Average Error: 0.5 → 0.1
Time: 32.8s
Precision: 64
Internal Precision: 128
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{1}{\sqrt{(\left(v \cdot v\right) \cdot -3 + 1)_* \cdot 2}} \cdot \frac{(\left(v \cdot v\right) \cdot -5 + 1)_*}{\pi}}{\left(1 - v \cdot v\right) \cdot t}\]

Error

Bits error versus v

Bits error versus t

Derivation

  1. Initial program 0.5

    \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]

  2. Simplified0.3

    \[\leadsto \color{blue}{\frac{\frac{(\left(v \cdot v\right) \cdot -5 + 1)_*}{\pi}}{\sqrt{2 \cdot (\left(v \cdot v\right) \cdot -3 + 1)_*} \cdot \left(t \cdot \left(1 - v \cdot v\right)\right)}}\]
  3. Using strategy rm

  4. Applied associate-/r*0.1

    \[\leadsto \color{blue}{\frac{\frac{\frac{(\left(v \cdot v\right) \cdot -5 + 1)_*}{\pi}}{\sqrt{2 \cdot (\left(v \cdot v\right) \cdot -3 + 1)_*}}}{t \cdot \left(1 - v \cdot v\right)}}\]
  5. Using strategy rm

  6. Applied div-inv0.1

    \[\leadsto \frac{\color{blue}{\frac{(\left(v \cdot v\right) \cdot -5 + 1)_*}{\pi} \cdot \frac{1}{\sqrt{2 \cdot (\left(v \cdot v\right) \cdot -3 + 1)_*}}}}{t \cdot \left(1 - v \cdot v\right)}\]

  7. Final simplification0.1

    \[\leadsto \frac{\frac{1}{\sqrt{(\left(v \cdot v\right) \cdot -3 + 1)_* \cdot 2}} \cdot \frac{(\left(v \cdot v\right) \cdot -5 + 1)_*}{\pi}}{\left(1 - v \cdot v\right) \cdot t}\]

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))

Details

Time bar (total: 32.3s)Debug log

sample131.0ms

Algorithm
intervals
Results
57.0ms287×body80nan
52.0ms256×body80valid

simplify148.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
148.0ms
(/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v))))

prune438.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize29.0ms

Local error

Found 2 expressions with local error:

0.4b
(* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v))))
0.4b
(/ (/ (fma (* v v) -5 1) PI) (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v)))))

rewrite27.0ms

Algorithm
rewrite-expression-head
Rules
9×add-exp-log
9×add-cbrt-cube
8×associate-*r/
8×distribute-lft-in
8×distribute-rgt-in
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×associate-/l*
4×associate-*l*
4×times-frac
4×sub-neg
4×pow1
3×div-inv
2×add-log-exp
2×flip--
2×log1p-expm1-u
2×cbrt-unprod
2×associate-/r/
2×prod-exp
2×div-exp
2×flip3--
2×cbrt-undiv
2×expm1-log1p-u
1×associate-/l/
1×sqrt-prod
1×*-commutative
1×pow-prod-down
1×associate-*r*
1×frac-2neg
1×associate-/r*
1×clear-num
Counts
2 → 56
Calls
2 calls:
Slowest
16.0ms
(/ (/ (fma (* v v) -5 1) PI) (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v)))))
9.0ms
(* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v))))

series465.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
307.0ms
(/ (/ (fma (* v v) -5 1) PI) (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v)))))
159.0ms
(* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v))))

simplify4.8s

Counts
42 → 62
Calls
42 calls:
Slowest
371.0ms
(- (* t (sqrt 2)) (* 5/2 (* t (* (sqrt 2) (pow v 2)))))
341.0ms
(- (log (/ (fma (* v v) -5 1) PI)) (+ (log (sqrt (* 2 (fma (* v v) -3 1)))) (log (* t (- 1 (* v v))))))
336.0ms
(- (/ 1 (* t (* (sqrt 2) PI))) (+ (* 53/8 (/ (pow v 4) (* t (* (sqrt 2) PI)))) (* 5/2 (/ (pow v 2) (* t (* (sqrt 2) PI))))))
262.0ms
(/ (* (* (/ (fma (* v v) -5 1) PI) (/ (fma (* v v) -5 1) PI)) (/ (fma (* v v) -5 1) PI)) (* (* (* (sqrt (* 2 (fma (* v v) -3 1))) (sqrt (* 2 (fma (* v v) -3 1)))) (sqrt (* 2 (fma (* v v) -3 1)))) (* (* (* t (- 1 (* v v))) (* t (- 1 (* v v)))) (* t (- 1 (* v v))))))
215.0ms
(/ (* (* (/ (fma (* v v) -5 1) PI) (/ (fma (* v v) -5 1) PI)) (/ (fma (* v v) -5 1) PI)) (* (* (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v)))) (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v))))) (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v))))))

prune756.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 2 expressions with local error:

