Average Error: 34.5 → 30.2
Time: 50.5s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le -1.0749396284737093 \cdot 10^{-175}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}} + \sqrt[3]{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}}\\ \mathbf{elif}\;g \le 5.247043520352711 \cdot 10^{+68}:\\ \;\;\;\;\sqrt[3]{\left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{1}{2 \cdot a}} + \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\left(-g\right) - g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt{\sqrt{g \cdot g - h \cdot h}}} + \frac{\sqrt[3]{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - g}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if g < -1.0749396284737093e-175

    1. Initial program 33.9

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
    2. Using strategy rm

    3. Applied cbrt-prod33.8

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \color{blue}{\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}\]

    4. Simplified33.8

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \color{blue}{\sqrt[3]{\frac{\frac{1}{2}}{a}}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\]
    5. Using strategy rm

    6. Applied cbrt-prod30.1

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}}} + \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\]

    7. Simplified30.1

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{2}}{a}}} \cdot \sqrt[3]{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}} + \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\]

    if -1.0749396284737093e-175 < g < 5.247043520352711e+68

    1. Initial program 18.3

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
    2. Using strategy rm

    3. Applied cbrt-prod14.5

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \color{blue}{\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}\]

    4. Simplified14.5

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \color{blue}{\sqrt[3]{\frac{\frac{1}{2}}{a}}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\]

    5. Taylor expanded around inf 11.5

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\left(-g\right) - \color{blue}{g}}\]

    if 5.247043520352711e+68 < g

    1. Initial program 46.9

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
    2. Using strategy rm

    3. Applied cbrt-prod43.5

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \color{blue}{\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}\]

    4. Simplified43.4

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \color{blue}{\sqrt[3]{\frac{\frac{1}{2}}{a}}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\]
    5. Using strategy rm

    6. Applied associate-*l/43.4

      \[\leadsto \sqrt[3]{\color{blue}{\frac{1 \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}{2 \cdot a}}} + \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\]

    7. Applied cbrt-div43.4

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{2 \cdot a}}} + \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\]

    8. Simplified43.4

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - g}}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\]
    9. Using strategy rm

    10. Applied add-sqr-sqrt43.4

      \[\leadsto \frac{\sqrt[3]{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{\color{blue}{\sqrt{g \cdot g - h \cdot h} \cdot \sqrt{g \cdot g - h \cdot h}}}}\]

    11. Applied sqrt-prod43.4

      \[\leadsto \frac{\sqrt[3]{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\left(-g\right) - \color{blue}{\sqrt{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt{\sqrt{g \cdot g - h \cdot h}}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification30.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le -1.0749396284737093 \cdot 10^{-175}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}} + \sqrt[3]{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}}\\ \mathbf{elif}\;g \le 5.247043520352711 \cdot 10^{+68}:\\ \;\;\;\;\sqrt[3]{\left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{1}{2 \cdot a}} + \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\left(-g\right) - g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt{\sqrt{g \cdot g - h \cdot h}}} + \frac{\sqrt[3]{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - g}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 +o rules:numerics
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  (+ (cbrt (* (/ 1 (* 2 a)) (+ (- g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1 (* 2 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))

Details

Time bar (total: 46.8s)Debug log

sample645.0ms

Algorithm
intervals

simplify452.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

452.0ms
(+ (cbrt (* (/ 1 (* 2 a)) (+ (- g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1 (* 2 a)) (- (- g) (sqrt (- (* g g) (* h h)))))))

prune27.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 35.3b

localize64.0ms

Local error

Found 4 expressions with local error:

28.9b
(sqrt (- (* g g) (* h h)))
28.9b
(sqrt (- (* g g) (* h h)))
10.2b
(cbrt (* (/ 1 (* 2 a)) (+ (- g) (sqrt (- (* g g) (* h h))))))
8.3b
(cbrt (* (/ 1 (* 2 a)) (- (- g) (sqrt (- (* g g) (* h h))))))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

7.0ms
(cbrt (* (/ 1 (* 2 a)) (- (- g) (sqrt (- (* g g) (* h h))))))
7.0ms
(cbrt (* (/ 1 (* 2 a)) (+ (- g) (sqrt (- (* g g) (* h h))))))
3.0ms
(sqrt (- (* g g) (* h h)))

series624.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

279.0ms
(cbrt (* (/ 1 (* 2 a)) (+ (- g) (sqrt (- (* g g) (* h h))))))
277.0ms
(cbrt (* (/ 1 (* 2 a)) (- (- g) (sqrt (- (* g g) (* h h))))))
42.0ms
(sqrt (- (* g g) (* h h)))
26.0ms
(sqrt (- (* g g) (* h h)))

simplify5.8s

Counts
38 → 78
Calls

38 calls. Slowest were:

