Average Error: 58.7 → 3.7
Time: 60.0s
Precision: 64
Internal Precision: 128
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.003997533940732 \cdot 10^{+224}:\\ \;\;\;\;\frac{a + b}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.7
Target14.4
Herbie3.7
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if b < -1.003997533940732e+224

    1. Initial program 49.3

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

    2. Taylor expanded around 0 61.8

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\varepsilon \cdot b + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right)\right)\right)}}\]

    3. Simplified55.1

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\varepsilon \cdot b + \left(\left(\varepsilon \cdot b\right) \cdot \left(\varepsilon \cdot b\right)\right) \cdot \left(b \cdot \left(\frac{1}{6} \cdot \varepsilon\right) + \frac{1}{2}\right)\right)}}\]

    4. Taylor expanded around 0 12.3

      \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
    5. Using strategy rm

    6. Applied frac-add20.7

      \[\leadsto \color{blue}{\frac{1 \cdot b + a \cdot 1}{a \cdot b}}\]

    7. Simplified20.7

      \[\leadsto \frac{\color{blue}{b + a}}{a \cdot b}\]

    if -1.003997533940732e+224 < b

    1. Initial program 59.2

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

    2. Taylor expanded around 0 56.1

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\varepsilon \cdot b + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right)\right)\right)}}\]

    3. Simplified54.8

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\varepsilon \cdot b + \left(\left(\varepsilon \cdot b\right) \cdot \left(\varepsilon \cdot b\right)\right) \cdot \left(b \cdot \left(\frac{1}{6} \cdot \varepsilon\right) + \frac{1}{2}\right)\right)}}\]

    4. Taylor expanded around 0 2.8

      \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.003997533940732 \cdot 10^{+224}:\\ \;\;\;\;\frac{a + b}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 323378853 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :pre (and (< -1 eps) (< eps 1))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))))

Details

Time bar (total: 58.8s)Debug log

sample846.0ms

Algorithm
intervals
Results
254.0ms80×body2560valid
236.0ms64×body10240exit
229.0ms112×body1280valid
51.0ms38×body640valid
16.0ms322×pre80true
8.0ms8×body320valid
6.0ms19×body80valid
1.0ms1×body160valid

simplify322.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
322.0ms
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)))

prune20.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 57.8b

localize100.0ms

Local error

Found 4 expressions with local error:

5.3b
(- (exp (* (+ a b) eps)) 1)
3.2b
(- (exp (* b eps)) 1)
2.5b
(- (exp (* a eps)) 1)
0.2b
(* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))

rewrite81.0ms

Algorithm
rewrite-expression-head
Rules
11×add-sqr-sqrt
9×flip--
9×flip3--
6×add-cube-cbrt
6×add-exp-log
6×add-cbrt-cube
6×*-un-lft-identity
6×pow1
5×difference-of-sqr-1
5×sub-neg
4×add-log-exp
4×associate-*l*
4×associate-*r*
4×frac-times
2×associate-*r/
2×associate-*l/
2×distribute-lft-in
2×distribute-rgt-in
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
Counts
4 → 67
Calls
4 calls:
Slowest
44.0ms
(* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))
14.0ms
(- (exp (* (+ a b) eps)) 1)
10.0ms
(- (exp (* a eps)) 1)
10.0ms
(- (exp (* b eps)) 1)

series250.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
76.0ms
(* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))
76.0ms
(- (exp (* (+ a b) eps)) 1)
51.0ms
(- (exp (* a eps)) 1)
47.0ms
(- (exp (* b eps)) 1)

