?

Average Error: 76.9% → 99.5%
Time: 15.4s
Precision: binary64
Cost: 39040.00

?

\[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
↓
\[\frac{r \cdot \sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \cos a\right)} \]
(FPCore (r a b) :precision binary64 (/ (* r (sin b)) (cos (+ a b))))
↓
(FPCore (r a b)
 :precision binary64
 (/ (* r (sin b)) (fma (sin b) (- (sin a)) (* (cos b) (cos 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(sin(b), -sin(a), (cos(b) * cos(a)));
}

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(sin(b), Float64(-sin(a)), Float64(cos(b) * cos(a))))
end

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

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

Error?

Derivation?

  1. Initial program 76.9

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

  2. Applied egg-rr99.5

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

  3. Simplified99.5

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

    [Start]99.5

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

    +-commutative [=>]99.5

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

    *-commutative [=>]99.5

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

    fma-def [=>]99.5

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

    *-commutative [=>]99.5

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

  4. Final simplification99.5

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

Alternatives

Alternative 1
Error99.4%
Cost32704.00
\[\frac{r}{\frac{\cos b \cdot \cos a - \sin b \cdot \sin a}{\sin b}} \]
Alternative 2
Error99.4%
Cost32704.00
\[\frac{\sin b}{\frac{\cos b \cdot \cos a - \sin b \cdot \sin a}{r}} \]
Alternative 3
Error99.5%
Cost32704.00
\[\frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
Alternative 4
Error99.4%
Cost32512.00
\[\frac{r}{\mathsf{fma}\left(\frac{\cos a}{\sin b}, \cos b, -\sin a\right)} \]
Alternative 5
Error99.4%
Cost32512.00
\[\frac{r}{\mathsf{fma}\left(\frac{\cos b}{\sin b}, \cos a, -\sin a\right)} \]
Alternative 6
Error76.7%
Cost13385.00
\[\begin{array}{l} \mathbf{if}\;a \leq -0.000122 \lor \neg \left(a \leq 6.1 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
Alternative 7
Error76.7%
Cost13384.00
\[\begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-7}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 1.1:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
Alternative 8
Error76.9%
Cost13248.00
\[r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]
Alternative 9
Error76.7%
Cost6985.00
\[\begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-7} \lor \neg \left(b \leq 1.1\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
Alternative 10
Error76.7%
Cost6985.00
\[\begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-7} \lor \neg \left(b \leq 1.1\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
Alternative 11
Error38.8%
Cost6592.00
\[r \cdot \sin b \]
Alternative 12
Error60.6%
Cost6592.00
\[r \cdot \tan b \]
Alternative 13
Error35.3%
Cost576.00
\[\frac{r}{b \cdot -0.3333333333333333 + \frac{1}{b}} \]
Alternative 14
Error34.4%
Cost192.00
\[r \cdot b \]

Error

Reproduce?

herbie shell --seed 2023093 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))