705.0ms
(cbrt (* (/ 1 (* 2 a)) (- (* (- g) (- g)) (* (sqrt (- (* g g) (* h h))) (sqrt (- (* g g) (* h h)))))))
665.0ms
(cbrt (* (/ 1 (* 2 a)) (- (* (- g) (- g)) (* (sqrt (- (* g g) (* h h))) (sqrt (- (* g g) (* h h)))))))
382.0ms
(* (exp (* 1/3 (- (+ (log 2) (log (/ 1 a))) (log (/ 1 g))))) (cbrt -1/2))

prune1.3s

Pruning

19 alts after pruning (19 fresh and 0 done)

Merged error: 28.7b

localize17.0ms

Local error

Found 4 expressions with local error:

28.9b
(sqrt (- (* g g) (* h h)))
28.9b
(sqrt (- (* g g) (* h h)))
10.2b
(cbrt (* (/ 1 (* 2 a)) (+ (- g) (sqrt (- (* g g) (* h h))))))
2.8b
(- (* g g) (* h h))

rewrite34.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

18.0ms
(cbrt (* (/ 1 (* 2 a)) (+ (- g) (sqrt (- (* g g) (* h h))))))
6.0ms
(sqrt (- (* g g) (* h h)))
5.0ms
(sqrt (- (* g g) (* h h)))

series432.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

335.0ms
(cbrt (* (/ 1 (* 2 a)) (+ (- g) (sqrt (- (* g g) (* h h))))))
39.0ms
(sqrt (- (* g g) (* h h)))
33.0ms
(sqrt (- (* g g) (* h h)))
25.0ms
(- (* g g) (* h h))

simplify3.6s

Counts
37 → 79
Calls

37 calls. Slowest were:

624.0ms
(cbrt (* (/ 1 (* 2 a)) (- (* (- g) (- g)) (* (sqrt (- (* g g) (* h h))) (sqrt (- (* g g) (* h h)))))))
346.0ms
(cbrt (* (/ 1 (* 2 a)) (+ (pow (- g) 3) (pow (sqrt (- (* g g) (* h h))) 3))))
343.0ms
(cbrt (* 1 (+ (pow (- g) 3) (pow (sqrt (- (* g g) (* h h))) 3))))

prune1.6s

Pruning

24 alts after pruning (24 fresh and 0 done)

Merged error: 28.3b

localize26.0ms

Local error

Found 4 expressions with local error:

28.9b
(sqrt (* (+ g h) (- g h)))
28.9b
(sqrt (- (* g g) (* h h)))
2.8b
(- (* g g) (* h h))
1.7b
(- (sqrt (* (+ g h) (- g h))) g)

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 70
Calls

4 calls. Slowest were:

6.0ms
(- (sqrt (* (+ g h) (- g h))) g)
3.0ms
(sqrt (- (* g g) (* h h)))
2.0ms
(sqrt (* (+ g h) (- g h)))

series129.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

53.0ms
(- (sqrt (* (+ g h) (- g h))) g)
31.0ms
(sqrt (* (+ g h) (- g h)))
27.0ms
(sqrt (- (* g g) (* h h)))
18.0ms
(- (* g g) (* h h))

simplify3.6s

Counts
36 → 82
Calls

36 calls. Slowest were:

617.0ms
(sqrt (* (+ g h) (- (* g g) (* h h))))
532.0ms
(* (cbrt (sqrt (* (+ g h) (- g h)))) (cbrt (sqrt (* (+ g h) (- g h)))))
510.0ms
(sqrt (* (+ (pow g 3) (pow h 3)) (- (pow g 3) (pow h 3))))

prune1.8s

Pruning

26 alts after pruning (26 fresh and 0 done)

Merged error: 28.3b

localize12.0ms

Local error

Found 4 expressions with local error:

28.9b
(sqrt (- (* g g) (* h h)))
28.9b
(sqrt (- (* g g) (* h h)))
2.8b
(- (* g g) (* h h))
2.8b
(- (* g g) (* h h))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

6.0ms
(sqrt (- (* g g) (* h h)))
5.0ms
(sqrt (- (* g g) (* h h)))
4.0ms
(- (* g g) (* h h))

series108.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

39.0ms
(sqrt (- (* g g) (* h h)))
33.0ms
(sqrt (- (* g g) (* h h)))
18.0ms
(- (* g g) (* h h))
17.0ms
(- (* g g) (* h h))

simplify1.6s

Counts
36 → 80
Calls

36 calls. Slowest were:

225.0ms
(sqrt (- (pow (* g g) 3) (pow (* h h) 3)))
214.0ms
(sqrt (- (* (* g g) (* g g)) (* (* h h) (* h h))))
207.0ms
(sqrt (- (* (* g g) (* g g)) (* (* h h) (* h h))))

prune1.5s

Pruning

26 alts after pruning (25 fresh and 1 done)

Merged error: 28.3b

regimes1.8s

Accuracy

36% (1.2b remaining)

Error of 30.2b against oracle of 29.0b and baseline of 30.8b

bsearch2.7s

end0.0ms

sample18.8s

Algorithm
intervals