0.1b
(/ (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1)))) (* t (- 1 (* v v))))
0.0b
(/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))

rewrite30.0ms

Algorithm
rewrite-expression-head
Rules
68×times-frac
30×add-cube-cbrt
30×*-un-lft-identity
30×add-sqr-sqrt
24×associate-/l*
17×div-inv
13×sqrt-prod
9×add-exp-log
9×add-cbrt-cube
5×associate-/r*
4×div-exp
4×cbrt-undiv
2×associate-/l/
2×add-log-exp
2×log1p-expm1-u
2×associate-/r/
2×associate-*r/
2×frac-2neg
2×clear-num
2×pow1
2×expm1-log1p-u
1×flip--
1×flip3--
Counts
2 → 99
Calls
2 calls:
Slowest
18.0ms
(/ (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1)))) (* t (- 1 (* v v))))
10.0ms
(/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))

series503.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
312.0ms
(/ (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1)))) (* t (- 1 (* v v))))
191.0ms
(/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))

simplify8.2s

Counts
115 → 105
Calls
115 calls:
Slowest
311.0ms
(- (/ 1 (* t (* (sqrt 2) PI))) (+ (* 53/8 (/ (pow v 4) (* t (* (sqrt 2) PI)))) (* 5/2 (/ (pow v 2) (* t (* (sqrt 2) PI))))))
265.0ms
(/ (/ (* (* (/ (fma (* v v) -5 1) PI) (/ (fma (* v v) -5 1) PI)) (/ (fma (* v v) -5 1) PI)) (* (* (sqrt (* 2 (fma (* v v) -3 1))) (sqrt (* 2 (fma (* v v) -3 1)))) (sqrt (* 2 (fma (* v v) -3 1))))) (* (* (* t (- 1 (* v v))) (* t (- 1 (* v v)))) (* t (- 1 (* v v)))))
248.0ms
(- (log (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))) (log (* t (- 1 (* v v)))))
245.0ms
(/ (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1)))) (- 1 (* v v)))
242.0ms
(/ (/ (/ (fma (* v v) -5 1) PI) (sqrt (sqrt (* 2 (fma (* v v) -3 1))))) (- 1 (* v v)))

prune1.4s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize8.0ms

Local error

Found 3 expressions with local error:

1.0b
(* (/ (fma (* v v) -5 1) PI) (/ 1 (sqrt (* 2 (fma (* v v) -3 1)))))
1.0b
(/ 1 (sqrt (* 2 (fma (* v v) -3 1))))
0.1b
(/ (* (/ (fma (* v v) -5 1) PI) (/ 1 (sqrt (* 2 (fma (* v v) -3 1))))) (* t (- 1 (* v v))))

rewrite17.0ms

Algorithm
rewrite-expression-head
Rules
16×add-exp-log
10×add-cbrt-cube
6×add-cube-cbrt
6×*-un-lft-identity
6×pow1
6×add-sqr-sqrt
5×associate-/r*
4×associate-/l/
4×div-inv
4×associate-*r/
4×prod-exp
4×associate-*l*
4×associate-*r*
3×add-log-exp
3×log1p-expm1-u
3×div-exp
3×rec-exp
3×expm1-log1p-u
2×pow-flip
2×cbrt-unprod
2×un-div-inv
2×associate-/r/
2×associate-*l/
2×frac-2neg
2×clear-num
2×cbrt-undiv
2×frac-times
1×inv-pow
1×associate-/l*
1×flip--
1×sqrt-prod
1×*-commutative
1×pow-prod-down
1×pow1/2
1×flip3--
1×times-frac
Counts
3 → 72
Calls
3 calls:
Slowest
9.0ms
(/ (* (/ (fma (* v v) -5 1) PI) (/ 1 (sqrt (* 2 (fma (* v v) -3 1))))) (* t (- 1 (* v v))))
6.0ms
(* (/ (fma (* v v) -5 1) PI) (/ 1 (sqrt (* 2 (fma (* v v) -3 1)))))
1.0ms
(/ 1 (sqrt (* 2 (fma (* v v) -3 1))))

series584.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
282.0ms
(/ (* (/ (fma (* v v) -5 1) PI) (/ 1 (sqrt (* 2 (fma (* v v) -3 1))))) (* t (- 1 (* v v))))
209.0ms
(* (/ (fma (* v v) -5 1) PI) (/ 1 (sqrt (* 2 (fma (* v v) -3 1)))))
93.0ms
(/ 1 (sqrt (* 2 (fma (* v v) -3 1))))

