?

Average Accuracy: 76.1% → 99.5%
Time: 19.6s
Precision: binary64
Cost: 39040

?

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

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

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]

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)}

Error?

Derivation?

  1. Initial program 76.1%

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

  2. Simplified76.1%

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

    [Start]76.1

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

    +-commutative [=>]76.1

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

  3. Applied egg-rr99.5%

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

    [Start]76.1

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

    cos-sum [=>]99.5

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

    cancel-sign-sub-inv [=>]99.5

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

    fma-def [=>]99.5

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

  4. Final simplification99.5%

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

Alternatives

Alternative 1
Accuracy99.4%
Cost32512
\[\frac{r}{\mathsf{fma}\left(\frac{\cos b}{\sin b}, \cos a, -\sin a\right)} \]
Alternative 2
Accuracy77.2%
Cost26048
\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, 0\right)} \]
Alternative 3
Accuracy75.6%
Cost13385
\[\begin{array}{l} \mathbf{if}\;a \leq -0.000185 \lor \neg \left(a \leq 13500000\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
Alternative 4
Accuracy75.6%
Cost13385
\[\begin{array}{l} \mathbf{if}\;a \leq -0.00039 \lor \neg \left(a \leq 13500000\right):\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
Alternative 5
Accuracy76.1%
Cost13248
\[r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]
Alternative 6
Accuracy75.9%
Cost13248
\[r \cdot \frac{\sin b}{\cos \left(b - a\right)} \]
Alternative 7
Accuracy75.8%
Cost6985
\[\begin{array}{l} \mathbf{if}\;b \leq -0.031 \lor \neg \left(b \leq 0.00043\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
Alternative 8
Accuracy75.7%
Cost6985
\[\begin{array}{l} \mathbf{if}\;b \leq -0.031 \lor \neg \left(b \leq 6 \cdot 10^{-5}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{b}}\\ \end{array} \]
Alternative 9
Accuracy75.8%
Cost6985
\[\begin{array}{l} \mathbf{if}\;b \leq -0.031 \lor \neg \left(b \leq 3.7 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \]
Alternative 10
Accuracy59.4%
Cost6592
\[r \cdot \tan b \]
Alternative 11
Accuracy33.9%
Cost192
\[r \cdot b \]

Error

Reproduce?

herbie shell --seed 2023151 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))