Average Error: 0.5 → 0.3
Time: 15.8s
Precision: binary64
Cost: 20416
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\cos th \cdot \left(\sqrt{a1 \cdot a1 + a2 \cdot a2} \cdot \sqrt{\frac{a1 \cdot a1 + a2 \cdot a2}{2}}\right)\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\cos th \cdot \left(\sqrt{a1 \cdot a1 + a2 \cdot a2} \cdot \sqrt{\frac{a1 \cdot a1 + a2 \cdot a2}{2}}\right)
(FPCore (a1 a2 th)
 :precision binary64
 (+
  (* (/ (cos th) (sqrt 2.0)) (* a1 a1))
  (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))
↓
(FPCore (a1 a2 th)
 :precision binary64
 (*
  (cos th)
  (* (sqrt (+ (* a1 a1) (* a2 a2))) (sqrt (/ (+ (* a1 a1) (* a2 a2)) 2.0)))))
double code(double a1, double a2, double th) {
	return ((cos(th) / sqrt(2.0)) * (a1 * a1)) + ((cos(th) / sqrt(2.0)) * (a2 * a2));
}
↓
double code(double a1, double a2, double th) {
	return cos(th) * (sqrt((a1 * a1) + (a2 * a2)) * sqrt(((a1 * a1) + (a2 * a2)) / 2.0));
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error1.2
Cost65984
\[\cos th \cdot \left(\frac{\sqrt[3]{a1 \cdot a1 + a2 \cdot a2} \cdot \sqrt[3]{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{\sqrt[3]{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}}}\right)\]
Alternative 2
Error0.5
Cost59072
\[\cos th \cdot \left(\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}}}\right)\]
Alternative 3
Error1.4
Cost53056
\[\cos th \cdot \left(\frac{\sqrt[3]{a1 \cdot a1 + a2 \cdot a2} \cdot \sqrt[3]{a1 \cdot a1 + a2 \cdot a2}}{\left|\sqrt[3]{2}\right|} \cdot \frac{\sqrt[3]{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{\sqrt[3]{2}}}\right)\]
Alternative 4
Error1.2
Cost53056
\[\cos th \cdot \left(\frac{\sqrt[3]{a1 \cdot a1 + a2 \cdot a2} \cdot \sqrt[3]{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{\sqrt{2}}} \cdot \frac{\sqrt[3]{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{\sqrt{2}}}\right)\]
Alternative 5
Error1.2
Cost53056
\[\frac{\sqrt[3]{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{\sqrt{2}}} \cdot \left(\cos th \cdot \frac{\sqrt[3]{a1 \cdot a1 + a2 \cdot a2} \cdot \sqrt[3]{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{\sqrt{2}}}\right)\]
Alternative 6
Error17.3
Cost52672
\[\left(\sqrt{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \cdot \sqrt{\cos th}\right) \cdot \left(\sqrt{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \cdot \sqrt{\cos th}\right)\]
Alternative 7
Error1.2
Cost46784
\[\cos th \cdot \left(\sqrt[3]{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \cdot \left(\sqrt[3]{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \cdot \sqrt[3]{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}}\right)\right)\]
Alternative 8
Error1.1
Cost46144
\[\cos th \cdot \left(\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\left|\sqrt[3]{2}\right|} \cdot \frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{\sqrt[3]{2}}}\right)\]
Alternative 9
Error0.6
Cost46144
\[\cos th \cdot \left(\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{\sqrt{2}}} \cdot \frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{\sqrt{2}}}\right)\]
Alternative 10
Error0.6
Cost45888
\[\cos th \cdot \left(\frac{1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt[3]{\sqrt{2}}}\right)\]
Alternative 11
Error0.5
Cost45760
\[\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt[3]{\sqrt{2}}} \cdot \frac{\cos th}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}\]
Alternative 12
Error0.8
Cost45760
\[\left(\sqrt[3]{\cos th} \cdot \sqrt[3]{\cos th}\right) \cdot \left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}} \cdot \sqrt[3]{\cos th}\right)\]
Alternative 13
Error0.4
Cost45760
\[\cos th \cdot \frac{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt{2}}}\]
Alternative 14
Error44.6
Cost40128
\[\cos th \cdot \frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\sqrt{2} \cdot \left({a1}^{4} + \left({a2}^{4} - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)\right)}\]
Alternative 15
Error1.3
Cost33728
\[\cos th \cdot \left(\left(\sqrt[3]{a1 \cdot a1 + a2 \cdot a2} \cdot \sqrt[3]{a1 \cdot a1 + a2 \cdot a2}\right) \cdot \frac{\sqrt[3]{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\]
Alternative 16
Error1.2
Cost33728
\[\frac{\sqrt[3]{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}} \cdot \left(\cos th \cdot \left(\sqrt[3]{a1 \cdot a1 + a2 \cdot a2} \cdot \sqrt[3]{a1 \cdot a1 + a2 \cdot a2}\right)\right)\]
Alternative 17
Error1.2
Cost33728
\[\cos th \cdot \frac{\sqrt[3]{a1 \cdot a1 + a2 \cdot a2} \cdot \sqrt[3]{a1 \cdot a1 + a2 \cdot a2}}{\frac{\sqrt{2}}{\sqrt[3]{a1 \cdot a1 + a2 \cdot a2}}}\]
Alternative 18
Error0.6
Cost33344
\[\cos th \cdot \left(\sqrt{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \cdot \sqrt{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}}\right)\]
Alternative 19
Error0.6
Cost33344
\[\sqrt{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \cdot \left(\cos th \cdot \sqrt{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}}\right)\]
Alternative 20
Error0.7
Cost32960
\[\cos th \cdot \left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{\sqrt{2}}} \cdot \frac{1}{\sqrt{\sqrt{2}}}\right)\]
Alternative 21
Error0.6
Cost32832
\[\frac{\cos th}{\sqrt{\sqrt{2}}} \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{\sqrt{2}}}\]
Alternative 22
Error17.2
Cost32832
