Average Error: 15.0 → 0.3
Time: 13.5s
Precision: binary64
Cost: 32704
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a}
(FPCore (r a b) :precision binary64 (* r (/ (sin b) (cos (+ a b)))))
↓
(FPCore (r a b)
 :precision binary64
 (/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin b) (sin a)))))
double code(double r, double a, double b) {
	return r * (sin(b) / cos(a + b));
}
↓
double code(double r, double a, double b) {
	return (r * sin(b)) / ((cos(a) * cos(b)) - (sin(b) * sin(a)));
}

Error

Bits error versus r

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
Error1.4
Cost136384
\[\left(r \cdot \frac{\sqrt[3]{\sin b} \cdot \sqrt[3]{\sin b}}{\sqrt[3]{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot \sqrt[3]{\cos a \cdot \cos b - \sin b \cdot \sin a}}\right) \cdot \frac{\sqrt[3]{\sin b}}{\sqrt[3]{\cos a \cdot \cos b - \sin b \cdot \sin a}}\]
Alternative 2
Error1.2
Cost117184
\[\sqrt[3]{\frac{\sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a}} \cdot \left(r \cdot \left(\sqrt[3]{\frac{\sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a}} \cdot \sqrt[3]{\frac{\sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a}}\right)\right)\]
Alternative 3
Error24.3
Cost103872
\[r \cdot \left(\frac{\sqrt[3]{\sin b} \cdot \sqrt[3]{\sin b}}{\sqrt{\cos a \cdot \cos b - \sin b \cdot \sin a}} \cdot \frac{\sqrt[3]{\sin b}}{\sqrt{\cos a \cdot \cos b - \sin b \cdot \sin a}}\right)\]
Alternative 4
Error1.3
Cost97856
\[\sqrt[3]{\frac{r}{\frac{\cos a}{\frac{\sin b}{\cos b}} - \sin a}} \cdot \left(\sqrt[3]{\frac{r}{\frac{\cos a}{\frac{\sin b}{\cos b}} - \sin a}} \cdot \sqrt[3]{\frac{r}{\frac{\cos a}{\frac{\sin b}{\cos b}} - \sin a}}\right)\]
Alternative 5
Error47.8
Cost97344
\[\frac{\sqrt[3]{r} \cdot \sqrt[3]{r}}{\sqrt{\frac{\cos a \cdot \cos b}{\sin b}} + \sqrt{\sin a}} \cdot \frac{\sqrt[3]{r}}{\sqrt{\frac{\cos a \cdot \cos b}{\sin b}} - \sqrt{\sin a}}\]
Alternative 6
Error0.5
Cost84672
\[r \cdot \left(\frac{\sin b}{{\cos a}^{2} \cdot {\cos b}^{2} - {\sin b}^{2} \cdot {\sin a}^{2}} \cdot \left(\cos a \cdot \cos b + \sin b \cdot \sin a\right)\right)\]
Alternative 7
Error0.5
Cost84672
\[\left(\cos a \cdot \cos b + \sin b \cdot \sin a\right) \cdot \left(r \cdot \frac{\sin b}{{\cos a}^{2} \cdot {\cos b}^{2} - {\sin b}^{2} \cdot {\sin a}^{2}}\right)\]
Alternative 8
Error1.1
Cost84544
\[\frac{r}{\sqrt[3]{\frac{\cos a \cdot \cos b}{\sin b}} \cdot \left(\sqrt[3]{\frac{\cos a \cdot \cos b}{\sin b}} \cdot \sqrt[3]{\frac{\cos a \cdot \cos b}{\sin b}}\right) - \sin a}\]
Alternative 9
Error31.4
Cost78144
\[r \cdot \left(\sqrt{\frac{\sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a}} \cdot \sqrt{\frac{\sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a}}\right)\]
Alternative 10
Error47.6
Cost78016
\[\frac{1}{\sqrt{\frac{\cos a \cdot \cos b}{\sin b}} + \sqrt{\sin a}} \cdot \frac{r}{\sqrt{\frac{\cos a \cdot \cos b}{\sin b}} - \sqrt{\sin a}}\]
Alternative 11
Error15.8
Cost78016
