Average Error: 19.7 → 13.6
Time: 19.2s
Precision: 64
Internal Precision: 128
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 9.876540340758696 \cdot 10^{+57}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{(\left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 9.876540340758696e+57

    1. Initial program 15.5

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

    2. Simplified15.5

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

    4. Applied add-sqr-sqrt15.5

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

    5. Applied sqrt-prod15.6

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

    if 9.876540340758696e+57 < b

    1. Initial program 36.9

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

    2. Simplified36.9

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

    3. Taylor expanded around 0 5.2

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

    5. Applied add-sqr-sqrt5.2

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

    6. Applied sqrt-prod5.2

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

    7. Applied fma-neg5.2

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

  4. Final simplification13.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 9.876540340758696 \cdot 10^{+57}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{(\left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019010 +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: 18.4s)Debug log

sample153.0ms

Algorithm
intervals

simplify120.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
120.0ms
(if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c))))))

prune18.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 18.1b

localize140.0ms

Local error

Found 4 expressions with local error:

23.4b
(sqrt (fma c (* a -4) (* b b)))
23.4b
(sqrt (fma c (* a -4) (* b b)))
1.7b
(- (sqrt (fma c (* a -4) (* b b))) b)
0.3b
(- (- 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)))

series251.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
76.0ms
(sqrt (fma c (* a -4) (* b b)))
67.0ms
(sqrt (fma c (* a -4) (* b b)))
66.0ms
(- (- b) (sqrt (fma c (* a -4) (* b b))))
43.0ms
(- (sqrt (fma c (* a -4) (* b b))) b)

simplify2.4s

Counts
81 → 105
Calls
81 calls:
Slowest
90.0ms
(fma 1 (- b) (- (* (sqrt (fma c (* a -4) (* b b))) (sqrt 1))))
76.0ms
(- (* 2 (/ (* a c) b)) (* 2 b))
69.0ms
(* -2 (/ (* a c) b))
66.0ms
(fma (* (cbrt (- b)) (cbrt (- b))) (cbrt (- b)) (- (* (sqrt (fma c (* a -4) (* b b))) (sqrt 1))))
60.0ms
(fma (- (sqrt (fma c (* a -4) (* b b)))) (sqrt 1) (* (sqrt (fma c (* a -4) (* b b))) (sqrt 1)))

prune1.6s

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 10.6b

localize113.0ms

Local error

Found 4 expressions with local error:

23.4b
(sqrt (fma c (* a -4) (* b b)))
1.7b
(- (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))

rewrite21.0ms

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
13.0ms
(/ (* 2 c) (- (sqrt (fma c (* a -4) (* b b))) b))
5.0ms
(- (sqrt (fma c (* a -4) (* b b))) b)
1.0ms
(sqrt (fma c (* a -4) (* b b)))
0.0ms
(fma c (* a -4) (* b b))

series320.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
167.0ms
(/ (* 2 c) (- (sqrt (fma c (* a -4) (* b b))) b))
59.0ms
(- (sqrt (fma c (* a -4) (* b b))) b)
48.0ms
(fma c (* a -4) (* b b))
45.0ms
(sqrt (fma c (* a -4) (* b b)))

simplify1.2s

Counts
34 → 78
Calls
34 calls:
Slowest
601.0ms
(/ (* (* (* 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)))
114.0ms
(+ (* (sqrt (fma c (* a -4) (* b b))) (sqrt (fma c (* a -4) (* b b)))) (+ (* b b) (* (sqrt (fma c (* a -4) (* b b))) b)))
73.0ms
(- (log (* 2 c)) (log (- (sqrt (fma c (* a -4) (* b b))) b)))
49.0ms
(- (pow b 2) (* 4 (* a c)))
47.0ms
(- (pow b 2) (* 4 (* a c)))

prune1.2s

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 10.6b

localize92.0ms

Local error

Found 4 expressions with local error:

23.4b
(sqrt (fma c (* a -4) (* b b)))
1.7b
(- (sqrt (fma c (* a -4) (* b b))) b)
0.1b
(* (* 2 c) (/ 1 (- (sqrt (fma c (* a -4) (* b b))) b)))
0.1b
(/ 1 (- (sqrt (fma c (* a -4) (* b b))) b))

rewrite31.0ms

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×fma-neg
6×add-cbrt-cube
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×sub-neg
1×clear-num
1×rem-sqrt-square
Counts
4 → 76
Calls
4 calls:
Slowest
10.0ms
(* (* 2 c) (/ 1 (- (sqrt (fma c (* a -4) (* b b))) b)))
10.0ms
(- (sqrt (fma c (* a -4) (* b b))) b)
8.0ms
(/ 1 (- (sqrt (fma c (* a -4) (* b b))) b))
2.0ms
(sqrt (fma c (* a -4) (* b b)))

series395.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
149.0ms
(* (* 2 c) (/ 1 (- (sqrt (fma c (* a -4) (* b b))) b)))
135.0ms
(/ 1 (- (sqrt (fma c (* a -4) (* b b))) b))
57.0ms
(- (sqrt (fma c (* a -4) (* b b))) b)
53.0ms
(sqrt (fma c (* a -4) (* b b)))

