?

Average Error: 15.3 → 0.4
Time: 17.9s
Precision: binary64
Cost: 39040

?

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

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

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

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

Error?

Derivation?

  1. Initial program 15.3

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

  2. Simplified15.3

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

    [Start]15.3

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

    *-commutative [=>]15.3

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

    associate-/r/ [<=]15.3

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

    +-commutative [=>]15.3

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

  3. Applied egg-rr0.4

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

  4. Final simplification0.4

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

Alternatives

Alternative 1
Error0.4
Cost32832
\[\frac{\sin b}{\cos a \cdot \frac{\cos b}{r} - \sin a \cdot \frac{\sin b}{r}} \]
Alternative 2
Error0.3
Cost32704
\[r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
Alternative 3
Error14.7
Cost19648
\[\frac{\sin b}{\frac{\cos b \cdot \cos a}{r}} \]
Alternative 4
Error15.8
Cost13385
\[\begin{array}{l} \mathbf{if}\;b \leq -1200 \lor \neg \left(b \leq 8000000000\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]
Alternative 5
Error15.8
Cost13385
\[\begin{array}{l} \mathbf{if}\;b \leq -1200 \lor \neg \left(b \leq 8000000000\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]
Alternative 6
Error15.3
Cost13248
\[r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]
Alternative 7
Error15.3
Cost13248
\[\sin b \cdot \frac{r}{\cos \left(b + a\right)} \]
Alternative 8
Error15.3
Cost13248
\[\sin b \cdot \frac{r}{\cos \left(b - a\right)} \]
Alternative 9
Error15.6
Cost6985
\[\begin{array}{l} \mathbf{if}\;b \leq -1200 \lor \neg \left(b \leq 53\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
Alternative 10
Error15.6
Cost6985
\[\begin{array}{l} \mathbf{if}\;b \leq -1200 \lor \neg \left(b \leq 53\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
Alternative 11
Error25.6
Cost6592
\[r \cdot \tan b \]
Alternative 12
Error42.2
Cost192
\[b \cdot r \]

Error

Reproduce?

herbie shell --seed 2023059 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))