Average Error: 19.5 → 13.8
Time: 20.7s
Precision: 64
Internal Precision: 128
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2.863205149176435 \cdot 10^{+62}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{(\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \left(\sqrt[3]{-b}\right) + \left(-\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}\right))_*}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 2.863205149176435e+62

    1. Initial program 15.6

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

    2. Simplified15.6

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

    4. Applied add-cube-cbrt15.7

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

    5. Applied fma-neg15.7

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

    if 2.863205149176435e+62 < b

    1. Initial program 37.0

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

    2. Simplified37.0

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

    3. Taylor expanded around 0 5.3

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

    4. Taylor expanded around 0 5.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

    5. Simplified5.3

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

  4. Final simplification13.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2.863205149176435 \cdot 10^{+62}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{(\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \left(\sqrt[3]{-b}\right) + \left(-\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}\right))_*}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))

Details

Time bar (total: 18.7s)Debug log

sample140.0ms

Algorithm
intervals
Results
368×(pre true 80)
277×(body real 80)
91×(body nan 80)

simplify117.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
117.0ms
(if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c))))))

prune27.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 20.6b

localize187.0ms

Local error

Found 4 expressions with local error:

24.0b
(sqrt (fma c (* a -4) (* b b)))
24.0b
(sqrt (fma c (* a -4) (* b b)))
2.2b
(- (sqrt (fma c (* a -4) (* b b))) b)
1.2b
(- (- b) (sqrt (fma c (* a -4) (* b b))))

rewrite13.0ms

Algorithm
rewrite-expression-head
Rules
29×add-sqr-sqrt
25×*-un-lft-identity
24×prod-diff
23×add-cube-cbrt
22×sqrt-prod
10×fma-neg
7×neg-mul-1
6×add-log-exp
4×log1p-expm1-u
4×add-exp-log
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
2×difference-of-squares
2×flip--
2×pow1/2
2×flip3--
2×sub-neg
2×rem-sqrt-square
1×distribute-lft-out--
1×diff-log
1×neg-sub0
1×associate--l-
Counts
4 → 93
Calls
4 calls:
Slowest
5.0ms
(- (sqrt (fma c (* a -4) (* b b))) b)
5.0ms
(- (- b) (sqrt (fma c (* a -4) (* b b))))
1.0ms
(sqrt (fma c (* a -4) (* b b)))
1.0ms
(sqrt (fma c (* a -4) (* b b)))

series201.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
58.0ms
(sqrt (fma c (* a -4) (* b b)))
53.0ms
(- (- b) (sqrt (fma c (* a -4) (* b b))))
45.0ms
(- (sqrt (fma c (* a -4) (* b b))) b)
44.0ms
(sqrt (fma c (* a -4) (* b b)))

simplify2.4s

Counts
81 → 105
Calls
81 calls:
Slowest
62.0ms
(/ (exp (- b)) (exp (sqrt (fma c (* a -4) (* b b)))))
59.0ms
(* -2 (/ (* a c) b))
59.0ms
(fma (- (sqrt (fma c (* a -4) (* b b)))) (sqrt 1) (* (sqrt (fma c (* a -4) (* b b))) (sqrt 1)))
57.0ms
(fma (- (sqrt (fma c (* a -4) (* b b)))) (sqrt 1) (* (sqrt (fma c (* a -4) (* b b))) (sqrt 1)))
57.0ms
(fma -1 b (- (* (sqrt (fma c (* a -4) (* b b))) 1)))

prune1.7s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 12.1b

localize126.0ms

Local error

Found 4 expressions with local error:

24.0b
(sqrt (fma c (* a -4) (* b b)))
2.2b
(- (sqrt (fma c (* a -4) (* b b))) b)
0.1b
(/ (* 2 c) (- (sqrt (fma c (* a -4) (* b b))) b))
0.0b
(fma c (* a -4) (* b b))

rewrite27.0ms

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×times-frac
3×associate-/r*
2×flip--
2×associate-/r/
2×flip3--
1×associate-/l*
1×div-inv
1×fma-udef
1×div-exp
1×pow1/2
1×frac-2neg
1×sub-neg
1×clear-num
1×rem-sqrt-square
1×cbrt-undiv
Counts
4 → 66
Calls
4 calls:
Slowest
14.0ms
(/ (* 2 c) (- (sqrt (fma c (* a -4) (* b b))) b))
10.0ms
(- (sqrt (fma c (* a -4) (* b b))) b)
2.0ms
(sqrt (fma c (* a -4) (* b b)))
0.0ms
(fma c (* a -4) (* b b))

series325.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
160.0ms
(/ (* 2 c) (- (sqrt (fma c (* a -4) (* b b))) b))
66.0ms
(sqrt (fma c (* a -4) (* b b)))
57.0ms
(- (sqrt (fma c (* a -4) (* b b))) b)
41.0ms
(fma c (* a -4) (* b b))

simplify1.1s

Counts
34 → 78
Calls
34 calls:
Slowest
577.0ms
(/ (* (* (* 2 c) (* 2 c)) (* 2 c)) (* (* (- (sqrt (fma c (* a -4) (* b b))) b) (- (sqrt (fma c (* a -4) (* b b))) b)) (- (sqrt (fma c (* a -4) (* b b))) b)))
99.0ms
(+ (* (sqrt (fma c (* a -4) (* b b))) (sqrt (fma c (* a -4) (* b b)))) (+ (* b b) (* (sqrt (fma c (* a -4) (* b b))) b)))
50.0ms
(- (pow b 2) (* 4 (* a c)))
37.0ms
(- (log (* 2 c)) (log (- (sqrt (fma c (* a -4) (* b b))) b)))
37.0ms
(- (pow b 2) (* 4 (* a c)))

prune1.2s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 12.1b

localize110.0ms

Local error

Found 4 expressions with local error:

