm1-log1p-u
3×flip--
3×flip3--
2×difference-of-squares
2×pow1/2
2×rem-sqrt-square
2×neg-sub0
1×distribute-lft-out--
1×distribute-lft-out
1×flip-+
1×associate-+l-
1×diff-log
1×flip3-+
1×sub-neg
1×sum-log
1×+-commutative
1×associate--l-
Counts
4 → 97
Calls
4 calls:
Slowest
11.0ms
(- (- b) (sqrt (- (* b b) (* (* 4 a) c))))
6.0ms
(+ (- b) (sqrt (- (* b b) (* (* 4 a) c))))
5.0ms
(sqrt (- (* b b) (* (* 4 a) c)))
3.0ms
(sqrt (- (* b b) (* (* 4 a) c)))

series185.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
52.0ms
(sqrt (- (* b b) (* (* 4 a) c)))
49.0ms
(+ (- b) (sqrt (- (* b b) (* (* 4 a) c))))
44.0ms
(- (- b) (sqrt (- (* b b) (* (* 4 a) c))))
40.0ms
(sqrt (- (* b b) (* (* 4 a) c)))

simplify5.6s

Counts
84 → 109
Calls
84 calls:
Slowest
470.0ms
(sqrt (- (pow (* b b) 3) (pow (* (* 4 a) c) 3)))
377.0ms
(sqrt (- (pow (* b b) 3) (pow (* (* 4 a) c) 3)))
168.0ms
(- b (* 2 (/ (* a c) b)))
160.0ms
(sqrt (- (* (* b b) (* b b)) (* (* (* 4 a) c) (* (* 4 a) c))))
126.0ms
(- b (* 2 (/ (* a c) b)))

prune1.8s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 6.9b

localize94.0ms

Local error

Found 4 expressions with local error:

23.2b
(sqrt (fma (* a -4) c (* b b)))
23.2b
(sqrt (- (* b b) (* (* 4 a) c)))
2.2b
(+ (- b) (fabs (sqrt (fma (* a -4) c (* b b)))))
1.2b
(- (- b) (sqrt (- (* b b) (* (* 4 a) c))))

rewrite24.0ms

Algorithm
rewrite-expression-head
Rules
28×add-sqr-sqrt
26×*-un-lft-identity
24×prod-diff
22×add-cube-cbrt
19×sqrt-prod
8×add-log-exp
8×neg-mul-1
4×log1p-expm1-u
4×add-exp-log
4×fma-neg
4×add-cbrt-cube
4×fma-def
4×pow1
4×expm1-log1p-u
2×difference-of-squares
2×flip--
2×sqrt-div
2×pow1/2
2×flip3--
2×rem-sqrt-square
2×neg-sub0
1×distribute-lft-out--
1×distribute-lft-out
1×flip-+
1×associate-+l-
1×diff-log
1×flip3-+
1×sub-neg
1×sum-log
1×+-commutative
1×associate--l-
Counts
4 → 95
Calls
4 calls:
Slowest
14.0ms
(- (- b) (sqrt (- (* b b) (* (* 4 a) c))))
4.0ms
(+ (- b) (fabs (sqrt (fma (* a -4) c (* b b)))))
3.0ms
(sqrt (- (* b b) (* (* 4 a) c)))
1.0ms
(sqrt (fma (* a -4) c (* b b)))

series187.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
80.0ms
(sqrt (fma (* a -4) c (* b b)))
57.0ms
(- (- b) (sqrt (- (* b b) (* (* 4 a) c))))
40.0ms
(sqrt (- (* b b) (* (* 4 a) c)))
9.0ms
(+ (- b) (fabs (sqrt (fma (* a -4) c (* b b)))))

simplify4.4s

Counts
82 → 107
Calls
82 calls:
Slowest
438.0ms
(sqrt (- (pow (* b b) 3) (pow (* (* 4 a) c) 3)))
163.0ms
(sqrt (- (* (* b b) (* b b)) (* (* (* 4 a) c) (* (* 4 a) c))))
121.0ms
(- b (* 2 (/ (* a c) b)))
97.0ms
(fma (* (cbrt (- b)) (cbrt (- b))) (cbrt (- b)) (- (* (sqrt (cbrt (- (* b b) (* (* 4 a) c)))) (sqrt (* (cbrt (- (* b b) (* (* 4 a) c))) (cbrt (- (* b b) (* (* 4 a) c))))))))
92.0ms
(fma (sqrt (- b)) (sqrt (- b)) (- (* (sqrt (- (* b b) (* (* 4 a) c))) (sqrt 1))))

prune2.1s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 5.2b

localize106.0ms

Local error

Found 4 expressions with local error:

