rsin A

Average Error: 15.1 → 0.3

Time: 18.5s

Precision: binary64

Cost: 39040

Math

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

↓

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

FPCore

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

↓

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

Julia

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

↓

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

Derivation

1. Initial program 15.1

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

2. Simplified 15.1

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

3. Applied egg-rr 0.3

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

4. Taylor expanded in r around 0 0.3

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

5. Simplified 0.3

   \[\leadsto \color{blue}{\frac{r}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot \sin b} \]
   Proof
   (*.f64 (/.f64 r (-.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (*.f64 (sin.f64 b) (sin.f64 a)))) (sin.f64 b)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 r (-.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 a) (sin.f64 b))))) (sin.f64 b)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 r (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (neg.f64 (*.f64 (sin.f64 a) (sin.f64 b)))))) (sin.f64 b)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 r (+.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (sin.f64 a) (sin.f64 b)))))) (sin.f64 b)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 r (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (*.f64 (sin.f64 a) (sin.f64 b))) (*.f64 (cos.f64 a) (cos.f64 b))))) (sin.f64 b)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r/_binary64 (/.f64 r (/.f64 (+.f64 (*.f64 -1 (*.f64 (sin.f64 a) (sin.f64 b))) (*.f64 (cos.f64 a) (cos.f64 b))) (sin.f64 b)))): 38 points increase in error, 28 points decrease in error
   (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 r (sin.f64 b)) (+.f64 (*.f64 -1 (*.f64 (sin.f64 a) (sin.f64 b))) (*.f64 (cos.f64 a) (cos.f64 b))))): 35 points increase in error, 43 points decrease in error
   (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 b) r)) (+.f64 (*.f64 -1 (*.f64 (sin.f64 a) (sin.f64 b))) (*.f64 (cos.f64 a) (cos.f64 b)))): 0 points increase in error, 0 points decrease in error

6. Applied egg-rr 0.3

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

7. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	39040

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

Alternative 2
Error	0.3
Cost	39040

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

Alternative 3
Error	0.4
Cost	32704

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

Alternative 4
Error	0.3
Cost	32704

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

Alternative 5
Error	0.3
Cost	32704

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

Alternative 6
Error	15.3
Cost	13512

\[\begin{array}{l} \mathbf{if}\;a \leq -12071.683251395545:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 2.8730953776777996 \cdot 10^{-8}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \end{array} \]

Alternative 7
Error	14.4
Cost	13512

\[\begin{array}{l} t_0 := \frac{r}{\frac{\cos a}{b} - \sin a}\\ \mathbf{if}\;a \leq -12071.683251395545:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 2.8730953776777996 \cdot 10^{-8}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b} - a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	15.4
Cost	13384

\[\begin{array}{l} t_0 := \frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{if}\;a \leq -12071.683251395545:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 2.8730953776777996 \cdot 10^{-8}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	15.4
Cost	13384

\[\begin{array}{l} t_0 := \frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{if}\;a \leq -12071.683251395545:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 2.8730953776777996 \cdot 10^{-8}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	15.4
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -12071.683251395545:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 2.8730953776777996 \cdot 10^{-8}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \end{array} \]

Alternative 11
Error	15.2
Cost	13248

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

Alternative 12
Error	15.1
Cost	13248

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

Alternative 13
Error	15.1
Cost	13248

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

Alternative 14
Error	28.5
Cost	13120

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

Alternative 15
Error	31.3
Cost	6976

\[\frac{1}{\cos \left(b + a\right)} \cdot \left(r \cdot b\right) \]

Alternative 16
Error	31.3
Cost	6848

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

Alternative 17
Error	31.3
Cost	6720

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

Alternative 18
Error	31.3
Cost	6720

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

Alternative 19
Error	31.3
Cost	6720

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

Alternative 20
Error	42.0
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2022300 
(FPCore (r a b)
  :name "rsin A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))