rsin B (should all be same)

Average Accuracy: 76.5% → 99.5%

Time: 19.3s

Precision: binary64

Cost: 39040

Math

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

↓

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

FPCore

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

↓

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

Julia

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

↓

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

Derivation

1. Initial program 76.5%

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

2. Simplified 76.5%

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

   [Start] 76.5	\[ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   +-commutative [=>] 76.5	\[ r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Applied egg-rr 99.5%

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

   [Start] 76.5	\[ r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]
   cos-sum [=>] 99.5	\[ r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]
   fma-neg [=>] 99.5	\[ r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, -\sin b \cdot \sin a\right)}} \]

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. Applied egg-rr 99.5%

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

   [Start] 99.5	\[ \frac{\sin b \cdot r}{\cos a \cdot \cos b - \sin a \cdot \sin b} \]
   sub-neg [=>] 99.5	\[ \frac{\sin b \cdot r}{\color{blue}{\cos a \cdot \cos b + \left(-\sin a \cdot \sin b\right)}} \]
   distribute-rgt-neg-in [=>] 99.5	\[ \frac{\sin b \cdot r}{\cos a \cdot \cos b + \color{blue}{\sin a \cdot \left(-\sin b\right)}} \]

6. Simplified 99.5%

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

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

7. Final simplification 99.5%

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

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	99.5%
Cost	32704

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

Alternative 3
Accuracy	99.5%
Cost	32704

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

Alternative 4
Accuracy	77.6%
Cost	19648

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

Alternative 5
Accuracy	76.3%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 0.13:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 6
Accuracy	76.5%
Cost	13248

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

Alternative 7
Accuracy	76.5%
Cost	13248

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

Alternative 8
Accuracy	76.3%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-6} \lor \neg \left(b \leq 0.13\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 9
Accuracy	76.3%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -3.9 \cdot 10^{-6} \lor \neg \left(b \leq 0.13\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 10
Accuracy	76.3%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-6} \lor \neg \left(b \leq 0.13\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \]

Alternative 11
Accuracy	39.1%
Cost	6592

\[\sin b \cdot r \]

Alternative 12
Accuracy	60.0%
Cost	6592

\[r \cdot \tan b \]

Alternative 13
Accuracy	34.6%
Cost	192

\[b \cdot r \]

Reproduce

herbie shell --seed 2023137 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))