Average Error: 0.5 → 0.3
Time: 28.4s
Precision: 64
Internal Precision: 128
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\left(\frac{1}{t} \cdot \frac{\frac{1}{\pi}}{\sqrt{2 + v \cdot \left(-6 \cdot v\right)}}\right) \cdot \frac{1 - \left(v \cdot v\right) \cdot 5}{1 - v \cdot v}\]

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 0.5

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

  3. Applied *-un-lft-identity0.5

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

  4. Applied times-frac0.5

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

  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

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

  8. Applied div-inv0.4

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

  9. Applied times-frac0.3

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019010 
(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: 27.9s)Debug log

sample145.0ms

Algorithm
intervals

simplify138.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
138.0ms
(/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v))))

prune14.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.4b

localize74.0ms

Local error

Found 4 expressions with local error:

0.5b
(* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v))))))
0.3b
(/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v))))
0.3b
(* PI t)
0.2b
(* 3 (* v v))

rewrite36.0ms

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
22.0ms
(/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v))))
10.0ms
(* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v))))))
2.0ms
(* 3 (* v v))
0.0ms
(* PI t)

series592.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
362.0ms
(/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v))))
180.0ms
(* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v))))))
39.0ms
(* PI t)
11.0ms
(* 3 (* v v))

simplify6.5s

Counts
47 → 81
Calls
47 calls:
Slowest
458.0ms
(* (* (* (* PI t) (* PI t)) (* PI t)) (* (* (sqrt (* 2 (- 1 (* 3 (* v v))))) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (sqrt (* 2 (- 1 (* 3 (* v v)))))))
402.0ms
(* (sqrt (+ (* 1 1) (+ (* (* 3 (* v v)) (* 3 (* v v))) (* 1 (* 3 (* v v)))))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))
398.0ms
(* (sqrt (+ (* 1 1) (+ (* (* 3 (* v v)) (* 3 (* v v))) (* 1 (* 3 (* v v)))))) (+ 1 (* v v)))
341.0ms
(/ (* (* (- 1 (* 5 (* v v))) (- 1 (* 5 (* v v)))) (- 1 (* 5 (* v v)))) (* (* (* (* (* PI t) (* PI t)) (* PI t)) (* (* (sqrt (* 2 (- 1 (* 3 (* v v))))) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (sqrt (* 2 (- 1 (* 3 (* v v))))))) (* (* (- 1 (* v v)) (- 1 (* v v))) (- 1 (* v v)))))
328.0ms
(* (sqrt (+ 1 (* 3 (* v v)))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))

prune1.1s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.0b

localize47.0ms

Local error

Found 4 expressions with local error:

0.5b
(/ (/ (/ 1 t) PI) (sqrt (+ 2 (* (* v -6) v))))
0.3b
(/ (/ 1 t) PI)
0.1b
(* (* v -6) v)
0.1b
(* 5 (* v v))

rewrite24.0ms

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
11.0ms
(/ (/ (/ 1 t) PI) (sqrt (+ 2 (* (* v -6) v))))
5.0ms
(/ (/ 1 t) PI)
4.0ms
(* (* v -6) v)
2.0ms
(* 5 (* v v))

series287.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
188.0ms
(/ (/ (/ 1 t) PI) (sqrt (+ 2 (* (* v -6) v))))
59.0ms
(/ (/ 1 t) PI)
20.0ms
(* 5 (* v v))
19.0ms
(* (* v -6) v)

simplify2.8s

Counts
76 → 94
Calls
76 calls:
Slowest
671.0ms
(/ (* (* (/ (/ 1 t) PI) (/ (/ 1 t) PI)) (/ (/ 1 t) PI)) (* (* (sqrt (+ 2 (* (* v -6) v))) (sqrt (+ 2 (* (* v -6) v)))) (sqrt (+ 2 (* (* v -6) v)))))
406.0ms
(sqrt (+ (* 2 2) (- (* (* (* v -6) v) (* (* v -6) v)) (* 2 (* (* v -6) v)))))
211.0ms
(+ (* 3/4 (/ (pow v 2) (* t (* PI (sqrt 1/2))))) (/ (sqrt 1/2) (* t PI)))
172.0ms
(- (/ (sqrt -1/6) (* t (* v PI))) (* 1/36 (/ 1 (* t (* (pow v 3) (* PI (sqrt -1/6)))))))
170.0ms
(- (log (/ (/ 1 t) PI)) (log (sqrt (+ 2 (* (* v -6) v)))))

prune1.2s

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0b

localize22.0ms

Local error

Found 4 expressions with local error:

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

rewrite26.0ms

Algorithm
rewrite-expression-head
Rules
33×times-frac
31×associate-*r*
28×add-cube-cbrt
28×*-un-lft-identity
28×add-sqr-sqrt
12×sqrt-prod
11×add-exp-log
11×add-cbrt-cube
9×div-inv
6×pow1
5×associate-*l*
4×add-log-exp
4×associate-/r/
3×associate-/l*
3×*-commutative
3×associate-/r*
2×associate-/l/
2×flip--
2×cbrt-unprod
2×prod-exp
2×div-exp
2×sqrt-div
2×flip3--
2×cbrt-undiv
1×flip-+
1×associate-*r/
1×associate-*l/
1×pow-prod-down
1×div-sub
1×flip3-+
1×frac-2neg
1×clear-num
1×frac-times
Counts
4 → 100
Calls
4 calls:
Slowest
11.0ms
(* (/ 1 t) (/ (/ 1 PI) (sqrt (+ 2 (* (* v -6) v)))))
6.0ms
(/ (- 1 (* 5 (* v v))) (- 1 (* v v)))
4.0ms
(* (* v -6) v)
2.0ms
(* 5 (* v v))

series219.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
169.0ms
(* (/ 1 t) (/ (/ 1 PI) (sqrt (+ 2 (* (* v -6) v)))))
28.0ms
(/ (- 1 (* 5 (* v v))) (- 1 (* v v)))
11.0ms
(* 5 (* v v))
11.0ms
(* (* v -6) v)

simplify4.8s

Counts
83 → 112
Calls
83 calls:
Slowest
604.0ms
(* (* (* (/ 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))))))
408.0ms
(/ (* (* (- 1 (* 5 (* v v))) (- 1 (* 5 (* v v)))) (- 1 (* 5 (* v v)))) (* (* (- 1 (* v v)) (- 1 (* v v))) (- 1 (* v v))))
336.0ms
(+ (log (/ 1 t)) (log (/ (/ 1 PI) (sqrt (+ 2 (* (* v -6) v))))))
293.0ms
(* (/ 1 t) (/ (/ 1 PI) (sqrt (+ 2 (* (* v -6) v)))))
273.0ms
(sqrt (+ (* 2 2) (- (* (* (* v -6) v) (* (* v -6) v)) (* 2 (* (* v -6) v)))))

prune1.5s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize13.0ms

Local error

Found 4 expressions with local error:

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

rewrite12.0ms

Algorithm
rewrite-expression-head
Rules
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
7×associate-*l*
6×add-exp-log
6×add-cbrt-cube
4×add-log-exp
4×associate-/l*
4×associate-*l/
4×sqrt-div
4×times-frac
4×pow1
3×div-inv
3×sqrt-prod
3×*-commutative
2×flip-+
2×associate-/r/
2×flip3-+
1×associate-/l/
1×associate-*r*
1×div-exp
1×frac-2neg
1×associate-/r*
1×clear-num
1×cbrt-undiv
Counts
4 → 58
Calls
4 calls:
Slowest
5.0ms
(/ (/ 1 t) (* (sqrt (+ 2 (* (* v -6) v))) PI))
3.0ms
(* (sqrt (+ 2 (* (* v -6) v))) PI)
2.0ms
(* (* v -6) v)
1.0ms
(* 5 (* v v))

series260.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
151.0ms
(/ (/ 1 t) (* (sqrt (+ 2 (* (* v -6) v))) PI))
69.0ms
(* (sqrt (+ 2 (* (* v -6) v))) PI)
20.0ms
(* 5 (* v v))
20.0ms
(* (* v -6) v)

simplify2.9s

Counts
33 → 70
Calls
33 calls:
Slowest
487.0ms
(* (sqrt (- (* 2 2) (* (* (* v -6) v) (* (* v -6) v)))) PI)
384.0ms
(sqrt (+ (* 2 2) (- (* (* (* v -6) v) (* (* v -6) v)) (* 2 (* (* v -6) v)))))
350.0ms
(* (sqrt (+ (pow 2 3) (pow (* (* v -6) v) 3))) PI)
256.0ms
(- (+ (/ PI (* v (sqrt -6))) (* v (* (sqrt -6) PI))) (* 1/2 (/ PI (* (pow v 3) (pow (sqrt -6) 3)))))
196.0ms
(- (* 1/2 (/ PI (* (pow v 3) (pow (sqrt -6) 3)))) (+ (/ PI (* v (sqrt -6))) (* v (* (sqrt -6) PI))))

prune989.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

regimes195.0ms

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