\[r \cdot \left(\frac{\sqrt[3]{\sin b} \cdot \sqrt[3]{\sin b}}{\sqrt[3]{\cos \left(a + b\right)} \cdot \sqrt[3]{\cos \left(a + b\right)}} \cdot \frac{\sqrt[3]{\sin b}}{\sqrt[3]{\cos \left(a + b\right)}}\right)\]
Alternative 12
Error47.6
Cost77888
\[\frac{\frac{r}{\sqrt{\frac{\cos a \cdot \cos b}{\sin b}} + \sqrt{\sin a}}}{\sqrt{\frac{\cos a \cdot \cos b}{\sin b}} - \sqrt{\sin a}}\]
Alternative 13
Error23.8
Cost71616
\[\frac{r}{\sqrt{\cos a \cdot \cos b - \sin b \cdot \sin a}} \cdot \frac{\sin b}{\sqrt{\cos a \cdot \cos b - \sin b \cdot \sin a}}\]
Alternative 14
Error29.0
Cost65216
\[\sqrt{\frac{r}{\frac{\cos a}{\frac{\sin b}{\cos b}} - \sin a}} \cdot \sqrt{\frac{r}{\frac{\cos a}{\frac{\sin b}{\cos b}} - \sin a}}\]
Alternative 15
Error1.3
Cost65216
\[\frac{\sqrt[3]{r \cdot \sin b} \cdot \left(\sqrt[3]{r \cdot \sin b} \cdot \sqrt[3]{r \cdot \sin b}\right)}{\cos a \cdot \cos b - \sin b \cdot \sin a}\]
Alternative 16
Error1.2
Cost64960
\[\left(r \cdot \left(\sqrt[3]{\sin b} \cdot \sqrt[3]{\sin b}\right)\right) \cdot \frac{\sqrt[3]{\sin b}}{\cos a \cdot \cos b - \sin b \cdot \sin a}\]
Alternative 17
Error15.7
Cost59072
\[\sqrt[3]{r \cdot \frac{\sin b}{\cos \left(a + b\right)}} \cdot \left(\sqrt[3]{r \cdot \frac{\sin b}{\cos \left(a + b\right)}} \cdot \sqrt[3]{r \cdot \frac{\sin b}{\cos \left(a + b\right)}}\right)\]
Alternative 18
Error15.7
Cost58816
\[r \cdot \left(\sqrt[3]{\frac{\sin b}{\cos \left(a + b\right)}} \cdot \left(\sqrt[3]{\frac{\sin b}{\cos \left(a + b\right)}} \cdot \sqrt[3]{\frac{\sin b}{\cos \left(a + b\right)}}\right)\right)\]
Alternative 19
Error1.2
Cost58432
\[\frac{r}{\frac{\cos a}{\sqrt[3]{\sin b} \cdot \sqrt[3]{\sin b}} \cdot \frac{\cos b}{\sqrt[3]{\sin b}} - \sin a}\]
Alternative 20
Error1.2
Cost52160
\[\frac{\left(\sqrt[3]{r} \cdot \sqrt[3]{r}\right) \cdot \left(\sin b \cdot \sqrt[3]{r}\right)}{\cos a \cdot \cos b - \sin b \cdot \sin a}\]
Alternative 21
Error51.6
Cost52160
\[\left(\sqrt{\frac{\sin b}{\cos \left(a + b\right)}} \cdot \sqrt{r}\right) \cdot \left(\sqrt{\frac{\sin b}{\cos \left(a + b\right)}} \cdot \sqrt{r}\right)\]
Alternative 22
Error47.3
Cost52032
\[r \cdot \left(\frac{\sqrt{\sin b}}{\sqrt{\cos \left(a + b\right)}} \cdot \frac{\sqrt{\sin b}}{\sqrt{\cos \left(a + b\right)}}\right)\]
Alternative 23
Error15.4
Cost45888
\[r \cdot \left(\frac{1}{\sqrt[3]{\cos \left(a + b\right)} \cdot \sqrt[3]{\cos \left(a + b\right)}} \cdot \frac{\sin b}{\sqrt[3]{\cos \left(a + b\right)}}\right)\]
Alternative 24
Error15.4
Cost45760
\[r \cdot \frac{\sin b}{\sqrt[3]{\cos \left(a + b\right)} \cdot \left(\sqrt[3]{\cos \left(a + b\right)} \cdot \sqrt[3]{\cos \left(a + b\right)}\right)}\]
Alternative 25
Error1.2
Cost45632
\[\left(\sqrt[3]{r} \cdot \sqrt[3]{r}\right) \cdot \frac{\sqrt[3]{r}}{\frac{\cos a}{\frac{\sin b}{\cos b}} - \sin a}\]
Alternative 26
Error35.1
Cost45568
\[\frac{\sqrt[3]{{\left(r \cdot \sin b\right)}^{3}}}{\cos a \cdot \cos b - \sin b \cdot \sin a}\]
