rsin B (should all be same)

Average Accuracy: 76.8% → 99.5%

Time: 20.6s

Precision: binary64

Cost: 39040

Math

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

↓

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

FPCore

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

↓

(FPCore (r a b)
 :precision binary64
 (* r (/ (sin b) (fma (sin b) (- (sin a)) (* (cos b) (cos 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(sin(b), -sin(a), (cos(b) * cos(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) / fma(sin(b), Float64(-sin(a)), Float64(cos(b) * cos(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[Sin[b], $MachinePrecision] * (-N[Sin[a], $MachinePrecision]) + N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

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

2. Simplified 76.8%

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

   [Start] 76.8	\[ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   +-commutative [=>] 76.8	\[ r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Applied egg-rr 99.5%

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

   [Start] 76.8	\[ r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]
   cos-sum [=>] 99.5	\[ r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

4. Applied egg-rr 99.5%

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

   [Start] 99.5	\[ r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
   prod-diff [=>] 99.5	\[ r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, -\sin a \cdot \sin b\right) + \mathsf{fma}\left(-\sin a, \sin b, \sin a \cdot \sin b\right)}} \]
   *-commutative [<=] 99.5	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, -\color{blue}{\sin b \cdot \sin a}\right) + \mathsf{fma}\left(-\sin a, \sin b, \sin a \cdot \sin b\right)} \]
   fma-neg [<=] 99.5	\[ r \cdot \frac{\sin b}{\color{blue}{\left(\cos b \cdot \cos a - \sin b \cdot \sin a\right)} + \mathsf{fma}\left(-\sin a, \sin b, \sin a \cdot \sin b\right)} \]
   cancel-sign-sub-inv [=>] 99.5	\[ r \cdot \frac{\sin b}{\color{blue}{\left(\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a\right)} + \mathsf{fma}\left(-\sin a, \sin b, \sin a \cdot \sin b\right)} \]
   associate-+l+ [=>] 99.5	\[ r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a + \left(\left(-\sin b\right) \cdot \sin a + \mathsf{fma}\left(-\sin a, \sin b, \sin a \cdot \sin b\right)\right)}} \]
   distribute-lft-neg-in [<=] 99.5	\[ r \cdot \frac{\sin b}{\cos b \cdot \cos a + \left(\color{blue}{\left(-\sin b \cdot \sin a\right)} + \mathsf{fma}\left(-\sin a, \sin b, \sin a \cdot \sin b\right)\right)} \]
   distribute-rgt-neg-in [=>] 99.5	\[ r \cdot \frac{\sin b}{\cos b \cdot \cos a + \left(\color{blue}{\sin b \cdot \left(-\sin a\right)} + \mathsf{fma}\left(-\sin a, \sin b, \sin a \cdot \sin b\right)\right)} \]
   *-commutative [<=] 99.5	\[ r \cdot \frac{\sin b}{\cos b \cdot \cos a + \left(\sin b \cdot \left(-\sin a\right) + \mathsf{fma}\left(-\sin a, \sin b, \color{blue}{\sin b \cdot \sin a}\right)\right)} \]

5. Simplified 99.5%

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

   [Start] 99.5	\[ r \cdot \frac{\sin b}{\cos b \cdot \cos a + \left(\sin b \cdot \left(-\sin a\right) + \mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right)\right)} \]
   associate-+r+ [=>] 99.5	\[ r \cdot \frac{\sin b}{\color{blue}{\left(\cos b \cdot \cos a + \sin b \cdot \left(-\sin a\right)\right) + \mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right)}} \]
   +-commutative [<=] 99.5	\[ r \cdot \frac{\sin b}{\color{blue}{\left(\sin b \cdot \left(-\sin a\right) + \cos b \cdot \cos a\right)} + \mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right)} \]
   fma-udef [=>] 99.5	\[ r \cdot \frac{\sin b}{\left(\sin b \cdot \left(-\sin a\right) + \cos b \cdot \cos a\right) + \color{blue}{\left(\left(-\sin a\right) \cdot \sin b + \sin b \cdot \sin a\right)}} \]
   *-commutative [<=] 99.5	\[ r \cdot \frac{\sin b}{\left(\sin b \cdot \left(-\sin a\right) + \cos b \cdot \cos a\right) + \left(\color{blue}{\sin b \cdot \left(-\sin a\right)} + \sin b \cdot \sin a\right)} \]
   associate-+r+ [=>] 99.4	\[ r \cdot \frac{\sin b}{\color{blue}{\left(\left(\sin b \cdot \left(-\sin a\right) + \cos b \cdot \cos a\right) + \sin b \cdot \left(-\sin a\right)\right) + \sin b \cdot \sin a}} \]

6. Taylor expanded in a around inf 99.5%

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

7. Simplified 99.5%

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

   [Start] 99.5	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \mathsf{fma}\left(\cos b, \cos a, -1 \cdot \left(\sin a \cdot \sin b\right) + \sin a \cdot \sin b\right)\right)} \]
   +-commutative [=>] 99.5	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \mathsf{fma}\left(\cos b, \cos a, \color{blue}{\sin a \cdot \sin b + -1 \cdot \left(\sin a \cdot \sin b\right)}\right)\right)} \]
   mul-1-neg [=>] 99.5	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \mathsf{fma}\left(\cos b, \cos a, \sin a \cdot \sin b + \color{blue}{\left(-\sin a \cdot \sin b\right)}\right)\right)} \]
   sub-neg [<=] 99.5	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \mathsf{fma}\left(\cos b, \cos a, \color{blue}{\sin a \cdot \sin b - \sin a \cdot \sin b}\right)\right)} \]
   +-inverses [=>] 99.5	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \mathsf{fma}\left(\cos b, \cos a, \color{blue}{0}\right)\right)} \]

8. Applied egg-rr 99.5%

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

   [Start] 99.5	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \mathsf{fma}\left(\cos b, \cos a, 0\right)\right)} \]
   fma-udef [=>] 99.5	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \color{blue}{\cos b \cdot \cos a + 0}\right)} \]
   +-rgt-identity [=>] 99.5	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \color{blue}{\cos b \cdot \cos a}\right)} \]

9. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	32704

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

Alternative 2
Accuracy	75.6%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{+15}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]

Alternative 3
Accuracy	75.7%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{+15}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]

Alternative 4
Accuracy	75.6%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]

Alternative 5
Accuracy	75.6%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]

Alternative 6
Accuracy	76.8%
Cost	13248

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

Alternative 7
Accuracy	55.0%
Cost	13120

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

Alternative 8
Accuracy	55.0%
Cost	13120

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

Alternative 9
Accuracy	55.3%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -2.3 \lor \neg \left(b \leq 12600000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 10
Accuracy	55.3%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -1.35 \lor \neg \left(b \leq 19000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 11
Accuracy	39.0%
Cost	6592

\[r \cdot \sin b \]

Alternative 12
Accuracy	34.5%
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2023136 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))