Average Error: 24.1 → 12.5
Time: 42.5s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{\log \left(\sqrt{e^{(\left(\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}\right) \cdot \left(\frac{\alpha + \beta}{\beta + (2 \cdot i + \alpha)_*}\right) + 1.0)_*}}\right) + \log \left(\sqrt{e^{(\left(\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}\right) \cdot \left(\frac{\alpha + \beta}{\beta + (2 \cdot i + \alpha)_*}\right) + 1.0)_*}}\right)}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 24.1

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

  2. Initial simplification12.4

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

  4. Applied add-log-exp12.5

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

  6. Applied add-sqr-sqrt12.5

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

  7. Applied log-prod12.5

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

  8. Final simplification12.5

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

Reproduce

herbie shell --seed 2018365 +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: 33.2s)Debug log

start151.0ms

Algorithm
intervals

setup747.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 13.5b

localize44.0ms

Local error

Found 4 expressions with local error:

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

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
4 → 100
Calls

4 calls. Slowest were:

10.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))
8.0ms
(/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha)))
7.0ms
(+ (+ beta 2.0) (fma 2 i alpha))

series256.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

126.0ms
(fma (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
63.0ms
(/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha)))
45.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))
21.0ms
(+ (+ beta 2.0) (fma 2 i alpha))

simplify3.8s

Counts
82 → 112
Calls

82 calls. Slowest were:

581.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 2.0) (* 2.0 (/ 1 beta)))
446.0ms
(+ (* (+ beta 2.0) (+ beta 2.0)) (- (* (fma 2 i alpha) (fma 2 i alpha)) (* (+ beta 2.0) (fma 2 i alpha))))
215.0ms
(/ (* (* (- 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))))

prune1.8s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 13.2b

localize9.0ms

Local error

Found 4 expressions with local error:

1.6b
(log (exp (fma (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)))
1.6b
(fma (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
0.0b
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))
0.0b
(/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha)))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 99
Calls

4 calls. Slowest were:

10.0ms
(/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha)))
8.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))
2.0ms
(log (exp (fma (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)))

series342.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

127.0ms
(log (exp (fma (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)))
102.0ms
(fma (/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
60.0ms
(/ (- beta alpha) (+ (+ beta 2.0) (fma 2 i alpha)))
52.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))

simplify4.6s

Counts
86 → 111
Calls

86 calls. Slowest were:

380.0ms
(+ (* (+ beta 2.0) (+ beta 2.0)) (- (* (fma 2 i alpha) (fma 2 i alpha)) (* (+ beta 2.0) (fma 2 i alpha))))
217.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 2.0) (* 2.0 (/ 1 beta)))
208.0ms
(- (+ (* 4.0 (/ 1 (pow beta 2))) 2.0) (* 2.0 (/ 1 beta)))

prune2.2s

Pruning

17 alts after pruning (17 fresh and 0 done)

Merged error: 13.2b

localize37.0ms

Local error

Found 4 expressions with local error:

1.6b
(log (exp (fma (/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)))
1.6b
(fma (/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
0.6b
(/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha)))
0.0b
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
4 → 96
Calls

4 calls. Slowest were:

11.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))
6.0ms
(/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha)))
4.0ms
(log (exp (fma (/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)))

series296.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

96.0ms
(log (exp (fma (/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)))
91.0ms
(fma (/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
69.0ms
(/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha)))
39.0ms
(/ (+ beta alpha) (+ (fma 2 i alpha) beta))

simplify4.6s

Counts
69 → 108
Calls

69 calls. Slowest were:

486.0ms
(log (sqrt (exp (fma (/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0))))
412.0ms
(fma (/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
379.0ms
(log (* (cbrt (exp (fma (/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0))) (cbrt (exp (fma (/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)))))

prune2.8s

Pruning

18 alts after pruning (18 fresh and 0 done)

Merged error: 13.2b

localize19.0ms

Local error

Found 4 expressions with local error:

3.2b
(log (log (exp (fma (/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0))))
1.6b
(log (exp (fma (/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)))
1.6b
(fma (/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
0.6b
(/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha)))

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 81
Calls

4 calls. Slowest were:

6.0ms
(/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha)))
4.0ms
(log (exp (fma (/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)))
4.0ms
(log (log (exp (fma (/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0))))

series461.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

178.0ms
(log (log (exp (fma (/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0))))
112.0ms
(log (exp (fma (/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)))
110.0ms
(fma (/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)
61.0ms
(/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha)))

simplify6.9s

Counts
48 → 93
Calls

48 calls. Slowest were:

450.0ms
(log (exp (fma (/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)))
436.0ms
(log (* (cbrt (log (exp (fma (/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)))) (cbrt (log (exp (fma (/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0))))))
428.0ms
(fma (/ 1 (/ (+ (+ beta 2.0) (fma 2 i alpha)) (- beta alpha))) (/ (+ beta alpha) (+ (fma 2 i alpha) beta)) 1.0)

prune2.4s

Pruning

17 alts after pruning (17 fresh and 0 done)

Merged error: 13.2b

regimes1.6s

Accuracy

0% (0.3b remaining)

Error of 12.5b against oracle of 12.1b and baseline of 12.5b

bsearch2.0ms