rsin A (should all be same)

Average Error: 14.9 → 0.3

Time: 13.3s

Precision: binary64

Cost: 39040

Math

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

↓

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

FPCore

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

↓

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

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) / fma(cos(b), cos(a), (sin(a) * -sin(b)))) * r;
}

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

Derivation

1. Initial program 14.9

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

2. Simplified 14.9

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

   \[\leadsto \color{blue}{\frac{\sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a} \cdot r} \]
   Proof
   (*.f64 (/.f64 (sin.f64 b) (-.f64 (*.f64 (cos.f64 a) (cos.f64 b)) (*.f64 (sin.f64 b) (sin.f64 a)))) r): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (sin.f64 b) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 b) (cos.f64 a))) (*.f64 (sin.f64 b) (sin.f64 a)))) r): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (sin.f64 b) (-.f64 (*.f64 (cos.f64 b) (cos.f64 a)) (Rewrite=> *-commutative_binary64 (*.f64 (sin.f64 a) (sin.f64 b))))) r): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (sin.f64 b) (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 (*.f64 (cos.f64 b) (cos.f64 a)) (*.f64 (neg.f64 (sin.f64 a)) (sin.f64 b))))) r): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (sin.f64 b) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (neg.f64 (sin.f64 a)) (sin.f64 b)) (*.f64 (cos.f64 b) (cos.f64 a))))) r): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (sin.f64 b) (+.f64 (Rewrite=> distribute-lft-neg-out_binary64 (neg.f64 (*.f64 (sin.f64 a) (sin.f64 b)))) (*.f64 (cos.f64 b) (cos.f64 a)))) r): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (sin.f64 b) (+.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (sin.f64 a) (sin.f64 b)))) (*.f64 (cos.f64 b) (cos.f64 a)))) r): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (sin.f64 b) (+.f64 (*.f64 -1 (*.f64 (sin.f64 a) (sin.f64 b))) (Rewrite=> *-commutative_binary64 (*.f64 (cos.f64 a) (cos.f64 b))))) r): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r/_binary64 (/.f64 (sin.f64 b) (/.f64 (+.f64 (*.f64 -1 (*.f64 (sin.f64 a) (sin.f64 b))) (*.f64 (cos.f64 a) (cos.f64 b))) r))): 51 points increase in error, 40 points decrease in error
   (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (sin.f64 b) r) (+.f64 (*.f64 -1 (*.f64 (sin.f64 a) (sin.f64 b))) (*.f64 (cos.f64 a) (cos.f64 b))))): 38 points increase in error, 44 points decrease in error

6. Applied egg-rr 0.3

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

7. Final simplification 0.3

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

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	14.2
Cost	26048

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

Alternative 3
Error	15.1
Cost	13384

\[\begin{array}{l} t_0 := \sin b \cdot \frac{r}{\cos a}\\ \mathbf{if}\;a \leq -0.33:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 8600000:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	15.1
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -0.33:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 8600000:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 5
Error	15.1
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -0.33:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 8600000:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \end{array} \]

Alternative 6
Error	14.9
Cost	13248

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

Alternative 7
Error	15.1
Cost	7112

\[\begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -14:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	15.1
Cost	6984

\[\begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -14:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-14}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	15.1
Cost	6984

\[\begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -14:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	25.3
Cost	6592

\[r \cdot \tan b \]

Alternative 11
Error	41.8
Cost	192

\[b \cdot r \]

Reproduce

herbie shell --seed 2022337 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))