Average Error: 23.8 → 12.6
Time: 40.0s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{\sqrt[3]{(\left(\frac{\beta - \alpha}{\sqrt{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)} \cdot \sqrt{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}}\right) \cdot \left(\frac{\alpha + \beta}{\beta + (2 \cdot i + \alpha)_*}\right) + 1.0)_* \cdot \left((\left(\frac{\beta - \alpha}{\sqrt{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)} \cdot \sqrt{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}}\right) \cdot \left(\frac{\alpha + \beta}{\beta + (2 \cdot i + \alpha)_*}\right) + 1.0)_* \cdot (\left(\frac{\beta - \alpha}{\sqrt{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)} \cdot \sqrt{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}}\right) \cdot \left(\frac{\alpha + \beta}{\beta + (2 \cdot i + \alpha)_*}\right) + 1.0)_*\right)}}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 23.8

    \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

  2. Simplified12.5

    \[\leadsto \color{blue}{\frac{(\left(\frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt12.6

    \[\leadsto \frac{(\left(\frac{\beta - \alpha}{\color{blue}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*} \cdot \sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\]
  5. Using strategy rm

  6. Applied add-cbrt-cube12.6

    \[\leadsto \frac{\color{blue}{\sqrt[3]{\left((\left(\frac{\beta - \alpha}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*} \cdot \sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_* \cdot (\left(\frac{\beta - \alpha}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*} \cdot \sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*\right) \cdot (\left(\frac{\beta - \alpha}{\sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*} \cdot \sqrt{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}}\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}}}{2.0}\]

  7. Final simplification12.6

    \[\leadsto \frac{\sqrt[3]{(\left(\frac{\beta - \alpha}{\sqrt{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)} \cdot \sqrt{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}}\right) \cdot \left(\frac{\alpha + \beta}{\beta + (2 \cdot i + \alpha)_*}\right) + 1.0)_* \cdot \left((\left(\frac{\beta - \alpha}{\sqrt{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)} \cdot \sqrt{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}}\right) \cdot \left(\frac{\alpha + \beta}{\beta + (2 \cdot i + \alpha)_*}\right) + 1.0)_* \cdot (\left(\frac{\beta - \alpha}{\sqrt{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)} \cdot \sqrt{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}}\right) \cdot \left(\frac{\alpha + \beta}{\beta + (2 \cdot i + \alpha)_*}\right) + 1.0)_*\right)}}{2.0}\]

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))

Details

Time bar (total: 38.4s)Debug log

sample97.0ms

Algorithm
intervals
Results
33.0ms207×body80valid
13.0ms256×pre80true
12.0ms19×body640valid
9.0ms12×body1280valid
7.0ms14×body320valid
1.0ms4×body160valid

simplify577.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
577.0ms
(/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0)

prune387.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 11.5b

localize27.0ms

Local error

Found 4 expressions with local error:

1.7b
(fma (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
0.0b
(/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha)))
0.0b
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))
0.0b
(+ (+ beta 2.0) (fma 2 i alpha))

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
32×*-un-lft-identity
22×add-cube-cbrt
22×add-sqr-sqrt
21×times-frac
8×add-exp-log
8×add-cbrt-cube
7×associate-/r*
6×add-log-exp
6×associate-/l*
5×distribute-lft-out
4×associate-/l/
4×log1p-expm1-u
4×flip-+
4×associate-/r/
4×flip3-+
4×pow1
4×expm1-log1p-u
3×fma-def
2×div-inv
2×fma-udef
2×div-exp
2×frac-2neg
2×clear-num
2×cbrt-undiv
1×flip--
1×div-sub
1×flip3--
1×sum-log
1×associate-+r+
1×associate-+l+
1×+-commutative
Counts
4 → 100
Calls
4 calls:
Slowest
6.0ms
(/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha)))
5.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))
4.0ms
(+ (+ beta 2.0) (fma 2 i alpha))
0.0ms
(fma (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)

series264.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
124.0ms
(fma (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
58.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))
55.0ms
(/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha)))
26.0ms
(+ (+ beta 2.0) (fma 2 i alpha))

simplify3.2s

Counts
82 → 112
Calls
82 calls:
Slowest
396.0ms
(+ (* (+ beta 2.0) (+ beta 2.0)) (- (* (fma 2 i alpha) (fma 2 i alpha)) (* (+ beta 2.0) (fma 2 i alpha))))
184.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 2.0) (* 2.0 (/ 1 beta)))
182.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 2.0) (* 2.0 (/ 1 beta)))
158.0ms
(/ (* (* (- beta alpha) (- beta alpha)) (- beta alpha)) (* (* (+ (+ beta 2.0) (fma 2 i alpha)) (+ (+ beta 2.0) (fma 2 i alpha))) (+ (+ beta 2.0) (fma 2 i alpha))))
146.0ms
(/ (* (* (+ beta alpha) (+ beta alpha)) (+ beta alpha)) (* (* (+ (fma 2 i alpha) beta) (+ (fma 2 i alpha) beta)) (+ (fma 2 i alpha) beta)))

prune1.3s

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 11.2b

localize9.0ms

Local error

Found 4 expressions with local error:

3.4b
(log (fma (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0))
1.7b
(fma (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
0.1b
(exp (log (fma (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)))
0.0b
(/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha)))

rewrite13.0ms

Algorithm
rewrite-expression-head
Rules
24×*-un-lft-identity
16×add-cube-cbrt
16×add-sqr-sqrt
12×times-frac
7×add-exp-log
6×add-cbrt-cube
6×log-prod
6×pow1
4×add-log-exp
4×distribute-lft-out
4×log1p-expm1-u
4×associate-/r*
4×exp-prod
4×expm1-log1p-u
3×associate-/l*
3×exp-sum
2×associate-/l/
2×log-pow
2×associate-/r/
1×rem-exp-log
1×flip--
1×div-inv
1×fma-udef
1×flip-+
1×rem-log-exp
1×div-exp
1×div-sub
1×flip3--
1×flip3-+
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
4 → 79
Calls
4 calls:
Slowest
6.0ms
(/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha)))
3.0ms
(log (fma (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0))
2.0ms
(exp (log (fma (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)))
0.0ms
(fma (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)

series509.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
215.0ms
(log (fma (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0))
122.0ms
(exp (log (fma (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)))
112.0ms
(fma (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
59.0ms
(/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha)))

simplify4.3s

Counts
61 → 91
Calls
61 calls:
Slowest
475.0ms
(+ (* (+ beta 2.0) (+ beta 2.0)) (- (* (fma 2 i alpha) (fma 2 i alpha)) (* (+ beta 2.0) (fma 2 i alpha))))
200.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 2.0) (* 2.0 (/ 1 beta)))
190.0ms
(- (+ (* 1.5 (/ 1 (pow beta 2))) (log 2.0)) (* 1.0 (/ 1 beta)))
181.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 2.0) (* 2.0 (/ 1 beta)))
168.0ms
(/ (* (* (- beta alpha) (- beta alpha)) (- beta alpha)) (* (* (+ (+ beta 2.0) (fma 2 i alpha)) (+ (+ beta 2.0) (fma 2 i alpha))) (+ (+ beta 2.0) (fma 2 i alpha))))

prune1.4s

Pruning

16 alts after pruning (16 fresh and 0 done)

Merged error: 11.1b

localize23.0ms

Local error

Found 4 expressions with local error:

6.3b
(fma (/ (- beta alpha) (* (sqrt (+ (+ beta 2.0) (fma 2 i alpha))) (sqrt (+ (+ beta 2.0) (fma 2 i alpha))))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
0.6b
(* (sqrt (+ (+ beta 2.0) (fma 2 i alpha))) (sqrt (+ (+ beta 2.0) (fma 2 i alpha))))
0.0b
(/ (- beta alpha) (* (sqrt (+ (+ beta 2.0) (fma 2 i alpha))) (sqrt (+ (+ beta 2.0) (fma 2 i alpha)))))
0.0b
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))

rewrite43.0ms

Algorithm
rewrite-expression-head
Rules
24×sqrt-div
22×*-un-lft-identity
18×add-cube-cbrt
18×add-sqr-sqrt
14×flip-+
14×flip3-+
13×add-exp-log
13×add-cbrt-cube
12×times-frac
10×associate-/r/
9×pow1
8×sqrt-prod
8×frac-times
7×associate-*l*
7×associate-*r*
6×associate-/l*
4×associate-/l/
4×add-log-exp
4×log1p-expm1-u
4×associate-*r/
4×associate-*l/
4×pow1/2
4×associate-/r*
4×expm1-log1p-u
3×div-exp
3×cbrt-undiv
2×distribute-lft-out
2×div-inv
2×cbrt-unprod
2×pow-prod-up
2×prod-exp
2×pow-prod-down
2×frac-2neg
2×clear-num
1×flip--
1×fma-udef
1×*-commutative
1×div-sub
1×sqrt-unprod
1×flip3--
1×pow-plus
1×rem-square-sqrt
1×pow2
Counts
4 → 119
Calls
4 calls:
Slowest
16.0ms
(/ (- beta alpha) (* (sqrt (+ (+ beta 2.0) (fma 2 i alpha))) (sqrt (+ (+ beta 2.0) (fma 2 i alpha)))))
13.0ms
(* (sqrt (+ (+ beta 2.0) (fma 2 i alpha))) (sqrt (+ (+ beta 2.0) (fma 2 i alpha))))
11.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))
0.0ms
(fma (/ (- beta alpha) (* (sqrt (+ (+ beta 2.0) (fma 2 i alpha))) (sqrt (+ (+ beta 2.0) (fma 2 i alpha))))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)

series217.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
112.0ms
(fma (/ (- beta alpha) (* (sqrt (+ (+ beta 2.0) (fma 2 i alpha))) (sqrt (+ (+ beta 2.0) (fma 2 i alpha))))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
51.0ms
(/ (- beta alpha) (* (sqrt (+ (+ beta 2.0) (fma 2 i alpha))) (sqrt (+ (+ beta 2.0) (fma 2 i alpha)))))
38.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))
16.0ms
(* (sqrt (+ (+ beta 2.0) (fma 2 i alpha))) (sqrt (+ (+ beta 2.0) (fma 2 i alpha))))