\[\sqrt{\cos th} \cdot \left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}} \cdot \sqrt{\cos th}\right)\]
Alternative 23
Error0.5
Cost32832
\[\cos th \cdot \frac{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}}\]
Alternative 24
Error0.5
Cost26816
\[\cos th \cdot \left(\sqrt{a1 \cdot a1 + a2 \cdot a2} \cdot \frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\]
Alternative 25
Error0.5
Cost26816
\[\cos th \cdot \left(\sqrt{a1 \cdot a1 + a2 \cdot a2} \cdot \left(\sqrt{a1 \cdot a1 + a2 \cdot a2} \cdot \sqrt{0.5}\right)\right)\]
Alternative 26
Error0.5
Cost26816
\[\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}} \cdot \left(\cos th \cdot \sqrt{a1 \cdot a1 + a2 \cdot a2}\right)\]
Alternative 27
Error0.5
Cost26816
\[\cos th \cdot \frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\frac{\sqrt{2}}{\sqrt{a1 \cdot a1 + a2 \cdot a2}}}\]
Alternative 28
Error34.3
Cost26688
\[\cos th \cdot \frac{\frac{{a1}^{4} - {a2}^{4}}{a1 \cdot a1 - a2 \cdot a2}}{\sqrt{2}}\]
Alternative 29
Error0.5
Cost26560
\[\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}}\]
Alternative 30
Error37.5
Cost26368
\[\cos th \cdot \sqrt[3]{{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}^{3}}\]
Alternative 31
Error28.0
Cost26368
\[\cos th \cdot \sqrt{\frac{{\left(\sqrt{a1 \cdot a1 + a2 \cdot a2}\right)}^{4}}{2}}\]
Alternative 32
Error37.5
Cost26368
\[\cos th \cdot \frac{\sqrt[3]{{\left(a1 \cdot a1 + a2 \cdot a2\right)}^{3}}}{\sqrt{2}}\]
Alternative 33
Error37.5
Cost26368
\[\sqrt[3]{{\left(\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}^{3}}\]
Alternative 34
Error0.3
Cost20416
\[\sqrt{\frac{a1 \cdot a1 + a2 \cdot a2}{2}} \cdot \left(\cos th \cdot \sqrt{a1 \cdot a1 + a2 \cdot a2}\right)\]
Alternative 35
Error41.7
Cost20096
\[\cos th \cdot \left(\sqrt{a1 \cdot a1 + a2 \cdot a2} \cdot \frac{-a2}{\sqrt{2}}\right)\]
Alternative 36
Error42.6
Cost20096
\[\cos th \cdot \left(\sqrt{a1 \cdot a1 + a2 \cdot a2} \cdot \frac{-a1}{\sqrt{2}}\right)\]
Alternative 37
Error41.7
Cost20096
\[\cos th \cdot \left(\sqrt{a1 \cdot a1 + a2 \cdot a2} \cdot \left(\left(-a2\right) \cdot \sqrt{0.5}\right)\right)\]
Alternative 38
Error0.5
Cost20032
\[\cos th \cdot \left(a1 \cdot \frac{a1}{\sqrt{2}} + a2 \cdot \frac{a2}{\sqrt{2}}\right)\]
Alternative 39
Error42.0
Cost20032
\[\cos th \cdot \left(\sqrt{a1 \cdot a1 + a2 \cdot a2} \cdot \frac{a1}{\sqrt{2}}\right)\]
Alternative 40
Error41.7
Cost20032
\[\cos th \cdot \left(\sqrt{a1 \cdot a1 + a2 \cdot a2} \cdot \frac{a2}{\sqrt{2}}\right)\]
Alternative 41
Error42.0
Cost20032
\[\cos th \cdot \left(\sqrt{a1 \cdot a1 + a2 \cdot a2} \cdot \left(a1 \cdot \sqrt{0.5}\right)\right)\]
Alternative 42
Error41.7
Cost20032
\[\cos th \cdot \left(\sqrt{a1 \cdot a1 + a2 \cdot a2} \cdot \left(a2 \cdot \sqrt{0.5}\right)\right)\]
Alternative 43
Error0.5
Cost20032
\[\cos th \cdot \left(\frac{a1 \cdot a1}{\sqrt{2}} + \frac{a2 \cdot a2}{\sqrt{2}}\right)\]
Alternative 44
Error34.2
Cost14528
\[\cos th \cdot \frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) - \left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right)}{\sqrt{2} \cdot \left(a1 \cdot a1 - a2 \cdot a2\right)}\]
Alternative 45
Error42.2
Cost14144
\[\cos th \cdot \left(\sqrt{\frac{a1 \cdot a1 + a2 \cdot a2}{2}} \cdot \left(\frac{a1 \cdot a1}{a2} \cdot -0.5 - a2\right)\right)\]
Alternative 46
Error41.7
Cost13696
\[\cos th \cdot \left(\left(-a2\right) \cdot \sqrt{\frac{a1 \cdot a1 + a2 \cdot a2}{2}}\right)\]
Alternative 47
Error0.5
Cost13632
\[\left(\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right) \cdot \frac{1}{\sqrt{2}}\]
Alternative 48
Error0.5
Cost13632
\[\cos th \cdot \left(\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \frac{1}{\sqrt{2}}\right)\]
Alternative 49
Error0.7
Cost13632
\[\cos th \cdot \frac{1}{\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}\]
Alternative 50
Error0.5
Cost13504
\[\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\]
Alternative 51
Error0.5
Cost13504
\[\sqrt{0.5} \cdot \left(\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\]
Alternative 52
Error0.5
Cost13504
\[\cos th \cdot \left(\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\]
Alternative 53
Error0.5
Cost13504
\[\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}\]
Alternative 54
Error28.6
Cost13248
\[\cos th \cdot \left(a1 \cdot \frac{a1}{\sqrt{2}}\right)\]
Alternative 55
Error27.4
Cost13248
\[\cos th \cdot \left(a2 \cdot \frac{a2}{\sqrt{2}}\right)\]
Alternative 56
Error28.6
Cost13248
\[\sqrt{0.5} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right)\right)\]
Alternative 57
Error27.4
Cost13248
\[\sqrt{0.5} \cdot \left(\cos th \cdot \left(a2 \cdot a2\right)\right)\]
Alternative 58
Error28.6
Cost13248
\[\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\]
Alternative 59
Error27.4
Cost13248
\[\cos th \cdot \left(a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\right)\]
Alternative 60
Error28.6
Cost13248
\[\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\]
Alternative 61
Error27.4
Cost13248
\[\cos th \cdot \frac{a2 \cdot a2}{\sqrt{2}}\]
Alternative 62
Error26.6
Cost6976
\[\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\]
Alternative 63
Error26.6
Cost6976
\[\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\]
Alternative 64
Error61.3
Cost64
\[1\]
Alternative 65
Error55.6
Cost64
\[0\]
Alternative 66
Error62.4
Cost64
\[-1\]

