Average Error: 25.1 → 18.9
Time: 1.1m
Precision: 64
Internal Precision: 128
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -2.3072532229286938 \cdot 10^{+76}:\\ \;\;\;\;\left(1 - \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(\sqrt{\frac{-1}{h}} \cdot {\left(\frac{-1}{d}\right)}^{\frac{-1}{2}}\right)\right)\\ \mathbf{elif}\;d \le -3.946078378106929 \cdot 10^{-292}:\\ \;\;\;\;\left(1 - \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left(\left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{\ell}}\right) \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{d} \cdot {\left(\frac{1}{h}\right)}^{\frac{1}{2}}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array}\]

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if d < -2.3072532229286938e+76

    1. Initial program 23.6

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

    2. Taylor expanded around -inf 16.5

      \[\leadsto \left(\color{blue}{e^{\frac{1}{2} \cdot \left(\log \left(\frac{-1}{h}\right) - \log \left(\frac{-1}{d}\right)\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

    3. Simplified12.2

      \[\leadsto \left(\color{blue}{\left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{h}}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

    if -2.3072532229286938e+76 < d < -3.946078378106929e-292

    1. Initial program 24.7

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

    2. Taylor expanded around -inf 23.5

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{e^{\frac{1}{2} \cdot \left(\log \left(\frac{-1}{\ell}\right) - \log \left(\frac{-1}{d}\right)\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

    3. Simplified20.4

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{\ell}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

    if -3.946078378106929e-292 < d

    1. Initial program 25.8

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
    2. Using strategy rm

    3. Applied div-inv25.9

      \[\leadsto \left({\color{blue}{\left(d \cdot \frac{1}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

    4. Applied unpow-prod-down20.6

      \[\leadsto \left(\color{blue}{\left({d}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

    5. Simplified20.6

      \[\leadsto \left(\left(\color{blue}{\sqrt{d}} \cdot {\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification18.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -2.3072532229286938 \cdot 10^{+76}:\\ \;\;\;\;\left(1 - \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(\sqrt{\frac{-1}{h}} \cdot {\left(\frac{-1}{d}\right)}^{\frac{-1}{2}}\right)\right)\\ \mathbf{elif}\;d \le -3.946078378106929 \cdot 10^{-292}:\\ \;\;\;\;\left(1 - \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left(\left({\left(\frac{-1}{d}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{\ell}}\right) \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{d} \cdot {\left(\frac{1}{h}\right)}^{\frac{1}{2}}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019010 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))))

Details

Time bar (total: 1.0m)Debug log

sample846.0ms

Algorithm
intervals

simplify185.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
185.0ms
(* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))

prune22.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 25.4b

localize63.0ms

Local error

Found 4 expressions with local error:

14.3b
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
13.4b
(pow (/ d l) (/ 1 2))
12.5b
(pow (/ d h) (/ 1 2))
6.4b
(/ (* M D) (* 2 d))

rewrite36.0ms

Algorithm
rewrite-expression-head
Rules
22×add-exp-log
11×add-cbrt-cube
9×add-cube-cbrt
9×prod-exp
9×*-un-lft-identity
9×add-sqr-sqrt
8×unpow-prod-down
8×pow-unpow
8×pow1
6×div-inv
4×add-log-exp
4×pow-exp
4×associate-*r*
3×cbrt-unprod
3×associate-*l/
3×pow-to-exp
2×div-exp
2×pow-pow
1×associate-/l*
1×*-commutative
1×associate-*r/
1×pow-prod-down
1×associate-*l*
1×times-frac
1×frac-2neg
1×associate-/r*
1×clear-num
1×cbrt-undiv
1×frac-times
Counts
4 → 75
Calls
4 calls:
Slowest
23.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
5.0ms
(/ (* M D) (* 2 d))
3.0ms
(pow (/ d h) (/ 1 2))
3.0ms
(pow (/ d l) (/ 1 2))

series297.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
106.0ms
(pow (/ d l) (/ 1 2))
105.0ms
(pow (/ d h) (/ 1 2))
67.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
19.0ms
(/ (* M D) (* 2 d))

simplify3.7s

Counts
52 → 87
Calls
52 calls:
Slowest
634.0ms
(* (* 1 (pow (/ (* M D) (* 2 d)) 2)) h)
477.0ms
(+ (+ (log (/ 1 2)) (log (pow (/ (* M D) (* 2 d)) 2))) (log (/ h l)))
335.0ms
(* (* (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2))) (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2))) (* (* (/ h l) (/ h l)) (/ h l)))
199.0ms
(/ (* (* (* M D) (* M D)) (* M D)) (* (* (* 2 d) (* 2 d)) (* 2 d)))
187.0ms
(exp (* 1/2 (- (log (/ 1 h)) (log (/ 1 d)))))

prune1.6s

Pruning

15 alts after pruning (15 fresh and 0 done)

Merged error: 13.2b

localize34.0ms

Local error

Found 4 expressions with local error:

