Average Error: 3.6 → 3.6
Time: 1.1m
Precision: 64
Internal Precision: 128
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\frac{x}{y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{a + t}}{t} - \left(b - c\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{0.6666666666666666}{t}\right)\right)} + x}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 3.6

    \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

  2. Taylor expanded around 0 3.6

    \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \color{blue}{\frac{0.6666666666666666}{t}}\right)\right)}}\]

  3. Final simplification3.6

    \[\leadsto \frac{x}{y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{a + t}}{t} - \left(b - c\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{0.6666666666666666}{t}\right)\right)} + x}\]

Reproduce

herbie shell --seed 2019004 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))

Details

Time bar (total: 1.1m)Debug log

sample347.0ms

Algorithm
intervals

simplify299.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

299.0ms
(/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))

prune23.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 2.7b

localize92.0ms

Local error

Found 4 expressions with local error:

3.5b
(/ (* z (sqrt (+ t a))) t)
1.7b
(- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))
0.6b
(/ 2.0 (* t 3.0))
0.3b
(* z (sqrt (+ t a)))

rewrite31.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

16.0ms
(/ (* z (sqrt (+ t a))) t)
11.0ms
(- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))
2.0ms
(* z (sqrt (+ t a)))

series259.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

168.0ms
(- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))
45.0ms
(/ (* z (sqrt (+ t a))) t)
37.0ms
(* z (sqrt (+ t a)))
9.0ms
(/ 2.0 (* t 3.0))

simplify7.2s

Counts
57 → 84
Calls

57 calls. Slowest were:

555.0ms
(- (* (* z (sqrt (+ t a))) (* (+ (* b b) (+ (* c c) (* b c))) (+ (* (+ a (/ 5.0 6.0)) (+ a (/ 5.0 6.0))) (+ (* (/ 2.0 (* t 3.0)) (/ 2.0 (* t 3.0))) (* (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))) (* t (* (- (pow b 3) (pow c 3)) (- (pow (+ a (/ 5.0 6.0)) 3) (pow (/ 2.0 (* t 3.0)) 3)))))
503.0ms
(* t (* (+ b c) (* (- a (/ 5.0 6.0)) (* t 3.0))))
397.0ms
(- (* (* z (sqrt (+ t a))) (* (+ b c) (+ (* (+ a (/ 5.0 6.0)) (+ a (/ 5.0 6.0))) (+ (* (/ 2.0 (* t 3.0)) (/ 2.0 (* t 3.0))) (* (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))) (* t (* (- (* b b) (* c c)) (- (pow (+ a (/ 5.0 6.0)) 3) (pow (/ 2.0 (* t 3.0)) 3)))))

prune1.9s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize25.0ms

Local error

Found 4 expressions with local error:

3.5b
(/ (* z (sqrt (+ t a))) t)
1.7b
(- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 0.6666666666666666 t))))
0.3b
(* z (sqrt (+ t a)))
0.1b
(* (- b c) (- (+ a (/ 5.0 6.0)) (/ 0.6666666666666666 t)))

rewrite36.0ms

Algorithm
rewrite-expression-head
Counts
4 → 100
Calls

4 calls. Slowest were:

16.0ms
(* (- b c) (- (+ a (/ 5.0 6.0)) (/ 0.6666666666666666 t)))
9.0ms
(- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 0.6666666666666666 t))))
7.0ms
(/ (* z (sqrt (+ t a))) t)

series337.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

169.0ms
(- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 0.6666666666666666 t))))
63.0ms
(/ (* z (sqrt (+ t a))) t)
55.0ms
(* (- b c) (- (+ a (/ 5.0 6.0)) (/ 0.6666666666666666 t)))
49.0ms
(* z (sqrt (+ t a)))

simplify14.9s

Counts
95 → 112
Calls

95 calls. Slowest were:

649.0ms
(* (- (pow b 3) (pow c 3)) (- (* (+ a (/ 5.0 6.0)) (+ a (/ 5.0 6.0))) (* (/ 0.6666666666666666 t) (/ 0.6666666666666666 t))))
562.0ms
(* t (* (+ (* a a) (- (* (/ 5.0 6.0) (/ 5.0 6.0)) (* a (/ 5.0 6.0)))) t))
524.0ms
(* (+ (* b b) (+ (* c c) (* b c))) (* (+ (* a a) (- (* (/ 5.0 6.0) (/ 5.0 6.0)) (* a (/ 5.0 6.0)))) t))

prune2.7s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize69.0ms

Local error

Found 4 expressions with local error:

39.6b
(cbrt (* (* (/ 2.0 (* t 3.0)) (/ 2.0 (* t 3.0))) (/ 2.0 (* t 3.0))))
3.5b
(/ (* z (sqrt (+ t a))) t)
1.7b
(- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (cbrt (* (* (/ 2.0 (* t 3.0)) (/ 2.0 (* t 3.0))) (/ 2.0 (* t 3.0)))))))
0.6b
(/ 2.0 (* t 3.0))

rewrite55.0ms

Algorithm
rewrite-expression-head
Counts
4 → 109
Calls

4 calls. Slowest were:

