rsin B (should all be same)

Average Error: 14.5 → 0.3

Time: 17.3s

Precision: binary64

Cost: 52160

Math

\[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]

↓

\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, 0\right) + \frac{{\sin b}^{2}}{\frac{\sin b}{-\sin a}}} \]

FPCore

(FPCore (r a b) :precision binary64 (* r (/ (sin b) (cos (+ a b)))))

↓

(FPCore (r a b)
 :precision binary64
 (*
  r
  (/
   (sin b)
   (+
    (fma (cos b) (cos a) 0.0)
    (/ (pow (sin b) 2.0) (/ (sin b) (- (sin a))))))))

C

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) / (fma(cos(b), cos(a), 0.0) + (pow(sin(b), 2.0) / (sin(b) / -sin(a)))));
}

Julia

function code(r, a, b)
	return Float64(r * Float64(sin(b) / cos(Float64(a + b))))
end

↓

function code(r, a, b)
	return Float64(r * Float64(sin(b) / Float64(fma(cos(b), cos(a), 0.0) + Float64((sin(b) ^ 2.0) / Float64(sin(b) / Float64(-sin(a)))))))
end

Wolfram

code[r_, a_, b_] := N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[r_, a_, b_] := N[(r * N[(N[Sin[b], $MachinePrecision] / N[(N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + 0.0), $MachinePrecision] + N[(N[Power[N[Sin[b], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Sin[b], $MachinePrecision] / (-N[Sin[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.5

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]

2. Simplified 14.5

   \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos \left(b + a\right)}} \]
   Proof

   [Start] 14.5	\[ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   +-commutative [=>] 14.5	\[ r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Applied egg-rr 0.3

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

4. Applied egg-rr 0.3

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a + \left(\sin b \cdot \left(-\sin a\right) + \mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right)\right)}} \]

5. Simplified 0.3

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)\right) - \sin a \cdot \sin b}} \]
   Proof

   [Start] 0.3	\[ r \cdot \frac{\sin b}{\cos b \cdot \cos a + \left(\sin b \cdot \left(-\sin a\right) + \mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right)\right)} \]
   associate-+r+ [=>] 0.3	\[ r \cdot \frac{\sin b}{\color{blue}{\left(\cos b \cdot \cos a + \sin b \cdot \left(-\sin a\right)\right) + \mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right)}} \]
   +-commutative [<=] 0.3	\[ r \cdot \frac{\sin b}{\color{blue}{\left(\sin b \cdot \left(-\sin a\right) + \cos b \cdot \cos a\right)} + \mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right)} \]
   +-commutative [<=] 0.3	\[ r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right) + \left(\sin b \cdot \left(-\sin a\right) + \cos b \cdot \cos a\right)}} \]
   +-commutative [=>] 0.3	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right) + \color{blue}{\left(\cos b \cdot \cos a + \sin b \cdot \left(-\sin a\right)\right)}} \]
   distribute-rgt-neg-out [=>] 0.3	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right) + \left(\cos b \cdot \cos a + \color{blue}{\left(-\sin b \cdot \sin a\right)}\right)} \]
   sub-neg [<=] 0.3	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right) + \color{blue}{\left(\cos b \cdot \cos a - \sin b \cdot \sin a\right)}} \]
   associate-+r- [=>] 0.3	\[ r \cdot \frac{\sin b}{\color{blue}{\left(\mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right) + \cos b \cdot \cos a\right) - \sin b \cdot \sin a}} \]

6. Applied egg-rr 0.3

   \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)\right) - \color{blue}{\frac{\left(0 - {\sin b}^{2}\right) \cdot \left(-\sin a\right)}{\sin b}}} \]

7. Simplified 0.3

   \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)\right) - \color{blue}{\frac{-{\sin b}^{2}}{\frac{\sin b}{-\sin a}}}} \]
   Proof

   [Start] 0.3	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)\right) - \frac{\left(0 - {\sin b}^{2}\right) \cdot \left(-\sin a\right)}{\sin b}} \]
   associate-/l* [=>] 0.3	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)\right) - \color{blue}{\frac{0 - {\sin b}^{2}}{\frac{\sin b}{-\sin a}}}} \]
   sub0-neg [=>] 0.3	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)\right) - \frac{\color{blue}{-{\sin b}^{2}}}{\frac{\sin b}{-\sin a}}} \]

8. Taylor expanded in b around inf 0.3

   \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{-1 \cdot \left(\sin a \cdot \sin b\right) + \sin a \cdot \sin b}\right) - \frac{-{\sin b}^{2}}{\frac{\sin b}{-\sin a}}} \]

9. Simplified 0.3

   \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{0}\right) - \frac{-{\sin b}^{2}}{\frac{\sin b}{-\sin a}}} \]
   Proof

   [Start] 0.3	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, -1 \cdot \left(\sin a \cdot \sin b\right) + \sin a \cdot \sin b\right) - \frac{-{\sin b}^{2}}{\frac{\sin b}{-\sin a}}} \]
   +-commutative [=>] 0.3	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\sin a \cdot \sin b + -1 \cdot \left(\sin a \cdot \sin b\right)}\right) - \frac{-{\sin b}^{2}}{\frac{\sin b}{-\sin a}}} \]
   *-commutative [<=] 0.3	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\sin b \cdot \sin a} + -1 \cdot \left(\sin a \cdot \sin b\right)\right) - \frac{-{\sin b}^{2}}{\frac{\sin b}{-\sin a}}} \]
   *-commutative [<=] 0.3	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \sin b \cdot \sin a + -1 \cdot \color{blue}{\left(\sin b \cdot \sin a\right)}\right) - \frac{-{\sin b}^{2}}{\frac{\sin b}{-\sin a}}} \]
   mul-1-neg [=>] 0.3	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \sin b \cdot \sin a + \color{blue}{\left(-\sin b \cdot \sin a\right)}\right) - \frac{-{\sin b}^{2}}{\frac{\sin b}{-\sin a}}} \]
   sub-neg [<=] 0.3	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{\sin b \cdot \sin a - \sin b \cdot \sin a}\right) - \frac{-{\sin b}^{2}}{\frac{\sin b}{-\sin a}}} \]
   +-inverses [=>] 0.3	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \color{blue}{0}\right) - \frac{-{\sin b}^{2}}{\frac{\sin b}{-\sin a}}} \]

10. Final simplification 0.3

    \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, 0\right) + \frac{{\sin b}^{2}}{\frac{\sin b}{-\sin a}}} \]

Alternatives

Alternative 1
Error	0.3
Cost	32704

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

Alternative 2
Error	0.3
Cost	32704

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

Alternative 3
Error	14.7
Cost	13385

\[\begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-5} \lor \neg \left(b \leq 0.018\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 4
Error	14.7
Cost	13384

\[\begin{array}{l} \mathbf{if}\;b \leq -1.48 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 0.018:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 5
Error	14.7
Cost	13384

\[\begin{array}{l} \mathbf{if}\;b \leq -5.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \mathbf{elif}\;b \leq 0.018:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 6
Error	14.7
Cost	13384

\[\begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{-5}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{elif}\;b \leq 0.018:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 7
Error	14.5
Cost	13248

\[r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]

Alternative 8
Error	28.9
Cost	13120

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

Alternative 9
Error	28.9
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -2.1 \lor \neg \left(b \leq 1.1 \cdot 10^{+36}\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 10
Error	39.3
Cost	6592

\[r \cdot \sin b \]

Alternative 11
Error	42.1
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2023066 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))