Average Error: 14.1 → 0.2
Time: 9.3s
Precision: binary64
Cost: 704
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{0.5 \cdot \frac{\pi}{a + b}}{a \cdot b}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{0.5 \cdot \frac{\pi}{a + b}}{a \cdot b}
(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))
↓
(FPCore (a b) :precision binary64 (/ (* 0.5 (/ PI (+ a b))) (* a b)))
double code(double a, double b) {
	return ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}
↓
double code(double a, double b) {
	return (0.5 * (((double) M_PI) / (a + b))) / (a * b);
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error46.2
Cost28096
\[\left(\sqrt{\frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}} \cdot \sqrt{\frac{1}{a} - \frac{1}{b}}\right) \cdot \left(\sqrt{\frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}} \cdot \sqrt{\frac{1}{a} - \frac{1}{b}}\right)\]
Alternative 2
Error14.5
Cost22976
\[\sqrt[3]{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}} \cdot \left(\sqrt[3]{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}} \cdot \sqrt[3]{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}}\right)\]
Alternative 3
Error0.8
Cost22208
\[\frac{\pi}{a + b} \cdot \left(\sqrt[3]{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{0.5}{b - a}} \cdot \left(\sqrt[3]{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{0.5}{b - a}} \cdot \sqrt[3]{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{0.5}{b - a}}\right)\right)\]
Alternative 4
Error0.9
Cost21184
\[\left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{0.5}{b - a}\right) \cdot \left(\sqrt[3]{\frac{\pi}{a + b}} \cdot \left(\sqrt[3]{\frac{\pi}{a + b}} \cdot \sqrt[3]{\frac{\pi}{a + b}}\right)\right)\]
Alternative 5
Error14.3
Cost21120
\[\frac{\frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}}{a} - \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot {\left(\frac{\sqrt[3]{1}}{\sqrt[3]{b}}\right)}^{3}\]
Alternative 6
Error1.3
Cost20928
\[\frac{\sqrt[3]{\frac{\pi}{2}} \cdot \sqrt[3]{\frac{\pi}{2}}}{a + b} \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\sqrt[3]{\frac{\pi}{2}}}{b - a}\right)\]
Alternative 7
Error14.3
Cost20480
\[\frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - {\left(\frac{\sqrt[3]{1}}{\sqrt[3]{b}}\right)}^{3}\right)\]
Alternative 8
Error37.8
Cost15296
\[\frac{\frac{\pi}{2} \cdot \left({\left(\frac{1}{a}\right)}^{3} - {\left(\frac{1}{b}\right)}^{3}\right)}{\left(b \cdot b - a \cdot a\right) \cdot \left(\frac{\frac{1}{a}}{a} + \left(\frac{\frac{1}{b}}{b} + \frac{\frac{1}{a}}{b}\right)\right)}\]
Alternative 9
Error28.3
Cost15296
\[\sqrt{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}} \cdot \sqrt{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}}\]
Alternative 10
Error27.7
Cost14912
\[\frac{\pi}{a + b} \cdot \left(\sqrt{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{0.5}{b - a}} \cdot \sqrt{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{0.5}{b - a}}\right)\]
Alternative 11
Error29.2
Cost14784
\[\sqrt{\frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}} \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \sqrt{\frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}}\right)\]
Alternative 12
Error39.2
Cost14656
\[\frac{1}{\sqrt{b \cdot b - a \cdot a}} \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\frac{\pi}{2}}{\sqrt{b \cdot b - a \cdot a}}\right)\]
Alternative 13
Error38.9
Cost14656
\[\frac{\frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}}{a} - \frac{\frac{\frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}}{\sqrt{b}}}{\sqrt{b}}\]
Alternative 14
Error32.3
Cost14400
\[\frac{\pi}{a + b} \cdot \left(\sqrt{\frac{0.5}{b - a}} \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \sqrt{\frac{0.5}{b - a}}\right)\right)\]
Alternative 15
Error0.5
Cost14272
\[\frac{\frac{1}{\sqrt{2}}}{a + b} \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\frac{\pi}{\sqrt{2}}}{b - a}\right)\]
Alternative 16
Error0.7
Cost14272
\[\frac{\sqrt{\frac{\pi}{2}}}{a + b} \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\sqrt{\frac{\pi}{2}}}{b - a}\right)\]
Alternative 17
Error32.3
Cost14272
\[\frac{\pi}{a + b} \cdot \left(\frac{\frac{1}{a} - \frac{1}{b}}{\sqrt{b - a}} \cdot \frac{0.5}{\sqrt{b - a}}\right)\]
Alternative 18
Error0.6
Cost14144
\[\frac{\sqrt{\pi}}{a + b} \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\frac{\sqrt{\pi}}{2}}{b - a}\right)\]
Alternative 19
Error0.6
Cost14144
\[\left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{0.5}{b - a}\right) \cdot \left(\sqrt{\pi} \cdot \frac{\sqrt{\pi}}{a + b}\right)\]
Alternative 20
Error0.4
Cost14144
\[\frac{\pi}{a + b} \cdot \left(\left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \sqrt{0.5}\right) \cdot \frac{\sqrt{0.5}}{b - a}\right)\]
Alternative 21
Error30.3
Cost14080
\[\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \sqrt[3]{\frac{{\left(\frac{\pi}{b \cdot b - a \cdot a}\right)}^{3}}{8}}\]
Alternative 22
Error30.4
Cost14080
\[\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\frac{\pi}{2}}{\sqrt[3]{{\left(b \cdot b - a \cdot a\right)}^{3}}}\]
Alternative 23
Error30.9
Cost14080
\[\sqrt[3]{{\left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}\right)}^{3}}\]
Alternative 24
Error17.6
Cost14080
\[\frac{\pi}{a + b} \cdot \sqrt[3]{\frac{0.125}{{\left(\frac{b - a}{\frac{1}{a} - \frac{1}{b}}\right)}^{3}}}\]