simplify6.3s

Counts
41 → 79
Calls
41 calls:
Slowest
796.0ms
(* (* (* (- (exp (* a eps)) 1) (- (exp (* a eps)) 1)) (- (exp (* a eps)) 1)) (* (* (- (exp (* b eps)) 1) (- (exp (* b eps)) 1)) (- (exp (* b eps)) 1)))
730.0ms
(* (+ (exp (* a eps)) 1) (+ (* (exp (* b eps)) (exp (* b eps))) (+ (* 1 1) (* (exp (* b eps)) 1))))
470.0ms
(+ (* eps b) (+ (* 1/2 (* (pow eps 2) (pow b 2))) (* 1/6 (* (pow eps 3) (pow b 3)))))
466.0ms
(+ (* 1/2 (* (pow a 2) (pow eps 2))) (+ (* 1/6 (* (pow a 3) (pow eps 3))) (* a eps)))
415.0ms
(* (- (* (exp (* a eps)) (exp (* a eps))) (* 1 1)) (- (exp (* b eps)) 1))

prune1.8s

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 45.5b

localize69.0ms

Local error

Found 4 expressions with local error:

5.3b
(- (exp (* (+ a b) eps)) 1)
2.9b
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)))))
2.5b
(- (exp (* a eps)) 1)
2.5b
(* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))

rewrite105.0ms

Algorithm
rewrite-expression-head
Rules
11×add-exp-log
11×add-cbrt-cube
9×flip--
9×flip3--
9×add-sqr-sqrt
8×associate-/r/
8×frac-times
6×flip-+
6×add-cube-cbrt
6×associate-*r/
6×flip3-+
6×*-un-lft-identity
6×pow1
4×add-log-exp
4×associate-*l/
4×associate-*l*
3×associate-*r*
3×difference-of-sqr-1
2×associate-/l/
2×cbrt-unprod
2×prod-exp
2×div-exp
2×sub-neg
2×cbrt-undiv
1×associate-/l*
1×div-inv
1×*-commutative
1×pow-prod-down
1×times-frac
1×distribute-lft-in
1×frac-2neg
1×associate-/r*
1×clear-num
1×distribute-rgt-in
Counts
4 → 79
Calls
4 calls:
Slowest
41.0ms
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)))))
32.0ms
(* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))
19.0ms
(- (exp (* (+ a b) eps)) 1)
10.0ms
(- (exp (* a eps)) 1)

series2.6s

Counts
4 → 12
Calls
4 calls:
Slowest
2.2s
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)))))
252.0ms
(* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))
72.0ms
(- (exp (* (+ a b) eps)) 1)
49.0ms
(- (exp (* a eps)) 1)

simplify14.3s

Counts
54 → 91
Calls
54 calls:
Slowest
892.0ms
(* (+ (exp (* a eps)) 1) (+ (* (* eps b) (* eps b)) (- (* (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))) (* (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))))
778.0ms
(/ (* (* (* eps (- (exp (* (+ a b) eps)) 1)) (* eps (- (exp (* (+ a b) eps)) 1))) (* eps (- (exp (* (+ a b) eps)) 1))) (* (* (* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)))) (* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))) (* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))))
700.0ms
(- (+ (/ (exp (* (+ a b) eps)) (* (- (* 1/6 (exp (* a eps))) 1/6) (* (pow b 3) (pow eps 2)))) (+ (* 1/2 (/ (exp (* a eps)) (* (pow (- (* 1/6 (exp (* a eps))) 1/6) 2) (* (pow b 4) (pow eps 3))))) (* 1/2 (/ (exp (* (+ a b) eps)) (* (pow (- (* 1/6 (exp (* a eps))) 1/6) 2) (* (pow b 4) (pow eps 3))))))) (+ (* 1/2 (/ (* (exp (* (+ a b) eps)) (exp (* a eps))) (* (pow eps 3) (* (pow (- (* 1/6 (exp (* a eps))) 1/6) 2) (pow b 4))))) (+ (/ 1 (* (- (* 1/6 (exp (* a eps))) 1/6) (* (pow eps 2) (pow b 3)))) (* 1/2 (/ 1 (* (pow (- (* 1/6 (exp (* a eps))) 1/6) 2) (* (pow eps 3) (pow b 4))))))))
692.0ms
(- (+ (/ (exp (* (+ a b) eps)) (* (- (* 1/6 (exp (* a eps))) 1/6) (* (pow b 3) (pow eps 2)))) (+ (* 1/2 (/ (exp (* a eps)) (* (pow (- (* 1/6 (exp (* a eps))) 1/6) 2) (* (pow b 4) (pow eps 3))))) (* 1/2 (/ (exp (* (+ a b) eps)) (* (pow (- (* 1/6 (exp (* a eps))) 1/6) 2) (* (pow b 4) (pow eps 3))))))) (+ (* 1/2 (/ (* (exp (* (+ a b) eps)) (exp (* a eps))) (* (pow eps 3) (* (pow (- (* 1/6 (exp (* a eps))) 1/6) 2) (pow b 4))))) (+ (/ 1 (* (- (* 1/6 (exp (* a eps))) 1/6) (* (pow eps 2) (pow b 3)))) (* 1/2 (/ 1 (* (pow (- (* 1/6 (exp (* a eps))) 1/6) 2) (* (pow eps 3) (pow b 4))))))))
592.0ms
(* (+ (* (exp (* a eps)) (exp (* a eps))) (+ (* 1 1) (* (exp (* a eps)) 1))) (+ (* (* eps b) (* eps b)) (- (* (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))) (* (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))))