simplify4.7s

Counts
46 → 81
Calls
46 calls:
Slowest
430.0ms
(/ (* (* (* (/ (fma (* v v) -5 1) PI) (/ (fma (* v v) -5 1) PI)) (/ (fma (* v v) -5 1) PI)) (* (* (/ 1 (sqrt (* 2 (fma (* v v) -3 1)))) (/ 1 (sqrt (* 2 (fma (* v v) -3 1))))) (/ 1 (sqrt (* 2 (fma (* v v) -3 1)))))) (* (* (* t (- 1 (* v v))) (* t (- 1 (* v v)))) (* t (- 1 (* v v)))))
425.0ms
(/ (* (* (* (/ (fma (* v v) -5 1) PI) (/ 1 (sqrt (* 2 (fma (* v v) -3 1))))) (* (/ (fma (* v v) -5 1) PI) (/ 1 (sqrt (* 2 (fma (* v v) -3 1)))))) (* (/ (fma (* v v) -5 1) PI) (/ 1 (sqrt (* 2 (fma (* v v) -3 1)))))) (* (* (* t (- 1 (* v v))) (* t (- 1 (* v v)))) (* t (- 1 (* v v)))))
370.0ms
(- (/ 1 (* t (* (sqrt 2) PI))) (+ (* 53/8 (/ (pow v 4) (* t (* (sqrt 2) PI)))) (* 5/2 (/ (pow v 2) (* t (* (sqrt 2) PI))))))
314.0ms
(- (+ (* 1/54 (/ 1 (* (sqrt 2) (* (sqrt -1/3) (pow v 5))))) (+ (* 1/18 (/ 1 (* (sqrt 2) (* (sqrt -1/3) (pow v 3))))) (* 1/648 (/ 1 (* (sqrt 2) (* (pow (sqrt -1/3) 3) (pow v 5))))))) (/ (sqrt -1/3) (* (sqrt 2) v)))
278.0ms
(- (/ (sqrt -1/3) (* (sqrt 2) v)) (+ (* 1/54 (/ 1 (* (sqrt 2) (* (sqrt -1/3) (pow v 5))))) (+ (* 1/18 (/ 1 (* (sqrt 2) (* (sqrt -1/3) (pow v 3))))) (* 1/648 (/ 1 (* (sqrt 2) (* (pow (sqrt -1/3) 3) (pow v 5))))))))

prune945.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize5.0ms

Local error

Found 2 expressions with local error:

0.4b
(* (* t (- 1 (* v v))) (sqrt (* 2 (fma (* v v) -3 1))))
0.4b
(/ (/ (fma (* v v) -5 1) PI) (* (* t (- 1 (* v v))) (sqrt (* 2 (fma (* v v) -3 1)))))

rewrite17.0ms

Algorithm
rewrite-expression-head
Rules
9×add-exp-log
9×add-cbrt-cube
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×associate-/l*
4×associate-*r/
4×associate-*l/
4×associate-*r*
4×times-frac
4×pow1
3×div-inv
2×add-log-exp
2×flip--
2×log1p-expm1-u
2×cbrt-unprod
2×associate-/r/
2×prod-exp
2×div-exp
2×flip3--
2×cbrt-undiv
2×expm1-log1p-u
1×associate-/l/
1×sqrt-prod
1×*-commutative
1×pow-prod-down
1×associate-*l*
1×frac-2neg
1×associate-/r*
1×clear-num
Counts
2 → 48
Calls
2 calls:
Slowest
9.0ms
(/ (/ (fma (* v v) -5 1) PI) (* (* t (- 1 (* v v))) (sqrt (* 2 (fma (* v v) -3 1)))))
7.0ms
(* (* t (- 1 (* v v))) (sqrt (* 2 (fma (* v v) -3 1))))

series499.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
369.0ms
(/ (/ (fma (* v v) -5 1) PI) (* (* t (- 1 (* v v))) (sqrt (* 2 (fma (* v v) -3 1)))))
130.0ms
(* (* t (- 1 (* v v))) (sqrt (* 2 (fma (* v v) -3 1))))

simplify3.9s

Counts
34 → 54
Calls
34 calls:
Slowest
409.0ms
(* (* t (- 1 (* v v))) (sqrt (* 2 (fma (* v v) -3 1))))
379.0ms
(* (* t (- (* 1 1) (* (* v v) (* v v)))) (sqrt (* 2 (fma (* v v) -3 1))))
349.0ms
(- (/ 1 (* t (* (sqrt 2) PI))) (+ (* 53/8 (/ (pow v 4) (* t (* (sqrt 2) PI)))) (* 5/2 (/ (pow v 2) (* t (* (sqrt 2) PI))))))
339.0ms
(- (log (/ (fma (* v v) -5 1) PI)) (log (* (* t (- 1 (* v v))) (sqrt (* 2 (fma (* v v) -3 1))))))
316.0ms
(- (log (/ (fma (* v v) -5 1) PI)) (+ (log (* t (- 1 (* v v)))) (log (sqrt (* 2 (fma (* v v) -3 1))))))

prune577.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

regimes181.0ms

Accuracy

0% (0.1b remaining)

Error of 0.1b against oracle of 0.0b and baseline of 0.1b

bsearch4.0ms

end0.0ms

sample4.0s

Algorithm
intervals
Results
2.0s8001×body80valid
1.6s7997×body80nan