?

Average Accuracy: 75.3% → 99.5%
Time: 20.5s
Precision: binary64
Cost: 39040

?

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

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(a) * Float64(-sin(b))))))
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[(r * 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]), $MachinePrecision]

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

Error?

Derivation?

  1. Initial program 75.3%

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

  2. Simplified75.3%

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

    [Start]75.3

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

    +-commutative [=>]75.3

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

  3. Applied egg-rr99.5%

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

    [Start]75.3

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

    cos-sum [=>]99.5

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

    fma-neg [=>]99.5

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

  4. Final simplification99.5%

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

Alternatives

Alternative 1
Accuracy99.5%
Cost32704
\[r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
Alternative 2
Accuracy75.1%
Cost19712
\[r \cdot {\left(\frac{\cos \left(b - a\right)}{\sin b}\right)}^{-1} \]
Alternative 3
Accuracy75.1%
Cost13385
\[\begin{array}{l} \mathbf{if}\;b \leq -0.00142 \lor \neg \left(b \leq 0.04\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{r}{\cos \left(b + a\right)}}{b \cdot 0.16666666666666666 + \frac{1}{b}}\\ \end{array} \]
Alternative 4
Accuracy75.1%
Cost13385
\[\begin{array}{l} \mathbf{if}\;b \leq -0.0155 \lor \neg \left(b \leq 0.04\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{r}{\cos \left(b + a\right)}}{b \cdot 0.16666666666666666 + \frac{1}{b}}\\ \end{array} \]
Alternative 5
Accuracy75.3%
Cost13248
\[r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]
Alternative 6
Accuracy75.1%
Cost13248
\[\frac{r}{\frac{\cos \left(b - a\right)}{\sin b}} \]
Alternative 7
Accuracy53.1%
Cost13120
\[r \cdot \frac{\sin b}{\cos a} \]
Alternative 8
Accuracy51.2%
Cost7497
\[\begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{-28} \lor \neg \left(a \leq 12000\right):\\ \;\;\;\;\frac{\frac{r}{\cos \left(b + a\right)}}{b \cdot 0.16666666666666666 + \frac{1}{b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \sin b\\ \end{array} \]
Alternative 9
Accuracy53.4%
Cost7241
\[\begin{array}{l} \mathbf{if}\;b \leq -9.2 \lor \neg \left(b \leq 27\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\cos \left(b + a\right)} \cdot \left(r \cdot b\right)\\ \end{array} \]
Alternative 10
Accuracy53.4%
Cost6985
\[\begin{array}{l} \mathbf{if}\;b \leq -4.8 \lor \neg \left(b \leq 17\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
Alternative 11
Accuracy53.4%
Cost6985
\[\begin{array}{l} \mathbf{if}\;b \leq -0.78 \lor \neg \left(b \leq 17\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \]
Alternative 12
Accuracy37.3%
Cost6592
\[r \cdot \sin b \]
Alternative 13
Accuracy32.6%
Cost192
\[r \cdot b \]

Error

Reproduce?

herbie shell --seed 2023135 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))