prune2.4s

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.0b

localize22.0ms

Local error

Found 1 expressions with local error:

0.0b
(+ (/ 1 a) (/ 1 b))

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
5×*-un-lft-identity
4×distribute-lft-out
4×div-inv
3×add-log-exp
1×flip-+
1×add-cube-cbrt
1×add-exp-log
1×frac-add
1×add-cbrt-cube
1×flip3-+
1×sum-log
1×+-commutative
1×pow1
1×add-sqr-sqrt
Counts
1 → 16
Calls
1 calls:
Slowest
8.0ms
(+ (/ 1 a) (/ 1 b))

series14.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
14.0ms
(+ (/ 1 a) (/ 1 b))

simplify53.0ms

Counts
10 → 19
Calls
10 calls:
Slowest
11.0ms
(* (exp (/ 1 a)) (exp (/ 1 b)))
5.0ms
(+ (/ 1 a) (/ 1 b))
5.0ms
(+ (/ 1 b) (/ 1 a))
5.0ms
(+ (/ 1 a) (/ 1 b))
5.0ms
(+ (/ 1 a) (/ 1 b))

prune278.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize21.0ms

Local error

Found 1 expressions with local error:

15.8b
(/ (+ b a) (* a b))

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
3×associate-/l*
3×add-cube-cbrt
3×add-exp-log
3×add-cbrt-cube
3×times-frac
3×*-un-lft-identity
3×add-sqr-sqrt
2×associate-/l/
1×add-log-exp
1×div-inv
1×flip-+
1×div-exp
1×flip3-+
1×frac-2neg
1×associate-/r*
1×clear-num
1×cbrt-undiv
1×pow1
Counts
1 → 21
Calls
1 calls:
Slowest
7.0ms
(/ (+ b a) (* a b))

series40.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
40.0ms
(/ (+ b a) (* a b))

simplify248.0ms

Counts
15 → 24
Calls
15 calls:
Slowest
126.0ms
(/ (* (* (+ b a) (+ b a)) (+ b a)) (* (* (* a b) (* a b)) (* a b)))
34.0ms
(- (log (+ b a)) (log (* a b)))
24.0ms
(- (* b b) (* a a))
13.0ms
(/ (* (cbrt (+ b a)) (cbrt (+ b a))) a)
7.0ms
(+ (/ 1 a) (/ 1 b))

prune297.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.0b

regimes282.0ms

Accuracy

-13.1% (3.6b remaining)

Error of 3.7b against oracle of 0.1b and baseline of 3.2b

bsearch757.0ms

end0.0ms

sample27.6s

Algorithm
intervals
Results
8.8s2595×body2560valid
7.7s3423×body1280valid
7.3s2006×body10240exit
1.7s1052×body640valid
819.0ms355×body320valid
647.0ms10009×pre80true
170.0ms433×body80valid
123.0ms145×body160valid