33.0ms
(- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (cbrt (* (* (/ 2.0 (* t 3.0)) (/ 2.0 (* t 3.0))) (/ 2.0 (* t 3.0)))))))
12.0ms
(cbrt (* (* (/ 2.0 (* t 3.0)) (/ 2.0 (* t 3.0))) (/ 2.0 (* t 3.0))))
7.0ms
(/ (* z (sqrt (+ t a))) t)

series481.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

317.0ms
(- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (cbrt (* (* (/ 2.0 (* t 3.0)) (/ 2.0 (* t 3.0))) (/ 2.0 (* t 3.0)))))))
96.0ms
(cbrt (* (* (/ 2.0 (* t 3.0)) (/ 2.0 (* t 3.0))) (/ 2.0 (* t 3.0))))
55.0ms
(/ (* z (sqrt (+ t a))) t)
13.0ms
(/ 2.0 (* t 3.0))

simplify21.7s

Counts
129 → 121
Calls

129 calls. Slowest were:

699.0ms
(- (* (* z (sqrt (+ t a))) (* (+ (* b b) (+ (* c c) (* b c))) (+ (+ a (/ 5.0 6.0)) (cbrt (* (* (/ 2.0 (* t 3.0)) (/ 2.0 (* t 3.0))) (/ 2.0 (* t 3.0))))))) (* t (* (- (pow b 3) (pow c 3)) (- (* (+ a (/ 5.0 6.0)) (+ a (/ 5.0 6.0))) (* (cbrt (* (* (/ 2.0 (* t 3.0)) (/ 2.0 (* t 3.0))) (/ 2.0 (* t 3.0)))) (cbrt (* (* (/ 2.0 (* t 3.0)) (/ 2.0 (* t 3.0))) (/ 2.0 (* t 3.0)))))))))
526.0ms
(- (* (* z (sqrt (+ t a))) (* (+ (* b b) (+ (* c c) (* b c))) (+ (* (+ a (/ 5.0 6.0)) (+ a (/ 5.0 6.0))) (+ (* (cbrt (* (* (/ 2.0 (* t 3.0)) (/ 2.0 (* t 3.0))) (/ 2.0 (* t 3.0)))) (cbrt (* (* (/ 2.0 (* t 3.0)) (/ 2.0 (* t 3.0))) (/ 2.0 (* t 3.0))))) (* (+ a (/ 5.0 6.0)) (cbrt (* (* (/ 2.0 (* t 3.0)) (/ 2.0 (* t 3.0))) (/ 2.0 (* t 3.0))))))))) (* t (* (- (pow b 3) (pow c 3)) (- (pow (+ a (/ 5.0 6.0)) 3) (pow (cbrt (* (* (/ 2.0 (* t 3.0)) (/ 2.0 (* t 3.0))) (/ 2.0 (* t 3.0)))) 3)))))
524.0ms
(* t (* (+ b c) (* (+ (* a a) (- (* (/ 5.0 6.0) (/ 5.0 6.0)) (* a (/ 5.0 6.0)))) (cbrt (* t 3.0)))))

prune3.2s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

localize23.0ms

Local error

Found 4 expressions with local error:

0.0b
(exp (* 2.0 (- (* 0.8333333333333334 c) (* (- b c) a))))
0.0b
(- (* 0.8333333333333334 c) (* (- b c) a))
0.0b
(* (- b c) a)
0.0b
(* y (exp (* 2.0 (- (* 0.8333333333333334 c) (* (- b c) a)))))

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 53
Calls

4 calls. Slowest were:

7.0ms
(* y (exp (* 2.0 (- (* 0.8333333333333334 c) (* (- b c) a)))))
3.0ms
(exp (* 2.0 (- (* 0.8333333333333334 c) (* (- b c) a))))
2.0ms
(* (- b c) a)

series185.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

84.0ms
(* y (exp (* 2.0 (- (* 0.8333333333333334 c) (* (- b c) a)))))
36.0ms
(exp (* 2.0 (- (* 0.8333333333333334 c) (* (- b c) a))))
35.0ms
(- (* 0.8333333333333334 c) (* (- b c) a))
30.0ms
(* (- b c) a)

simplify3.1s

Counts
30 → 65
Calls

30 calls. Slowest were:

512.0ms
(exp (* (- (* (- b c) a)) 2.0))
310.0ms
(* (- (* b b) (* c c)) a)
211.0ms
(exp (* 2.0 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b))))

prune810.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0.0b

regimes516.0ms

Accuracy

0% (2.5b remaining)

Error of 3.6b against oracle of 1.1b and baseline of 3.6b

bsearch4.0ms

end0.0ms

sample6.2s

Algorithm
intervals