rsin A (should all be same)

Average Accuracy: 77.1% → 99.5%

Time: 20.7s

Precision: binary64

Cost: 39040

Math

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

↓

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

FPCore

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

↓

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

Derivation

1. Initial program 73.6%

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

2. Simplified 73.6%

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

   [Start] 73.6	\[ \frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   associate-*r/ [<=] 73.6	\[ \color{blue}{r \cdot \frac{\sin b}{\cos \left(a + b\right)}} \]
   *-commutative [<=] 73.6	\[ \color{blue}{\frac{\sin b}{\cos \left(a + b\right)} \cdot r} \]
   +-commutative [=>] 73.6	\[ \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \cdot r \]

3. Applied egg-rr 99.5%

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

   [Start] 73.6	\[ \frac{\sin b}{\cos \left(b + a\right)} \cdot r \]
   cos-sum [=>] 99.4	\[ \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \cdot r \]
   cancel-sign-sub-inv [=>] 99.4	\[ \frac{\sin b}{\color{blue}{\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a}} \cdot r \]
   fma-def [=>] 99.5	\[ \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \cdot r \]

4. Final simplification 99.5%

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

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	99.4%
Cost	32512

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

Alternative 3
Accuracy	78.1%
Cost	26048

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

Alternative 4
Accuracy	78.7%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;a \leq -0.034 \lor \neg \left(a \leq 0.056\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{b} - \sin a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\mathsf{fma}\left(-0.5, a \cdot a, 1\right)}{\tan b} - a}\\ \end{array} \]

Alternative 5
Accuracy	76.8%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -1650 \lor \neg \left(a \leq 0.0105\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]

Alternative 6
Accuracy	76.8%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -1650 \lor \neg \left(a \leq 0.0102\right):\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]

Alternative 7
Accuracy	77.1%
Cost	13248

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

Alternative 8
Accuracy	77.1%
Cost	13248

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

Alternative 9
Accuracy	77.0%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-5} \lor \neg \left(b \leq 0.01\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 10
Accuracy	39.2%
Cost	6592

\[\sin b \cdot r \]

Alternative 11
Accuracy	60.7%
Cost	6592

\[r \cdot \tan b \]

Alternative 12
Accuracy	35.6%
Cost	576

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

Alternative 13
Accuracy	35.1%
Cost	192

\[b \cdot r \]

Reproduce

herbie shell --seed 2023159 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))