Average Error: 0.4 → 0.3
Time: 28.8s
Precision: 64
Internal Precision: 128
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v} \cdot \frac{\frac{\frac{1}{\pi}}{t}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}\]

Error

Bits error versus v

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.4

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

  3. Applied *-un-lft-identity0.4

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

  4. Applied times-frac0.4

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

  5. Simplified0.4

    \[\leadsto \color{blue}{\frac{\frac{\frac{1}{t}}{\pi}}{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity0.4

    \[\leadsto \frac{\frac{\frac{1}{t}}{\pi}}{\sqrt{\color{blue}{1 \cdot \left(2 + \left(v \cdot -6\right) \cdot v\right)}}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\]

  8. Applied sqrt-prod0.4

    \[\leadsto \frac{\frac{\frac{1}{t}}{\pi}}{\color{blue}{\sqrt{1} \cdot \sqrt{2 + \left(v \cdot -6\right) \cdot v}}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\]

  9. Applied div-inv0.4

    \[\leadsto \frac{\color{blue}{\frac{1}{t} \cdot \frac{1}{\pi}}}{\sqrt{1} \cdot \sqrt{2 + \left(v \cdot -6\right) \cdot v}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\]

  10. Applied times-frac0.3

    \[\leadsto \color{blue}{\left(\frac{\frac{1}{t}}{\sqrt{1}} \cdot \frac{\frac{1}{\pi}}{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}\right)} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\]

  11. Simplified0.3

    \[\leadsto \left(\color{blue}{\frac{1}{t}} \cdot \frac{\frac{1}{\pi}}{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}\right) \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\]
  12. Using strategy rm

  13. Applied pow10.3

    \[\leadsto \left(\frac{1}{t} \cdot \color{blue}{{\left(\frac{\frac{1}{\pi}}{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}\right)}^{1}}\right) \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\]

  14. Applied pow10.3

    \[\leadsto \left(\color{blue}{{\left(\frac{1}{t}\right)}^{1}} \cdot {\left(\frac{\frac{1}{\pi}}{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}\right)}^{1}\right) \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\]

  15. Applied pow-prod-down0.3

    \[\leadsto \color{blue}{{\left(\frac{1}{t} \cdot \frac{\frac{1}{\pi}}{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}\right)}^{1}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\]

  16. Simplified0.3

    \[\leadsto {\color{blue}{\left(\frac{\frac{\frac{1}{\pi}}{t}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}\right)}}^{1} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\]

  17. Final simplification0.3

    \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v} \cdot \frac{\frac{\frac{1}{\pi}}{t}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}\]

Reproduce

herbie shell --seed 2019016 
(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: 28.3s)Debug log

sample203.0ms

Algorithm
intervals
Results
544×(pre true 80)
288×(body nan 80)
256×(body real 80)

simplify146.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
146.0ms
(/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v))))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.4b

localize31.0ms

Local error

Found 4 expressions with local error:

0.5b
(* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v))))))
0.3b
(* PI t)
0.3b
(/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v))))
0.1b
(* 5 (* v v))

rewrite42.0ms

Algorithm
rewrite-expression-head
Rules
18×associate-*r/
15×add-exp-log
15×add-cbrt-cube
8×flip--
8×associate-/r/
8×sqrt-div
8×flip3--
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×pow1
5×associate-*r*
4×add-log-exp
4×cbrt-unprod
4×prod-exp
4×frac-times
3×associate-/l*
3×*-commutative
3×div-exp
3×times-frac
3×cbrt-undiv
2×associate-/l/
2×associate-*l/
1×div-inv
1×sqrt-prod
1×pow-prod-down
1×associate-*l*
1×div-sub
1×frac-2neg
1×associate-/r*
1×clear-num
Counts
4 → 69
Calls
4 calls:
Slowest
21.0ms
(/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v))))
17.0ms
(* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v))))))
2.0ms
(* 5 (* v v))
0.0ms
(* PI t)

series526.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
307.0ms
(/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v))))
157.0ms
(* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v))))))
43.0ms
(* PI t)
19.0ms
(* 5 (* v v))

simplify6.9s

Counts
47 → 81
Calls
47 calls:
Slowest
531.0ms
(* (* (* (* PI t) (* PI t)) (* PI t)) (* (* (sqrt (* 2 (- 1 (* 3 (* v v))))) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (sqrt (* 2 (- 1 (* 3 (* v v)))))))
506.0ms
(- (log (- 1 (* 5 (* v v)))) (+ (log (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v))))))) (log (- 1 (* v v)))))
391.0ms
(- (/ 1 (* t (* (sqrt 2) PI))) (+ (* 53/8 (/ (pow v 4) (* t (* (sqrt 2) PI)))) (* 5/2 (/ (pow v 2) (* t (* (sqrt 2) PI))))))
380.0ms
(* (sqrt (+ (* 1 1) (+ (* (* 3 (* v v)) (* 3 (* v v))) (* 1 (* 3 (* v v)))))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))
361.0ms
(* (sqrt (+ 1 (* 3 (* v v)))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))

prune1.1s

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.0b

localize50.0ms

Local error

Found 4 expressions with local error:

