rsin B (should all be same)

Average Error: 14.8 → 0.3

Time: 15.9s

Precision: binary64

Cost: 65152

Math

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

↓

\[\begin{array}{l} t_0 := \sin b \cdot \sin a\\ r \cdot \frac{\sin b}{\left(\mathsf{fma}\left(-\sin a, \sin b, t_0\right) + \cos b \cdot \cos a\right) - t_0} \end{array} \]

FPCore

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

↓

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (sin a))))
   (*
    r
    (/
     (sin b)
     (- (+ (fma (- (sin a)) (sin b) t_0) (* (cos b) (cos a))) t_0)))))

C

double code(double r, double a, double b) {
	return r * (sin(b) / cos((a + b)));
}

↓

double code(double r, double a, double b) {
	double t_0 = sin(b) * sin(a);
	return r * (sin(b) / ((fma(-sin(a), sin(b), t_0) + (cos(b) * cos(a))) - t_0));
}

Julia

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

↓

function code(r, a, b)
	t_0 = Float64(sin(b) * sin(a))
	return Float64(r * Float64(sin(b) / Float64(Float64(fma(Float64(-sin(a)), sin(b), t_0) + Float64(cos(b) * cos(a))) - t_0)))
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_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]}, N[(r * N[(N[Sin[b], $MachinePrecision] / N[(N[(N[((-N[Sin[a], $MachinePrecision]) * N[Sin[b], $MachinePrecision] + t$95$0), $MachinePrecision] + N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 14.8

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

2. Simplified 14.8

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

   [Start] 14.8	\[ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   +-commutative [=>] 14.8	\[ r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Applied egg-rr 0.3

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

4. Applied egg-rr 0.3

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

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	32704

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

Alternative 2
Error	0.4
Cost	32512

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

Alternative 3
Error	14.3
Cost	26496

\[r \cdot \frac{-\sin b}{\sin b \cdot \left(\sin a \cdot 2\right) - \cos \left(b + a\right)} \]

Alternative 4
Error	13.9
Cost	20553

\[\begin{array}{l} \mathbf{if}\;a \leq -0.035 \lor \neg \left(a \leq 7600\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{b} - \sin a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b \cdot \left(-0.5 \cdot \left(a \cdot a\right) + 1\right) - \sin b \cdot a}\\ \end{array} \]

Alternative 5
Error	15.1
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -520000 \lor \neg \left(a \leq 0.00072\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 6
Error	15.1
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -520000:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{-5}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]

Alternative 7
Error	14.8
Cost	13248

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

Alternative 8
Error	28.5
Cost	13120

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

Alternative 9
Error	29.6
Cost	7364

\[\begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{r}{\frac{1}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{r}{\cos \left(b + a\right)}}{b \cdot 0.16666666666666666 + \frac{1}{b}}\\ \end{array} \]

Alternative 10
Error	29.9
Cost	6852

\[\begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{r}{\frac{1}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 11
Error	31.1
Cost	6720

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

Alternative 12
Error	41.6
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2023067 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))