24.0b
(sqrt (fma c (* a -4) (* b b)))
24.0b
(sqrt (fma c (* a -4) (* b b)))
24.0b
(sqrt (fma c (* a -4) (* b b)))
12.9b
(- (- b) (* (sqrt (sqrt (fma c (* a -4) (* b b)))) (sqrt (sqrt (fma c (* a -4) (* b b))))))

rewrite15.0ms

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×fma-neg
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
3×pow1/2
3×rem-sqrt-square
2×neg-mul-1
1×difference-of-squares
1×flip--
1×diff-log
1×flip3--
1×sub-neg
1×neg-sub0
1×associate--l-
Counts
4 → 66
Calls
4 calls:
Slowest
8.0ms
(- (- b) (* (sqrt (sqrt (fma c (* a -4) (* b b)))) (sqrt (sqrt (fma c (* a -4) (* b b))))))
2.0ms
(sqrt (fma c (* a -4) (* b b)))
2.0ms
(sqrt (fma c (* a -4) (* b b)))
2.0ms
(sqrt (fma c (* a -4) (* b b)))

series280.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
75.0ms
(sqrt (fma c (* a -4) (* b b)))
71.0ms
(- (- b) (* (sqrt (sqrt (fma c (* a -4) (* b b)))) (sqrt (sqrt (fma c (* a -4) (* b b))))))
67.0ms
(sqrt (fma c (* a -4) (* b b)))
66.0ms
(sqrt (fma c (* a -4) (* b b)))

simplify603.0ms

Counts
37 → 78
Calls
37 calls:
Slowest
75.0ms
(- (* 2 (/ (* a c) b)) (* 2 b))
55.0ms
(/ (exp (- b)) (exp (* (sqrt (sqrt (fma c (* a -4) (* b b)))) (sqrt (sqrt (fma c (* a -4) (* b b)))))))
50.0ms
(fma 1 (- b) (- (* (sqrt (sqrt (fma c (* a -4) (* b b)))) (sqrt (sqrt (fma c (* a -4) (* b b)))))))
43.0ms
(* -2 (/ (* a c) b))
40.0ms
(fma (sqrt (- b)) (sqrt (- b)) (- (* (sqrt (sqrt (fma c (* a -4) (* b b)))) (sqrt (sqrt (fma c (* a -4) (* b b)))))))

prune1.4s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 12.1b

localize122.0ms

Local error

Found 4 expressions with local error:

24.7b
(cbrt (fma c (* a -4) (* b b)))
24.7b
(cbrt (fma (* c -4) a (* b b)))
18.0b
(fma (fabs (cbrt (fma (* c -4) a (* b b)))) (sqrt (cbrt (fma c (* a -4) (* b b)))) (- b))
0.1b
(sqrt (cbrt (fma c (* a -4) (* b b))))

rewrite5.0ms

Algorithm
rewrite-expression-head
Rules
10×add-sqr-sqrt
10×cbrt-prod
8×add-cube-cbrt
8×*-un-lft-identity
6×sqrt-prod
4×add-log-exp
4×log1p-expm1-u
4×add-exp-log
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
2×pow1/3
2×rem-sqrt-square
1×fma-udef
1×pow1/2
Counts
4 → 54
Calls
4 calls:
Slowest
2.0ms
(sqrt (cbrt (fma c (* a -4) (* b b))))
1.0ms
(cbrt (fma (* c -4) a (* b b)))
1.0ms
(cbrt (fma c (* a -4) (* b b)))
0.0ms
(fma (fabs (cbrt (fma (* c -4) a (* b b)))) (sqrt (cbrt (fma c (* a -4) (* b b)))) (- b))

series962.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
268.0ms
(fma (fabs (cbrt (fma (* c -4) a (* b b)))) (sqrt (cbrt (fma c (* a -4) (* b b)))) (- b))
240.0ms
(cbrt (fma (* c -4) a (* b b)))
238.0ms
(cbrt (fma c (* a -4) (* b b)))
215.0ms
(sqrt (cbrt (fma c (* a -4) (* b b))))

simplify2.0s

Counts
26 → 66
Calls
26 calls:
Slowest
224.0ms
(exp (* 1/3 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c))))))
205.0ms
(exp (* 1/3 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c))))))
196.0ms
(exp (* 1/6 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c))))))
180.0ms
(- (* (exp (* 1/6 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c)))))) (fabs (pow (fma (* -4 c) a (pow b 2)) 1/3))) b)
177.0ms
(- (pow b 1/3) (* 2/3 (* (* a c) (pow (/ 1 (pow b 5)) 1/3))))

prune1.6s

Pruning

13 alts after pruning (11 fresh and 2 done)

Merged error: 12.1b

regimes529.0ms

Accuracy

87.3% (0.8b remaining)

Error of 13.8b against oracle of 13.0b and baseline of 19.5b

bsearch231.0ms

end0.0ms

sample3.4s

Algorithm
intervals
Results
11455×(pre true 80)
8717×(body real 80)
2738×(body nan 80)