Average Error: 15.1 → 0.3
Time: 14.4s
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 15.1

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

  2. Simplified15.1

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

    [Start]15.1

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

    +-commutative [=>]15.1

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

  3. Applied egg-rr0.3

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

  4. Final simplification0.3

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

Alternatives

Alternative 1
Error0.3
Cost32704
\[r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
Alternative 2
Error15.5
Cost13385
\[\begin{array}{l} \mathbf{if}\;b \leq -1750 \lor \neg \left(b \leq 2.75 \cdot 10^{-11}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\cos \left(b + a\right)} \cdot \left(r \cdot b\right)\\ \end{array} \]
Alternative 3
Error15.5
Cost13385
\[\begin{array}{l} \mathbf{if}\;b \leq -1750 \lor \neg \left(b \leq 2.75 \cdot 10^{-11}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\cos \left(b + a\right)} \cdot \left(r \cdot b\right)\\ \end{array} \]
Alternative 4
Error15.5
Cost13384
\[\begin{array}{l} \mathbf{if}\;b \leq -1750:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 2.75 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\cos \left(b + a\right)} \cdot \left(r \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
Alternative 5
Error15.1
Cost13248
\[r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]
Alternative 6
Error29.0
Cost13120
\[r \cdot \frac{\sin b}{\cos a} \]
Alternative 7
Error28.9
Cost6985
\[\begin{array}{l} \mathbf{if}\;b \leq -2.05 \lor \neg \left(b \leq 22000000000000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
Alternative 8
Error28.9
Cost6985
\[\begin{array}{l} \mathbf{if}\;b \leq -0.88 \lor \neg \left(b \leq 22000000000000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
Alternative 9
Error39.4
Cost6592
\[r \cdot \sin b \]
Alternative 10
Error42.3
Cost192
\[r \cdot b \]

Error

Reproduce

herbie shell --seed 2022356 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))