0.4b
(/ (/ (/ 1 t) PI) (sqrt (+ 2 (* (* v -6) v))))
0.3b
(/ (/ 1 t) PI)
0.1b
(* 5 (* v v))
0.1b
(* (* v -6) v)

rewrite22.0ms

Algorithm
rewrite-expression-head
Rules
24×times-frac
22×add-cube-cbrt
22×*-un-lft-identity
22×add-sqr-sqrt
15×sqrt-prod
10×div-inv
8×associate-/l*
6×add-exp-log
6×add-cbrt-cube
6×associate-/r*
4×add-log-exp
4×pow1
2×associate-/l/
2×*-commutative
2×associate-/r/
2×sqrt-div
2×frac-2neg
2×clear-num
1×flip-+
1×associate-*l*
1×associate-*r*
1×div-exp
1×flip3-+
1×cbrt-undiv
Counts
4 → 82
Calls
4 calls:
Slowest
10.0ms
(/ (/ (/ 1 t) PI) (sqrt (+ 2 (* (* v -6) v))))
4.0ms
(/ (/ 1 t) PI)
4.0ms
(* (* v -6) v)
2.0ms
(* 5 (* v v))

series261.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
156.0ms
(/ (/ (/ 1 t) PI) (sqrt (+ 2 (* (* v -6) v))))
76.0ms
(/ (/ 1 t) PI)
14.0ms
(* (* v -6) v)
14.0ms
(* 5 (* v v))

simplify2.5s

Counts
76 → 94
Calls
76 calls:
Slowest
587.0ms
(/ (* (* (/ (/ 1 t) PI) (/ (/ 1 t) PI)) (/ (/ 1 t) PI)) (* (* (sqrt (+ 2 (* (* v -6) v))) (sqrt (+ 2 (* (* v -6) v)))) (sqrt (+ 2 (* (* v -6) v)))))
356.0ms
(sqrt (+ (* 2 2) (- (* (* (* v -6) v) (* (* v -6) v)) (* 2 (* (* v -6) v)))))
162.0ms
(+ (* 3/4 (/ (pow v 2) (* t (* PI (sqrt 1/2))))) (/ (sqrt 1/2) (* t PI)))
146.0ms
(- (log (/ (/ 1 t) PI)) (log (sqrt (+ 2 (* (* v -6) v)))))
146.0ms
(- (/ (sqrt -1/6) (* t (* v PI))) (* 1/36 (/ 1 (* t (* (pow v 3) (* PI (sqrt -1/6)))))))

prune1.3s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0b

localize21.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (/ 1 t) (/ (/ 1 PI) (sqrt (+ 2 (* (* v -6) v)))))
0.1b
(* 5 (* v v))
0.1b
(* (* v -6) v)
0.0b
(/ (/ 1 PI) (sqrt (+ 2 (* (* v -6) v))))

rewrite27.0ms

Algorithm
rewrite-expression-head
Rules
48×times-frac
37×add-cube-cbrt
37×*-un-lft-identity
37×add-sqr-sqrt
31×associate-*r*
27×sqrt-prod
16×div-inv
11×add-exp-log
11×add-cbrt-cube
6×associate-/r*
6×pow1
5×associate-*l*
4×add-log-exp
4×associate-/l*
4×associate-/r/
4×sqrt-div
3×*-commutative
2×flip-+
2×cbrt-unprod
2×prod-exp
2×div-exp
2×flip3-+
2×cbrt-undiv
1×associate-/l/
1×associate-*r/
1×associate-*l/
1×pow-prod-down
1×frac-2neg
1×clear-num
1×frac-times
Counts
4 → 117
Calls
4 calls:
Slowest
11.0ms
(* (/ 1 t) (/ (/ 1 PI) (sqrt (+ 2 (* (* v -6) v)))))
7.0ms
(/ (/ 1 PI) (sqrt (+ 2 (* (* v -6) v))))
4.0ms
(* (* v -6) v)
2.0ms
(* 5 (* v v))

