rsin A (should all be same)

Percentage Accurate: 76.1% → 99.5%

Time: 20.4s

Precision: binary64

Cost: 39040

Math

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

↓

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

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 a) (cos b)))))

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(a) * cos(b)));
}

Julia

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 80.2%

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

2. Simplified 80.2%

   \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
   Step-by-step derivation

   [Start] 80.2%	\[ \frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   +-commutative [=>] 80.2%	\[ \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Applied egg-rr 99.5%

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

   [Start] 80.2%	\[ \frac{r \cdot \sin b}{\cos \left(b + a\right)} \]
   cos-sum [=>] 99.5%	\[ \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

4. Applied egg-rr 99.5%

   \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)}} \]
   Step-by-step derivation

   [Start] 99.5%	\[ \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
   log1p-expm1-u [=>] 99.5%	\[ \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)}} \]

5. Taylor expanded in b around inf 99.5%

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

6. Simplified 99.5%

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

   [Start] 99.5%	\[ \frac{r \cdot \sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b} \]
   cancel-sign-sub-inv [=>] 99.5%	\[ \frac{r \cdot \sin b}{\color{blue}{\cos a \cdot \cos b + \left(-\sin a\right) \cdot \sin b}} \]
   *-commutative [=>] 99.5%	\[ \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a} + \left(-\sin a\right) \cdot \sin b} \]
   *-commutative [<=] 99.5%	\[ \frac{r \cdot \sin b}{\cos b \cdot \cos a + \color{blue}{\sin b \cdot \left(-\sin a\right)}} \]
   +-commutative [=>] 99.5%	\[ \frac{r \cdot \sin b}{\color{blue}{\sin b \cdot \left(-\sin a\right) + \cos b \cdot \cos a}} \]
   fma-def [=>] 99.5%	\[ \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \cos a\right)}} \]
   *-commutative [<=] 99.5%	\[ \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin b, -\sin a, \color{blue}{\cos a \cdot \cos b}\right)} \]

7. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	39040

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

Alternative 2
Accuracy	99.5%
Cost	32704

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

Alternative 3
Accuracy	99.5%
Cost	32704

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

Alternative 4
Accuracy	76.1%
Cost	13248

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

Alternative 5
Accuracy	76.1%
Cost	13248

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

Alternative 6
Accuracy	75.9%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-5} \lor \neg \left(b \leq 1.35 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 7
Accuracy	75.8%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -11.6 \lor \neg \left(b \leq 1.06 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 8
Accuracy	75.8%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -11.6 \lor \neg \left(b \leq 7.5 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \]

Alternative 9
Accuracy	60.2%
Cost	6592

\[r \cdot \tan b \]

Alternative 10
Accuracy	34.2%
Cost	576

\[r \cdot \frac{1}{\frac{1}{b} - a} \]

Alternative 11
Accuracy	35.1%
Cost	576

\[\frac{r}{b \cdot -0.3333333333333333 + \frac{1}{b}} \]

Alternative 12
Accuracy	34.2%
Cost	448

\[\frac{r}{\frac{1}{b} - a} \]

Alternative 13
Accuracy	34.6%
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2023271 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))