Alternative 25
Error30.9
Cost1984
\[\frac{\frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{\frac{1}{a}}{a} - \frac{\frac{1}{b}}{b}\right)}{\frac{1}{a} + \frac{1}{b}}\]
Alternative 26
Error30.7
Cost1984
\[\frac{\frac{\pi}{2} \cdot \left(\frac{\frac{1}{a}}{a} - \frac{\frac{1}{b}}{b}\right)}{\left(b \cdot b - a \cdot a\right) \cdot \left(\frac{1}{a} + \frac{1}{b}\right)}\]
Alternative 27
Error14.1
Cost1728
\[\frac{\frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}}{a} - \frac{\frac{\frac{\pi}{2}}{b}}{b \cdot b - a \cdot a}\]
Alternative 28
Error14.1
Cost1344
\[\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}\]
Alternative 29
Error0.3
Cost1344
\[\frac{1}{a + b} \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b - a}\right)\]
Alternative 30
Error14.1
Cost1344
\[\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right)\]
Alternative 31
Error0.4
Cost1344
\[\left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{0.5}{b - a}\right) \cdot \left(\pi \cdot \frac{1}{a + b}\right)\]
Alternative 32
Error0.4
Cost1344
\[\frac{1}{\frac{a + b}{\pi}} \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{0.5}{b - a}\right)\]
Alternative 33
Error14.1
Cost1216
\[\frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \frac{b - a}{a \cdot b}\]
Alternative 34
Error14.1
Cost1216
\[\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \left(\pi \cdot \frac{0.5}{b \cdot b - a \cdot a}\right)\]
Alternative 35
Error14.1
Cost1216
\[\frac{\pi}{2} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b \cdot b - a \cdot a}\]
Alternative 36
Error0.3
Cost1216
\[\frac{\pi}{a + b} \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{0.5}{b - a}\right)\]
Alternative 37
Error14.1
Cost1216
\[\frac{\left(b - a\right) \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}}{a \cdot b}\]
Alternative 38
Error14.1
Cost1216
\[\frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\pi}{2}}{b \cdot b - a \cdot a}\]
Alternative 39
Error18.8
Cost1216
\[\frac{\left(b - a\right) \cdot \frac{\pi}{2}}{\left(a \cdot b\right) \cdot \left(b \cdot b - a \cdot a\right)}\]
Alternative 40
Error0.3
Cost1216
\[\frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \left(0.5 \cdot \frac{\pi}{a + b}\right)}{b - a}\]
Alternative 41
Error0.3
Cost1216
\[\frac{\pi \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{0.5}{b - a}\right)}{a + b}\]
Alternative 42
Error0.3
Cost1216
\[\frac{\pi}{a + b} \cdot \frac{0.5 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}\]
Alternative 43
Error0.9
Cost1216
\[\frac{\pi}{a + b} \cdot \frac{\left(b - a\right) \cdot 0.5}{\left(a \cdot b\right) \cdot \left(b - a\right)}\]
Alternative 44
Error14.1
Cost1216
\[\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}\]
Alternative 45
Error25.1
Cost1088
\[\frac{\pi}{b} \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{0.5}{b - a}\right)\]
Alternative 46
Error25.1
Cost1088
\[\left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{0.5}{b - a}\right) \cdot \frac{\pi}{a}\]
Alternative 47
Error31.8
Cost1024
\[\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\frac{\pi}{2}}{-a \cdot a}\]
Alternative 48
Error18.7
Cost960
\[\frac{\pi}{a + b} \cdot \left(\frac{-1}{b} \cdot \frac{0.5}{b - a}\right)\]
Alternative 49
Error18.6
Cost960
\[\frac{\pi}{a + b} \cdot \left(\frac{1}{a} \cdot \frac{0.5}{b - a}\right)\]
Alternative 50
Error31.8
Cost960
\[\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \left(-0.5 \cdot \frac{\pi}{a \cdot a}\right)\]
Alternative 51
Error32.1
Cost960
\[\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b \cdot b}\]
Alternative 52
Error0.3
Cost832
\[\frac{0.5}{a \cdot b} \cdot \frac{1}{\frac{a + b}{\pi}}\]
Alternative 53
Error0.3
Cost832
\[\frac{\pi}{a + b} \cdot \left(\frac{1}{a} \cdot \frac{0.5}{b}\right)\]
Alternative 54
Error32.1
Cost832
\[\frac{\pi}{b \cdot b} \cdot \left(\frac{0.5}{a} - \frac{0.5}{b}\right)\]
Alternative 55
Error31.8
Cost832
\[\frac{\pi}{a \cdot a} \cdot \left(\frac{0.5}{b} - \frac{0.5}{a}\right)\]
Alternative 56
Error19.0
Cost704
\[\frac{\frac{\pi}{a \cdot b} \cdot -0.5}{b - a}\]
Alternative 57
Error4.3
Cost704
\[\frac{\frac{\pi}{a + b}}{a} \cdot \frac{0.5}{b}\]
Alternative 58
Error0.8
Cost704
\[\pi \cdot \frac{0.5}{\left(a \cdot b\right) \cdot \left(a + b\right)}\]
Alternative 59
Error0.3
Cost704
\[\frac{\pi \cdot \frac{0.5}{a \cdot b}}{a + b}\]
Alternative 60
Error0.3
Cost704
\[\frac{\pi}{a + b} \cdot \frac{\frac{0.5}{a}}{b}\]
Alternative 61
Error0.8
Cost704
\[\frac{\pi \cdot 0.5}{\left(a \cdot b\right) \cdot \left(a + b\right)}\]
Alternative 62
Error0.3
Cost704
\[\frac{0.5}{a \cdot b} \cdot \frac{\pi}{a + b}\]
Alternative 63
Error25.1
Cost576
\[\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\]
Alternative 64
Error25.2
Cost576
\[0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\]
Alternative 65
Error29.3
Cost576
\[0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)}\]
Alternative 66
Error29.6
Cost576
\[0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\]
Alternative 67
Error25.2
Cost576
\[0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\]
Alternative 68
Error25.0
Cost576
\[\frac{0.5}{a \cdot b} \cdot \frac{\pi}{a}\]
Alternative 69
Error61.8
Cost64
\[1\]
Alternative 70
Error36.6
Cost64
\[0\]
Alternative 71
Error61.8
Cost64
\[-1\]

