rsin A

Average Error: 14.5 → 0.3

Time: 14.7s

Precision: binary64

Cost: 39040

Math

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

↓

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

FPCore

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

↓

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

Derivation

1. Initial program 14.5

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

2. Simplified 14.5

   \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
   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. Final simplification 0.3

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

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

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

Alternative 3
Error	13.9
Cost	26048

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

Alternative 4
Error	13.8
Cost	26048

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

Alternative 5
Error	14.7
Cost	13384

\[\begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos b}\\ \mathbf{if}\;b \leq -1.85:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-8}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	14.6
Cost	13384

\[\begin{array}{l} t_0 := \frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{if}\;a \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	14.6
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{elif}\;a \leq 6.1 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \end{array} \]

Alternative 8
Error	14.6
Cost	13384

\[\begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;a \leq -6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_0}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\cos a}\\ \end{array} \]

Alternative 9
Error	14.5
Cost	13376

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

Alternative 10
Error	14.5
Cost	13248

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

Alternative 11
Error	14.5
Cost	13248

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

Alternative 12
Error	14.5
Cost	13248

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

Alternative 13
Error	31.6
Cost	6848

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

Alternative 14
Error	31.6
Cost	6848

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

Alternative 15
Error	31.6
Cost	6720

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

Alternative 16
Error	31.6
Cost	6720

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

Alternative 17
Error	42.0
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2022331 
(FPCore (r a b)
  :name "rsin A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))