Alternative 27
Error0.4
Cost45504
\[r \cdot \frac{\sin b}{\cos a \cdot \cos b - \log \left(e^{\sin b \cdot \sin a}\right)}\]
Alternative 28
Error22.6
Cost45504
\[r \cdot \log \left(e^{\frac{\sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a}}\right)\]
Alternative 29
Error15.7
Cost45504
\[r \cdot \left(\left(\sqrt[3]{\sin b} \cdot \sqrt[3]{\sin b}\right) \cdot \frac{\sqrt[3]{\sin b}}{\cos \left(a + b\right)}\right)\]
Alternative 30
Error15.7
Cost45504
\[\left(r \cdot \left(\sqrt[3]{\sin b} \cdot \sqrt[3]{\sin b}\right)\right) \cdot \frac{\sqrt[3]{\sin b}}{\cos \left(a + b\right)}\]
Alternative 31
Error27.8
Cost39360
\[r \cdot \frac{\sin b}{\cos b} + r \cdot \left({\sin b}^{2} \cdot \frac{a}{{\cos b}^{2}}\right)\]
Alternative 32
Error36.1
Cost39360
\[\sqrt{r \cdot \frac{\sin b}{\cos \left(a + b\right)}} \cdot \sqrt{r \cdot \frac{\sin b}{\cos \left(a + b\right)}}\]
Alternative 33
Error38.7
Cost39232
\[r \cdot \left(\sqrt{\frac{\sin b}{\cos \left(a + b\right)}} \cdot \sqrt{\frac{\sin b}{\cos \left(a + b\right)}}\right)\]
Alternative 34
Error13.9
Cost39040
\[\frac{r}{\sqrt[3]{{\left(\frac{\cos a \cdot \cos b}{\sin b}\right)}^{3}} - \sin a}\]
Alternative 35
Error35.1
Cost39040
\[\sqrt[3]{{\left(\frac{r}{\frac{\cos a}{\frac{\sin b}{\cos b}} - \sin a}\right)}^{3}}\]
Alternative 36
Error31.2
Cost38976
\[e^{\log \left(\frac{r}{\frac{\cos a}{\frac{\sin b}{\cos b}} - \sin a}\right)}\]
Alternative 37
Error32.3
Cost38976
\[\frac{r}{e^{\log \left(\frac{\cos a \cdot \cos b}{\sin b} - \sin a\right)}}\]
Alternative 38
Error0.4
Cost32832
\[r \cdot \left(\sin b \cdot \frac{1}{\cos a \cdot \cos b - \sin b \cdot \sin a}\right)\]
Alternative 39
Error0.4
Cost32832
\[\frac{1}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot \left(r \cdot \sin b\right)\]
Alternative 40
Error31.4
Cost32832
\[r \cdot \left(\frac{1}{\sqrt{\cos \left(a + b\right)}} \cdot \frac{\sin b}{\sqrt{\cos \left(a + b\right)}}\right)\]
Alternative 41
Error0.3
Cost32704
\[r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a}\]
Alternative 42
Error39.5
Cost32576
\[\left(r \cdot \sqrt{\sin b}\right) \cdot \frac{\sqrt{\sin b}}{\cos \left(a + b\right)}\]
Alternative 43
Error39.5
Cost32576
\[r \cdot \left(\sqrt{\sin b} \cdot \frac{\sqrt{\sin b}}{\cos \left(a + b\right)}\right)\]
Alternative 44
Error27.4
Cost26560
\[r \cdot \frac{\sin b}{\cos b - a \cdot \left(\sin b + a \cdot \left(\cos b \cdot 0.5\right)\right)}\]
Alternative 45
Error31.9
Cost26496
\[r \cdot \left(\frac{b}{\cos a} + \frac{b \cdot \left(b \cdot \sin a\right)}{{\cos a}^{2}}\right)\]
Alternative 46
Error0.4
Cost26304
\[\frac{r}{\left(\cos a \cdot \cos b\right) \cdot \frac{1}{\sin b} - \sin a}\]
Alternative 47
Error0.4
Cost26304
\[r \cdot \frac{1}{\frac{\cos a}{\frac{\sin b}{\cos b}} - \sin a}\]
Alternative 48
Error0.4
Cost26304
\[\frac{r}{\frac{1}{\frac{\sin b}{\cos a \cdot \cos b}} - \sin a}\]
Alternative 49
Error0.7
Cost26304
\[\frac{1}{\frac{\frac{\cos a \cdot \cos b}{\sin b} - \sin a}{r}}\]