series274.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
141.0ms
(* (/ 1 t) (/ (/ 1 PI) (sqrt (+ 2 (* (* v -6) v)))))
93.0ms
(/ (/ 1 PI) (sqrt (+ 2 (* (* v -6) v))))
20.0ms
(* (* v -6) v)
20.0ms
(* 5 (* v v))

simplify4.5s

Counts
113 → 129
Calls
113 calls:
Slowest
523.0ms
(* (* (* (/ 1 t) (/ 1 t)) (/ 1 t)) (* (* (/ (/ 1 PI) (sqrt (+ 2 (* (* v -6) v)))) (/ (/ 1 PI) (sqrt (+ 2 (* (* v -6) v))))) (/ (/ 1 PI) (sqrt (+ 2 (* (* v -6) v))))))
384.0ms
(sqrt (+ (* 2 2) (- (* (* (* v -6) v) (* (* v -6) v)) (* 2 (* (* v -6) v)))))
353.0ms
(+ (log (/ 1 t)) (log (/ (/ 1 PI) (sqrt (+ 2 (* (* v -6) v))))))
298.0ms
(sqrt (+ (* 2 2) (- (* (* (* v -6) v) (* (* v -6) v)) (* 2 (* (* v -6) v)))))
261.0ms
(/ (* (* (/ 1 PI) (/ 1 PI)) (/ 1 PI)) (* (* (sqrt (+ 2 (* (* v -6) v))) (sqrt (+ 2 (* (* v -6) v)))) (sqrt (+ 2 (* (* v -6) v)))))

prune1.6s

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0b

localize23.0ms

Local error

Found 4 expressions with local error:

0.4b
(/ (/ (/ 1 PI) t) (sqrt (+ 2 (* (* v v) -6))))
0.4b
(/ (/ 1 PI) t)
0.2b
(* (* v v) -6)
0.1b
(* 5 (* v v))

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
24×times-frac
22×add-cube-cbrt
22×*-un-lft-identity
22×add-sqr-sqrt
15×sqrt-prod
10×div-inv
8×associate-/l*
6×add-exp-log
6×add-cbrt-cube
6×associate-/r*
4×add-log-exp
4×pow1
2×associate-/l/
2×*-commutative
2×associate-/r/
2×sqrt-div
2×frac-2neg
2×clear-num
1×flip-+
1×associate-*l*
1×associate-*r*
1×div-exp
1×flip3-+
1×cbrt-undiv
Counts
4 → 82
Calls
4 calls:
Slowest
8.0ms
(/ (/ (/ 1 PI) t) (sqrt (+ 2 (* (* v v) -6))))
2.0ms
(/ (/ 1 PI) t)
2.0ms
(* (* v v) -6)
1.0ms
(* 5 (* v v))

series273.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
147.0ms
(/ (/ (/ 1 PI) t) (sqrt (+ 2 (* (* v v) -6))))
89.0ms
(/ (/ 1 PI) t)
19.0ms
(* 5 (* v v))
18.0ms
(* (* v v) -6)

simplify2.8s

Counts
76 → 94
Calls
76 calls:
Slowest
579.0ms
(/ (* (* (/ (/ 1 PI) t) (/ (/ 1 PI) t)) (/ (/ 1 PI) t)) (* (* (sqrt (+ 2 (* (* v v) -6))) (sqrt (+ 2 (* (* v v) -6)))) (sqrt (+ 2 (* (* v v) -6)))))
350.0ms
(sqrt (+ (* 2 2) (- (* (* (* v v) -6) (* (* v v) -6)) (* 2 (* (* v v) -6)))))
208.0ms
(+ (* 3/4 (/ (pow v 2) (* t (* PI (sqrt 1/2))))) (/ (sqrt 1/2) (* t PI)))
141.0ms
(- (/ (sqrt -1/6) (* t (* v PI))) (* 1/36 (/ 1 (* t (* (pow v 3) (* PI (sqrt -1/6)))))))
141.0ms
(- (* 1/36 (/ 1 (* t (* (pow v 3) (* PI (sqrt -1/6)))))) (/ (sqrt -1/6) (* t (* v PI))))

prune1.3s

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0b

regimes259.0ms

Accuracy

0% (0.3b remaining)

Error of 0.3b against oracle of 0.0b and baseline of 0.3b

bsearch2.0ms

end0.0ms

sample4.1s

Algorithm
intervals
Results
16050×(pre true 80)
8048×(body nan 80)
8002×(body real 80)