rsin B

Average Error: 14.8 → 0.3

Time: 21.6s

Precision: binary64

Cost: 39040

Math

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

↓

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

FPCore

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

↓

(FPCore (r a b)
 :precision binary64
 (* r (/ (sin b) (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 r * (sin(b) / 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(r * Float64(sin(b) / 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[(r * N[(N[Sin[b], $MachinePrecision] / N[(N[Sin[a], $MachinePrecision] * (-N[Sin[b], $MachinePrecision]) + N[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision]), $MachinePrecision]), $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
   (*.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. 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

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

5. Taylor expanded in b around inf 0.3

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

6. Simplified 0.3

   \[\leadsto r \cdot \color{blue}{\frac{\sin b}{\mathsf{fma}\left(\sin a, -\sin b, \cos a \cdot \cos b\right)}} \]
   Proof
   (/.f64 (sin.f64 b) (fma.f64 (sin.f64 a) (neg.f64 (sin.f64 b)) (*.f64 (cos.f64 a) (cos.f64 b)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (sin.f64 b) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (sin.f64 a) (neg.f64 (sin.f64 b))) (*.f64 (cos.f64 a) (cos.f64 b))))): 7 points increase in error, 10 points decrease in error
   (/.f64 (sin.f64 b) (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (sin.f64 a) (sin.f64 b)))) (*.f64 (cos.f64 a) (cos.f64 b)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (sin.f64 b) (+.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 (sin.f64 a)) (sin.f64 b))) (*.f64 (cos.f64 a) (cos.f64 b)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (sin.f64 b) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (*.f64 (neg.f64 (sin.f64 a)) (sin.f64 b))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (sin.f64 b) (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (*.f64 (sin.f64 a) (sin.f64 b))))): 0 points increase in error, 0 points decrease in error

7. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	39040

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

Alternative 2
Error	0.3
Cost	32704

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

Alternative 3
Error	0.3
Cost	32704

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

Alternative 4
Error	0.3
Cost	32704

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

Alternative 5
Error	14.0
Cost	27272

\[\begin{array}{l} t_0 := r \cdot {\left(\frac{\cos a}{b} - \sin a\right)}^{-1}\\ \mathbf{if}\;a \leq -120634.49979016042:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 0.002714775271232511:\\ \;\;\;\;r \cdot \frac{\sin b}{\sin b \cdot \left(0.16666666666666666 \cdot {a}^{3} - a\right) + \cos b \cdot \left(-0.5 \cdot \left(a \cdot a\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	14.0
Cost	26952

\[\begin{array}{l} t_0 := r \cdot {\left(\frac{\cos a}{b} - \sin a\right)}^{-1}\\ \mathbf{if}\;a \leq -120634.49979016042:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 0.002714775271232511:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b \cdot \left(-0.5 \cdot \left(a \cdot a\right) + 1\right) - \sin b \cdot \sin a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	13.9
Cost	26568

\[\begin{array}{l} t_0 := r \cdot {\left(\frac{\cos a}{b} - \sin a\right)}^{-1}\\ \mathbf{if}\;a \leq -120634.49979016042:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 0.002714775271232511:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	14.0
Cost	20552

\[\begin{array}{l} t_0 := r \cdot {\left(\frac{\cos a}{b} - \sin a\right)}^{-1}\\ \mathbf{if}\;a \leq -120634.49979016042:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 0.002714775271232511:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b \cdot \left(-0.5 \cdot \left(a \cdot a\right) + 1\right) - \sin b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	14.9
Cost	13384

\[\begin{array}{l} t_0 := \frac{r \cdot \sin b}{\cos b}\\ \mathbf{if}\;b \leq -7.443575294864509 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 4.0839894579162385 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	15.1
Cost	13384

\[\begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos a}\\ \mathbf{if}\;a \leq -120634.49979016042:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 30.794776646147486:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	14.9
Cost	13248

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

Alternative 12
Error	14.9
Cost	13248

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

Alternative 13
Error	31.5
Cost	6720

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

Alternative 14
Error	31.5
Cost	6720

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

Alternative 15
Error	31.5
Cost	6720

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

Alternative 16
Error	31.5
Cost	6720

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

Alternative 17
Error	42.1
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2022295 
(FPCore (r a b)
  :name "rsin B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))