Alternative 50
Error0.4
Cost26176
\[\frac{r}{\frac{\cos a}{\frac{\sin b}{\cos b}} - \sin a}\]
Alternative 51
Error0.4
Cost26176
\[\frac{r}{\frac{\cos a \cdot \cos b}{\sin b} - \sin a}\]
Alternative 52
Error0.4
Cost26176
\[\frac{r}{\cos a \cdot \frac{\cos b}{\sin b} - \sin a}\]
Alternative 53
Error15.1
Cost26112
\[r \cdot \frac{\sin b}{\sqrt[3]{{\cos \left(a + b\right)}^{3}}}\]
Alternative 54
Error40.2
Cost26112
\[\sqrt[3]{{\left(r \cdot \frac{\sin b}{\cos \left(a + b\right)}\right)}^{3}}\]
Alternative 55
Error28.3
Cost26112
\[r \cdot \sqrt[3]{{\left(\frac{\sin b}{\cos \left(a + b\right)}\right)}^{3}}\]
Alternative 56
Error15.1
Cost26048
\[r \cdot \frac{\sin b}{\log \left(e^{\cos \left(a + b\right)}\right)}\]
Alternative 57
Error30.5
Cost19776
\[r \cdot \frac{\sin b}{\cos a - b \cdot \sin a}\]
Alternative 58
Error30.5
Cost19776
\[\frac{r \cdot \sin b}{\cos a - b \cdot \sin a}\]
Alternative 59
Error27.0
Cost19776
\[r \cdot \frac{\sin b}{\cos b - a \cdot \sin b}\]
Alternative 60
Error24.8
Cost19648
\[\frac{r}{\frac{\cos b}{\sin b} - \sin a}\]
Alternative 61
Error15.0
Cost13376
\[\left(r \cdot \sin b\right) \cdot \frac{1}{\cos \left(a + b\right)}\]
Alternative 62
Error15.0
Cost13376
\[r \cdot \left(\sin b \cdot \frac{1}{\cos \left(a + b\right)}\right)\]
Alternative 63
Error15.1
Cost13376
\[r \cdot \frac{1}{\frac{\cos \left(a + b\right)}{\sin b}}\]
Alternative 64
Error27.1
Cost13248
\[\frac{r}{\frac{\cos b}{\sin b} - a}\]
Alternative 65
Error29.5
Cost13248
\[\frac{r}{\frac{\cos a}{b} - \sin a}\]
Alternative 66
Error15.0
Cost13248
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
Alternative 67
Error15.0
Cost13248
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
Alternative 68
Error25.6
Cost13120
\[r \cdot \frac{\sin b}{\cos b}\]
Alternative 69
Error25.6
Cost13120
\[\frac{r}{\frac{\cos b}{\sin b}}\]
Alternative 70
Error25.6
Cost13120
\[\frac{r \cdot \sin b}{\cos b}\]
Alternative 71
Error29.0
Cost13120
\[r \cdot \frac{\sin b}{\cos a}\]
Alternative 72
Error31.7
Cost6720
\[r \cdot \frac{b}{\cos a}\]
Alternative 73
Error31.8
Cost6720
\[\frac{r}{\frac{\cos a}{b}}\]
Alternative 74
Error31.7
Cost6720
\[\frac{r \cdot b}{\cos a}\]
Alternative 75
Error61.7
Cost64
\[1\]
Alternative 76
Error54.1
Cost64
\[0\]
Alternative 77
Error61.7
Cost64
\[-1\]

Error

Derivation

  1. Initial program 15.0

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  2. Using strategy rm

  3. Applied cos-sum_binary64_5380.3

    \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]
  4. Using strategy rm

  5. Applied associate-*r/_binary64_3460.3

    \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]

  6. Simplified0.3

    \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a}}\]

  7. Final simplification0.3

    \[\leadsto \frac{r \cdot \sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (r a b)
  :name "rsin B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))