rsin B

Average Error: 14.4 → 0.3

Time: 13.8s

Precision: binary64

Cost: 71488

Math

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

↓

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

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 b)
    (/
     r
     (+
      (fma (cos b) (cos a) (* (sin a) t_0))
      (fma t_0 (sin a) (* (sin b) (sin a))))))))

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);
	return sin(b) * (r / (fma(cos(b), cos(a), (sin(a) * t_0)) + fma(t_0, sin(a), (sin(b) * sin(a)))));
}

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))
	return Float64(sin(b) * Float64(r / Float64(fma(cos(b), cos(a), Float64(sin(a) * t_0)) + fma(t_0, sin(a), Float64(sin(b) * sin(a))))))
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[Sin[b], $MachinePrecision])}, N[(N[Sin[b], $MachinePrecision] * N[(r / N[(N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + N[(N[Sin[a], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Sin[a], $MachinePrecision] + N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 14.4

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

2. Simplified 14.4

   \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos \left(b + a\right)}} \]
   Proof
   (*.f64 r (/.f64 (sin.f64 b) (cos.f64 (+.f64 b a)))): 0 points increase in error, 0 points decrease in error
   (*.f64 r (/.f64 (sin.f64 b) (cos.f64 (Rewrite<= +-commutative_binary64 (+.f64 a b))))): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in r around 0 14.4

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

4. Simplified 14.4

   \[\leadsto \color{blue}{\sin b \cdot \frac{r}{\cos \left(a + b\right)}} \]
   Proof
   (*.f64 (sin.f64 b) (/.f64 r (cos.f64 (+.f64 a b)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sin.f64 b) (/.f64 r (cos.f64 (Rewrite<= +-commutative_binary64 (+.f64 b a))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 r (cos.f64 (+.f64 b a))) (sin.f64 b))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 r (sin.f64 b)) (cos.f64 (+.f64 b a)))): 35 points increase in error, 30 points decrease in error
   (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 b) r)) (cos.f64 (+.f64 b a))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (sin.f64 b) r) (cos.f64 (Rewrite=> +-commutative_binary64 (+.f64 a b)))): 0 points increase in error, 0 points decrease in error

5. Applied egg-rr 0.3

   \[\leadsto \sin b \cdot \frac{r}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, -\sin b \cdot \left(\sin a \cdot 1\right)\right) + \mathsf{fma}\left(-\sin b, \sin a \cdot 1, \sin b \cdot \left(\sin a \cdot 1\right)\right)}} \]

6. Taylor expanded in b around inf 0.3

   \[\leadsto \sin b \cdot \frac{r}{\mathsf{fma}\left(\cos b, \cos a, -\sin b \cdot \left(\sin a \cdot 1\right)\right) + \mathsf{fma}\left(-\sin b, \sin a \cdot 1, \color{blue}{\sin a \cdot \sin b}\right)} \]

7. Taylor expanded in b around inf 0.3

   \[\leadsto \sin b \cdot \frac{r}{\mathsf{fma}\left(\cos b, \cos a, -\color{blue}{\sin a \cdot \sin b}\right) + \mathsf{fma}\left(-\sin b, \sin a \cdot 1, \sin a \cdot \sin b\right)} \]

8. Final simplification 0.3

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

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	0.3
Cost	32704

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

Alternative 3
Error	13.8
Cost	26048

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

Alternative 4
Error	15.3
Cost	13384

\[\begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos a}\\ \mathbf{if}\;a \leq -1.42 \cdot 10^{+32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-18}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	15.3
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -1.42 \cdot 10^{+32}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-18}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 6
Error	14.4
Cost	13248

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

Alternative 7
Error	14.4
Cost	13248

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

Alternative 8
Error	14.6
Cost	6984

\[\begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -5.5 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-12}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	14.6
Cost	6984

\[\begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -5.5 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-12}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	25.0
Cost	6592

\[r \cdot \tan b \]

Alternative 11
Error	41.5
Cost	192

\[b \cdot r \]

Reproduce

herbie shell --seed 2022330 
(FPCore (r a b)
  :name "rsin B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))