Error

Derivation

  1. Initial program 0.5

    \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]

  2. Simplified0.5

    \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity_binary64_7600.5

    \[\leadsto \cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{\color{blue}{1 \cdot 2}}}\]

  5. Applied sqrt-prod_binary64_7760.5

    \[\leadsto \cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\color{blue}{\sqrt{1} \cdot \sqrt{2}}}\]

  6. Applied add-sqr-sqrt_binary64_7820.5

    \[\leadsto \cos th \cdot \frac{\color{blue}{\sqrt{a1 \cdot a1 + a2 \cdot a2} \cdot \sqrt{a1 \cdot a1 + a2 \cdot a2}}}{\sqrt{1} \cdot \sqrt{2}}\]

  7. Applied times-frac_binary64_7660.5

    \[\leadsto \cos th \cdot \color{blue}{\left(\frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{1}} \cdot \frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)}\]

  8. Simplified0.5

    \[\leadsto \cos th \cdot \left(\color{blue}{\sqrt{a1 \cdot a1 + a2 \cdot a2}} \cdot \frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\]
  9. Using strategy rm

  10. Applied sqrt-undiv_binary64_7810.3

    \[\leadsto \cos th \cdot \left(\sqrt{a1 \cdot a1 + a2 \cdot a2} \cdot \color{blue}{\sqrt{\frac{a1 \cdot a1 + a2 \cdot a2}{2}}}\right)\]

  11. Simplified0.3

    \[\leadsto \color{blue}{\cos th \cdot \left(\sqrt{a1 \cdot a1 + a2 \cdot a2} \cdot \sqrt{\frac{a1 \cdot a1 + a2 \cdot a2}{2}}\right)}\]

  12. Final simplification0.3

    \[\leadsto \cos th \cdot \left(\sqrt{a1 \cdot a1 + a2 \cdot a2} \cdot \sqrt{\frac{a1 \cdot a1 + a2 \cdot a2}{2}}\right)\]

Reproduce

herbie shell --seed 2021022 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2.0)) (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))