simplify10.0s

Counts
102 → 131
Calls
102 calls:
Slowest
1.0s
(* (* (* (sqrt (+ (+ beta 2.0) (fma 2 i alpha))) (sqrt (+ (+ beta 2.0) (fma 2 i alpha)))) (sqrt (+ (+ beta 2.0) (fma 2 i alpha)))) (* (* (sqrt (+ (+ beta 2.0) (fma 2 i alpha))) (sqrt (+ (+ beta 2.0) (fma 2 i alpha)))) (sqrt (+ (+ beta 2.0) (fma 2 i alpha)))))
751.0ms
(* (+ (+ beta 2.0) (fma 2 i alpha)) (+ (+ beta 2.0) (fma 2 i alpha)))
719.0ms
(* (+ (+ beta 2.0) (fma 2 i alpha)) (+ (+ beta 2.0) (fma 2 i alpha)))
477.0ms
(sqrt (+ (* (+ beta 2.0) (+ beta 2.0)) (- (* (fma 2 i alpha) (fma 2 i alpha)) (* (+ beta 2.0) (fma 2 i alpha)))))
469.0ms
(* (sqrt (+ (pow (+ beta 2.0) 3) (pow (fma 2 i alpha) 3))) (sqrt (- (* (+ beta 2.0) (+ beta 2.0)) (* (fma 2 i alpha) (fma 2 i alpha)))))

prune2.0s

Pruning

15 alts after pruning (15 fresh and 0 done)

Merged error: 11.1b

localize16.0ms

Local error

Found 4 expressions with local error:

1.6b
(+ (* (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta))) 1.0)
0.1b
(* (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)))
0.0b
(/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha)))
0.0b
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))

rewrite40.0ms

Algorithm
rewrite-expression-head
Rules
50×*-un-lft-identity
42×times-frac
36×add-cube-cbrt
36×add-sqr-sqrt
20×add-exp-log
20×add-cbrt-cube
18×associate-*l*
15×associate-*r*
8×associate-/r/
7×distribute-lft-out
7×associate-/r*
6×associate-/l*
6×flip-+
6×div-exp
6×flip3-+
6×cbrt-undiv
6×pow1
4×associate-/l/
4×add-log-exp
4×div-inv
4×log1p-expm1-u
4×cbrt-unprod
4×prod-exp
4×expm1-log1p-u
2×frac-2neg
2×clear-num
1×flip--
1×*-commutative
1×associate-*r/
1×associate-*l/
1×pow-prod-down
1×div-sub
1×flip3--
1×fma-def
1×+-commutative
1×frac-times
Counts
4 → 139
Calls
4 calls:
Slowest
15.0ms
(* (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)))
13.0ms
(+ (* (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta))) 1.0)
6.0ms
(/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha)))
5.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))

series437.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
157.0ms
(* (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)))
133.0ms
(+ (* (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta))) 1.0)
81.0ms
(/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha)))
66.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))

simplify7.9s

Counts
124 → 151
Calls
124 calls:
Slowest
718.0ms
(* (/ (* (* (- beta alpha) (- beta alpha)) (- beta alpha)) (* (* (+ (+ beta 2.0) (fma 2 i alpha)) (+ (+ beta 2.0) (fma 2 i alpha))) (+ (+ beta 2.0) (fma 2 i alpha)))) (* (* (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) (/ (+ beta alpha) (+ (fma 2 i alpha) beta))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta))))
606.0ms
(* (+ (+ beta 2.0) (fma 2 i alpha)) (+ (fma 2 i alpha) beta))
433.0ms
(+ (* (+ beta 2.0) (+ beta 2.0)) (- (* (fma 2 i alpha) (fma 2 i alpha)) (* (+ beta 2.0) (fma 2 i alpha))))
413.0ms
(* (/ (* (* (- beta alpha) (- beta alpha)) (- beta alpha)) (* (* (+ (+ beta 2.0) (fma 2 i alpha)) (+ (+ beta 2.0) (fma 2 i alpha))) (+ (+ beta 2.0) (fma 2 i alpha)))) (/ (* (* (+ beta alpha) (+ beta alpha)) (+ beta alpha)) (* (* (+ (fma 2 i alpha) beta) (+ (fma 2 i alpha) beta)) (+ (fma 2 i alpha) beta))))
379.0ms
(* (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)))

prune1.8s

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 11.1b

regimes357.0ms

Accuracy

0% (0.4b remaining)

Error of 12.6b against oracle of 12.2b and baseline of 12.6b

bsearch3.0ms

end0.0ms

sample3.3s

Algorithm
intervals
Results
1.0s6335×body80valid
947.0ms605×body640valid
455.0ms521×body1280valid
358.0ms8000×pre80true
259.0ms345×body320valid
59.0ms194×body160valid