simplify1.3s

Counts
39 → 88
Calls
39 calls:
Slowest
402.0ms
(* (* (* (* 2 c) (* 2 c)) (* 2 c)) (* (* (/ 1 (- (sqrt (fma c (* a -4) (* b b))) b)) (/ 1 (- (sqrt (fma c (* a -4) (* b b))) b))) (/ 1 (- (sqrt (fma c (* a -4) (* b b))) b))))
146.0ms
(* (* 2 c) (/ 1 (- (sqrt (fma c (* a -4) (* b b))) b)))
120.0ms
(+ (log (* 2 c)) (log (/ 1 (- (sqrt (fma c (* a -4) (* b b))) b))))
89.0ms
(+ (* (sqrt (fma c (* a -4) (* b b))) (sqrt (fma c (* a -4) (* b b)))) (+ (* b b) (* (sqrt (fma c (* a -4) (* b b))) b)))
84.0ms
(+ (* (sqrt (fma c (* a -4) (* b b))) (sqrt (fma c (* a -4) (* b b)))) (+ (* b b) (* (sqrt (fma c (* a -4) (* b b))) b)))

prune1.3s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 10.6b

localize118.0ms

Local error

Found 4 expressions with local error:

23.4b
(sqrt (fma c (* a -4) (* b b)))
23.4b
(sqrt (fma c (* a -4) (* b b)))
19.2b
(fma (sqrt (sqrt (fma c (* a -4) (* b b)))) (sqrt (sqrt (fma c (* a -4) (* b b)))) (- b))
0.1b
(/ (* 2 c) (fma (sqrt (sqrt (fma c (* a -4) (* b b)))) (sqrt (sqrt (fma c (* a -4) (* b b)))) (- b)))

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
10×add-sqr-sqrt
8×add-cube-cbrt
8×*-un-lft-identity
6×sqrt-prod
6×add-exp-log
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×pow1/2
2×rem-sqrt-square
1×associate-/l*
1×div-inv
1×fma-udef
1×div-exp
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
4 → 59
Calls
4 calls:
Slowest
3.0ms
(/ (* 2 c) (fma (sqrt (sqrt (fma c (* a -4) (* b b)))) (sqrt (sqrt (fma c (* a -4) (* b b)))) (- b)))
1.0ms
(sqrt (fma c (* a -4) (* b b)))
1.0ms
(sqrt (fma c (* a -4) (* b b)))
0.0ms
(fma (sqrt (sqrt (fma c (* a -4) (* b b)))) (sqrt (sqrt (fma c (* a -4) (* b b)))) (- b))

series803.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
357.0ms
(/ (* 2 c) (fma (sqrt (sqrt (fma c (* a -4) (* b b)))) (sqrt (sqrt (fma c (* a -4) (* b b)))) (- b)))
338.0ms
(fma (sqrt (sqrt (fma c (* a -4) (* b b)))) (sqrt (sqrt (fma c (* a -4) (* b b)))) (- b))
62.0ms
(sqrt (fma c (* a -4) (* b b)))
45.0ms
(sqrt (fma c (* a -4) (* b b)))

simplify1.8s

Counts
31 → 71
Calls
31 calls:
Slowest
653.0ms
(- (* 1/2 (/ (pow (exp (* 1/4 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c)))))) 4) (* a b))) (+ (* 2 (/ c b)) (* 2 (/ (* (pow (exp (* 1/4 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c)))))) 2) c) (pow b 2)))))
331.0ms
(- (pow (exp (* 1/4 (- (log -4) (+ (log (/ 1 a)) (log (/ 1 c)))))) 2) b)
286.0ms
(- (* 1/2 (/ (pow (exp (* 1/4 (- (log -4) (+ (log (/ -1 c)) (log (/ -1 a)))))) 4) (* a b))) (+ (* 2 (/ c b)) (* 2 (/ (* c (pow (exp (* 1/4 (- (log -4) (+ (log (/ -1 c)) (log (/ -1 a)))))) 2)) (pow b 2)))))
135.0ms
(- (pow (exp (* 1/4 (- (log -4) (+ (log (/ -1 c)) (log (/ -1 a)))))) 2) b)
87.0ms
(/ (* (* (* 2 c) (* 2 c)) (* 2 c)) (* (* (fma (sqrt (sqrt (fma c (* a -4) (* b b)))) (sqrt (sqrt (fma c (* a -4) (* b b)))) (- b)) (fma (sqrt (sqrt (fma c (* a -4) (* b b)))) (sqrt (sqrt (fma c (* a -4) (* b b)))) (- b))) (fma (sqrt (sqrt (fma c (* a -4) (* b b)))) (sqrt (sqrt (fma c (* a -4) (* b b)))) (- b))))

prune1.3s

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 10.6b

regimes336.0ms

Accuracy

90.6% (0.8b remaining)

Error of 13.6b against oracle of 12.8b and baseline of 21.1b

bsearch269.0ms

end0.0ms

sample3.1s

Algorithm
intervals