rsin A (should all be same)

Average Accuracy: 76.6% → 99.5%

Time: 15.5s

Precision: binary64

Cost: 39040

Math

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

↓

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

FPCore

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

↓

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

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 (sin(b) / fma(cos(a), cos(b), (sin(b) * -sin(a)))) * r;
}

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

Derivation

1. Initial program 76.6%

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

2. Applied egg-rr 99.5%

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

   [Start] 76.6	\[ \frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   cos-sum [=>] 99.5	\[ \frac{r \cdot \sin b}{\color{blue}{\cos a \cdot \cos b - \sin a \cdot \sin b}} \]
   fma-neg [=>] 99.5	\[ \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos a, \cos b, -\sin a \cdot \sin b\right)}} \]
   *-commutative [=>] 99.5	\[ \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos a, \cos b, -\color{blue}{\sin b \cdot \sin a}\right)} \]

3. Applied egg-rr 99.5%

   \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos a, \cos b, -\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)}\right)} \]
   Proof

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

4. Taylor expanded in r around 0 99.5%

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

5. Simplified 99.5%

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

   [Start] 99.5	\[ \frac{\sin b \cdot r}{\cos a \cdot \cos b - \sin a \cdot \sin b} \]
   associate-/l* [=>] 99.4	\[ \color{blue}{\frac{\sin b}{\frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{r}}} \]
   associate-/r/ [=>] 99.5	\[ \color{blue}{\frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b} \cdot r} \]
   *-commutative [<=] 99.5	\[ \frac{\sin b}{\cos a \cdot \cos b - \color{blue}{\sin b \cdot \sin a}} \cdot r \]
   fma-neg [=>] 99.5	\[ \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos a, \cos b, -\sin b \cdot \sin a\right)}} \cdot r \]
   distribute-rgt-neg-in [=>] 99.5	\[ \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, \color{blue}{\sin b \cdot \left(-\sin a\right)}\right)} \cdot r \]

6. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	32704

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

Alternative 2
Accuracy	99.4%
Cost	32512

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

Alternative 3
Accuracy	77.6%
Cost	19648

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

Alternative 4
Accuracy	76.4%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-6} \lor \neg \left(b \leq 6.5 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 5
Accuracy	76.4%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \end{array} \]

Alternative 6
Accuracy	76.4%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \mathbf{elif}\;b \leq 2.95 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \end{array} \]

Alternative 7
Accuracy	76.6%
Cost	13248

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

Alternative 8
Accuracy	76.6%
Cost	13248

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

Alternative 9
Accuracy	76.6%
Cost	13248

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

Alternative 10
Accuracy	76.5%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-6} \lor \neg \left(b \leq 4.5 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 11
Accuracy	38.3%
Cost	6592

\[\sin b \cdot r \]

Alternative 12
Accuracy	60.3%
Cost	6592

\[r \cdot \tan b \]

Alternative 13
Accuracy	34.7%
Cost	576

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

Alternative 14
Accuracy	33.8%
Cost	192

\[b \cdot r \]

Reproduce

herbie shell --seed 2023129 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))