23.5b
(cbrt (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))
23.5b
(cbrt (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))
23.5b
(cbrt (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))
14.3b
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))

rewrite58.0ms

Algorithm
rewrite-expression-head
Rules
18×add-exp-log
12×associate-*l/
9×prod-exp
9×add-cbrt-cube
9×cbrt-div
6×pow1
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×add-log-exp
4×associate-*r/
4×associate-*r*
4×frac-times
3×cbrt-unprod
3×pow1/3
3×cbrt-prod
2×pow-exp
1×div-inv
1×*-commutative
1×pow-prod-down
1×associate-*l*
1×div-exp
1×pow-to-exp
Counts
4 → 60
Calls
4 calls:
Slowest
19.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
13.0ms
(cbrt (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))
13.0ms
(cbrt (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))
12.0ms
(cbrt (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))

series680.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
226.0ms
(cbrt (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))
193.0ms
(cbrt (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))
191.0ms
(cbrt (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))
69.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))

simplify10.2s

Counts
40 → 72
Calls
40 calls:
Slowest
832.0ms
(cbrt (* (* 1 (pow (/ (* M D) (* 2 d)) 2)) h))
811.0ms
(cbrt (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)))
723.0ms
(cbrt (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)))
634.0ms
(cbrt (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)))
616.0ms
(cbrt (* (* 1 (pow (/ (* M D) (* 2 d)) 2)) h))

prune1.8s

Pruning

15 alts after pruning (14 fresh and 1 done)

Merged error: 13.2b

localize20.0ms

Local error

Found 4 expressions with local error:

14.3b
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
13.4b
(pow (/ d l) (/ 1 2))
12.5b
(pow (/ d h) (/ 1 2))
12.5b
(pow (/ d h) (/ 1 2))

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
21×add-exp-log
12×unpow-prod-down
12×pow-unpow
11×add-cube-cbrt
11×*-un-lft-identity
11×add-sqr-sqrt
9×prod-exp
9×add-cbrt-cube
9×pow1
7×div-inv
5×pow-exp
4×add-log-exp
4×associate-*r*
4×pow-to-exp
3×cbrt-unprod
3×associate-*l/
3×pow-pow
1×*-commutative
1×associate-*r/
1×pow-prod-down
1×associate-*l*
1×div-exp
1×frac-times
Counts
4 → 78
Calls
4 calls:
Slowest
10.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
1.0ms
(pow (/ d h) (/ 1 2))
1.0ms
(pow (/ d h) (/ 1 2))
1.0ms
(pow (/ d l) (/ 1 2))

series319.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
88.0ms
(pow (/ d h) (/ 1 2))
85.0ms
(pow (/ d l) (/ 1 2))
74.0ms
(pow (/ d h) (/ 1 2))
71.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))

simplify3.9s

Counts
58 → 90
Calls
58 calls:
Slowest
687.0ms
(* (* 1 (pow (/ (* M D) (* 2 d)) 2)) h)
552.0ms
(+ (+ (log (/ 1 2)) (log (pow (/ (* M D) (* 2 d)) 2))) (log (/ h l)))
359.0ms
(* (* (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2))) (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2))) (* (* (/ h l) (/ h l)) (/ h l)))
205.0ms
(exp (* 1/2 (- (log (/ 1 l)) (log (/ 1 d)))))
188.0ms
(exp (* 1/2 (- (log (/ 1 h)) (log (/ 1 d)))))

prune2.0s

Pruning

18 alts after pruning (16 fresh and 2 done)

Merged error: 13.2b

localize13.0ms

Local error

Found 4 expressions with local error:

14.3b
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
13.4b
(pow (/ d l) (/ 1 2))
13.4b
(pow (/ d l) (/ 1 2))
12.5b
(pow (/ d h) (/ 1 2))

rewrite18.0ms

Algorithm
rewrite-expression-head
Rules
21×add-exp-log
12×unpow-prod-down
12×pow-unpow
11×add-cube-cbrt
11×*-un-lft-identity
11×add-sqr-sqrt
9×prod-exp
9×add-cbrt-cube
9×pow1
7×div-inv
5×pow-exp
4×add-log-exp
4×associate-*r*
4×pow-to-exp
3×cbrt-unprod
3×associate-*l/
3×pow-pow
1×*-commutative
1×associate-*r/
1×pow-prod-down
1×associate-*l*
1×div-exp
1×frac-times
Counts
4 → 78
Calls
4 calls:
Slowest
13.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
1.0ms
(pow (/ d l) (/ 1 2))
1.0ms
(pow (/ d l) (/ 1 2))
1.0ms
(pow (/ d h) (/ 1 2))

series356.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
106.0ms
(pow (/ d h) (/ 1 2))
97.0ms
(pow (/ d l) (/ 1 2))
82.0ms
(* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))
71.0ms
(pow (/ d l) (/ 1 2))

simplify4.0s

Counts
58 → 90
Calls
58 calls:
Slowest
766.0ms
(* (* 1 (pow (/ (* M D) (* 2 d)) 2)) h)
500.0ms
(+ (+ (log (/ 1 2)) (log (pow (/ (* M D) (* 2 d)) 2))) (log (/ h l)))
410.0ms
(* (* (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2))) (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2))) (* (* (/ h l) (/ h l)) (/ h l)))
175.0ms
(exp (* 1/2 (- (log (/ 1 l)) (log (/ 1 d)))))
153.0ms
(exp (* 1/2 (- (log (/ 1 l)) (log (/ 1 d)))))

prune1.9s

Pruning

20 alts after pruning (18 fresh and 2 done)

Merged error: 13.1b

regimes1.9s

Accuracy

47.7% (6.8b remaining)

Error of 18.9b against oracle of 12.1b and baseline of 25.1b

bsearch2.5s

end0.0ms

sample25.3s

Algorithm
intervals