rsin B (should all be same)

Average Error: 15.0 → 0.3

Time: 16.1s

Precision: binary64

Cost: 39040

Math

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

↓

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

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(r * Float64(sin(b) / cos(Float64(a + b))))
end

↓

function code(r, a, b)
	return Float64(r * Float64(sin(b) / fma(cos(b), cos(a), Float64(sin(b) * Float64(-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_] := N[(r * N[(N[Sin[b], $MachinePrecision] / 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 15.0

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

2. Simplified 15.0

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

   [Start] 15.0	\[ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   +-commutative [=>] 15.0	\[ r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Applied egg-rr 0.3

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

4. Final simplification 0.3

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

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	14.5
Cost	26496

\[r \cdot \frac{-\sin b}{\sin a \cdot \left(\sin b \cdot 2\right) - \cos \left(b + a\right)} \]

Alternative 3
Error	14.4
Cost	19648

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

Alternative 4
Error	15.2
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -22 \lor \neg \left(a \leq 9.5 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]

Alternative 5
Error	15.0
Cost	13248

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

Alternative 6
Error	15.1
Cost	13248

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

Alternative 7
Error	15.2
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -7.3 \cdot 10^{-6} \lor \neg \left(b \leq 0.003\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 8
Error	39.8
Cost	6592

\[r \cdot \sin b \]

Alternative 9
Error	26.0
Cost	6592

\[r \cdot \tan b \]

Alternative 10
Error	42.7
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2023041 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))