23.2b
(sqrt (- (* b b) (* (* 4 a) c)))
5.0b
(* (/ c b) a)
3.8b
(/ (* 2 c) (- (- b) (fma (* (/ c b) a) -2 b)))
2.2b
(+ (- b) (sqrt (- (* b b) (* (* 4 a) c))))

rewrite19.0ms

Algorithm
rewrite-expression-head
Rules
15×*-un-lft-identity
14×add-sqr-sqrt
9×add-cube-cbrt
6×add-log-exp
6×add-exp-log
6×add-cbrt-cube
5×times-frac
5×associate-/r*
4×log1p-expm1-u
4×associate-*l*
4×fma-def
4×pow1
4×expm1-log1p-u
3×sqrt-prod
2×difference-of-squares
2×distribute-lft-out--
2×flip--
2×div-inv
2×associate-/r/
2×sqrt-div
2×flip3--
1×associate-/l*
1×distribute-lft-out
1×flip-+
1×associate-+l-
1×*-commutative
1×associate-*l/
1×div-exp
1×pow1/2
1×flip3-+
1×frac-2neg
1×sum-log
1×neg-mul-1
1×clear-num
1×rem-sqrt-square
1×+-commutative
1×cbrt-undiv
1×neg-sub0
Counts
4 → 77
Calls
4 calls:
Slowest
7.0ms
(/ (* 2 c) (- (- b) (fma (* (/ c b) a) -2 b)))
6.0ms
(+ (- b) (sqrt (- (* b b) (* (* 4 a) c))))
3.0ms
(sqrt (- (* b b) (* (* 4 a) c)))
2.0ms
(* (/ c b) a)

series194.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
77.0ms
(/ (* 2 c) (- (- b) (fma (* (/ c b) a) -2 b)))
48.0ms
(sqrt (- (* b b) (* (* 4 a) c)))
45.0ms
(+ (- b) (sqrt (- (* b b) (* (* 4 a) c))))
23.0ms
(* (/ c b) a)

simplify3.1s

Counts
44 → 89
Calls
44 calls:
Slowest
363.0ms
(sqrt (- (pow (* b b) 3) (pow (* (* 4 a) c) 3)))
271.0ms
(+ (* (- b) (- b)) (+ (* (fma (* (/ c b) a) -2 b) (fma (* (/ c b) a) -2 b)) (* (- b) (fma (* (/ c b) a) -2 b))))
209.0ms
(sqrt (- (* (* b b) (* b b)) (* (* (* 4 a) c) (* (* 4 a) c))))
165.0ms
(- b (* 2 (/ (* a c) b)))
142.0ms
(sqrt (- (- b) (fma (* (/ c b) a) -2 b)))

prune1.6s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 5.2b

localize135.0ms

Local error

Found 4 expressions with local error:

23.2b
(sqrt (fma (* a -4) c (* b b)))
23.2b
(sqrt (- (* b b) (* (* 4 a) c)))
23.2b
(sqrt (- (* b b) (* (* 4 a) c)))
9.3b
(- (- b) (* (sqrt (sqrt (- (* b b) (* (* 4 a) c)))) (sqrt (sqrt (- (* b b) (* (* 4 a) c))))))

rewrite12.0ms

Algorithm
rewrite-expression-head
Rules
13×add-sqr-sqrt
9×sqrt-prod
9×add-cube-cbrt
9×*-un-lft-identity
6×add-log-exp
4×log1p-expm1-u
4×add-exp-log
4×prod-diff
4×sqrt-div
4×fma-neg
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
3×flip--
3×pow1/2
3×flip3--
3×rem-sqrt-square
2×neg-mul-1
1×difference-of-squares
1×diff-log
1×sub-neg
1×neg-sub0
1×associate--l-
Counts
4 → 70
Calls
4 calls:
Slowest
4.0ms
(- (- b) (* (sqrt (sqrt (- (* b b) (* (* 4 a) c)))) (sqrt (sqrt (- (* b b) (* (* 4 a) c))))))
3.0ms
(sqrt (- (* b b) (* (* 4 a) c)))
3.0ms
(sqrt (- (* b b) (* (* 4 a) c)))
1.0ms
(sqrt (fma (* a -4) c (* b b)))

series267.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
75.0ms
(sqrt (- (* b b) (* (* 4 a) c)))
70.0ms
(sqrt (- (* b b) (* (* 4 a) c)))
67.0ms
(sqrt (fma (* a -4) c (* b b)))
56.0ms
(- (- b) (* (sqrt (sqrt (- (* b b) (* (* 4 a) c)))) (sqrt (sqrt (- (* b b) (* (* 4 a) c))))))

simplify2.4s

Counts
41 → 82
Calls
41 calls:
Slowest
377.0ms
(sqrt (- (pow (* b b) 3) (pow (* (* 4 a) c) 3)))
366.0ms
(sqrt (- (pow (* b b) 3) (pow (* (* 4 a) c) 3)))
192.0ms
(sqrt (- (* (* b b) (* b b)) (* (* (* 4 a) c) (* (* 4 a) c))))
139.0ms
(sqrt (- (* (* b b) (* b b)) (* (* (* 4 a) c) (* (* 4 a) c))))
134.0ms
(- b (* 2 (/ (* a c) b)))

prune1.9s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 5.2b

regimes401.0ms

Accuracy

91.9% (1.1b remaining)

Error of 6.9b against oracle of 5.8b and baseline of 19.7b

bsearch454.0ms

end0.0ms

sample3.1s

Algorithm
intervals
Results
11500×(pre true 80)
8754×(body real 80)
2746×(body nan 80)