Error

Derivation

  1. Initial program 14.1

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  2. Simplified14.1

    \[\leadsto \color{blue}{\frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}\]
  3. Using strategy rm

  4. Applied difference-of-squares_binary64_7069.1

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

  5. Applied div-inv_binary64_7349.1

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

  6. Applied times-frac_binary64_7438.7

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

  7. Applied associate-*l*_binary64_6780.3

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

  8. Simplified0.3

    \[\leadsto \frac{\pi}{b + a} \cdot \color{blue}{\left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{0.5}{b - a}\right)}\]
  9. Using strategy rm

  10. Applied frac-sub_binary64_7460.4

    \[\leadsto \frac{\pi}{b + a} \cdot \left(\color{blue}{\frac{1 \cdot b - a \cdot 1}{a \cdot b}} \cdot \frac{0.5}{b - a}\right)\]

  11. Applied associate-*l/_binary64_6800.3

    \[\leadsto \frac{\pi}{b + a} \cdot \color{blue}{\frac{\left(1 \cdot b - a \cdot 1\right) \cdot \frac{0.5}{b - a}}{a \cdot b}}\]

  12. Applied associate-*r/_binary64_6790.3

    \[\leadsto \color{blue}{\frac{\frac{\pi}{b + a} \cdot \left(\left(1 \cdot b - a \cdot 1\right) \cdot \frac{0.5}{b - a}\right)}{a \cdot b}}\]

  13. Simplified0.2

    \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{\pi}{a + b}}}{a \cdot b}\]

  14. Simplified0.2

    \[\leadsto \color{blue}{\frac{0.5 \cdot \frac{\pi}{a + b}}{a \cdot b}}\]

  15. Final simplification0.2

    \[\leadsto \frac{0.5 \cdot \frac{\pi}{a + b}}{a \cdot b}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))