rsin A

Average Error: 15.6 → 0.3

Time: 12.6s

Precision: binary64

Cost: 39040

Math

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

↓

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

FPCore

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

↓

(FPCore (r a b)
 :precision binary64
 (/ (* (sin b) r) (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 (sin(b) * r) / 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(sin(b) * r) / 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[(N[Sin[b], $MachinePrecision] * r), $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 15.6

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

2. Applied egg-rr 0.3

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

3. Applied egg-rr 0.3

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

4. Taylor expanded in r around 0 0.3

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

5. Applied egg-rr 0.3

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

6. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	32704

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

Alternative 2
Error	0.3
Cost	32704

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

Alternative 3
Error	16.4
Cost	13384

\[\begin{array}{l} t_0 := \sin b \cdot \frac{r}{\cos a}\\ \mathbf{if}\;a \leq -124.45381542025724:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 8.659823856620301 \cdot 10^{-39}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	16.4
Cost	13384

\[\begin{array}{l} t_0 := \frac{\sin b \cdot r}{\cos a}\\ \mathbf{if}\;a \leq -124.45381542025724:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 8.659823856620301 \cdot 10^{-39}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	15.6
Cost	13248

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

Alternative 6
Error	15.7
Cost	13248

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

Alternative 7
Error	16.0
Cost	6984

\[\begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -176672074.5232325:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.1492680206531809 \cdot 10^{-11}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	40.0
Cost	6592

\[\sin b \cdot r \]

Alternative 9
Error	26.2
Cost	6592

\[r \cdot \tan b \]

Alternative 10
Error	42.3
Cost	704

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

Alternative 11
Error	42.3
Cost	576

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

Alternative 12
Error	42.9
Cost	192

\[b \cdot r \]

Reproduce

herbie shell --seed 2022228 
(FPCore (r a b)
  :name "rsin A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))