Average Error: 14.9 → 0.4
Time: 18.2s
Precision: binary64
Cost: 71552
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
↓
\[\begin{array}{l} t_0 := \sin b \cdot \sin a\\ \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, t_0\right) + \mathsf{fma}\left(t_0, -1, \sin b \cdot \left(-\sin a\right)\right)} \end{array} \]
(FPCore (r a b) :precision binary64 (/ (* r (sin b)) (cos (+ a b))))
↓
(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (sin a))))
   (/
    (* r (sin b))
    (+ (fma (cos b) (cos a) t_0) (fma t_0 -1.0 (* (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) {
	double t_0 = sin(b) * sin(a);
	return (r * sin(b)) / (fma(cos(b), cos(a), t_0) + fma(t_0, -1.0, (sin(b) * -sin(a))));
}

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / cos(Float64(a + b)))
end
↓
function code(r, a, b)
	t_0 = Float64(sin(b) * sin(a))
	return Float64(Float64(r * sin(b)) / Float64(fma(cos(b), cos(a), t_0) + fma(t_0, -1.0, Float64(sin(b) * Float64(-sin(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_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]}, N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + t$95$0), $MachinePrecision] + N[(t$95$0 * -1.0 + N[(N[Sin[b], $MachinePrecision] * (-N[Sin[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\begin{array}{l}
t_0 := \sin b \cdot \sin a\\
\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, t_0\right) + \mathsf{fma}\left(t_0, -1, \sin b \cdot \left(-\sin a\right)\right)}
\end{array}

Error

Derivation

  1. Initial program 14.9

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

  2. Simplified14.9

    \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
    Proof
    (/.f64 (*.f64 r (sin.f64 b)) (cos.f64 (+.f64 b a))): 0 points increase in error, 0 points decrease in error
    (/.f64 (*.f64 r (sin.f64 b)) (cos.f64 (Rewrite<= +-commutative_binary64 (+.f64 a b)))): 0 points increase in error, 0 points decrease in error

  3. Applied egg-rr0.3

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

  4. Applied egg-rr0.4

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, -\left(\sin b \cdot \sin a\right) \cdot -1\right) + \mathsf{fma}\left(\sin b \cdot \sin a, -1, \left(\sin b \cdot \sin a\right) \cdot -1\right)}} \]

  5. Final simplification0.4

    \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \sin b \cdot \sin a\right) + \mathsf{fma}\left(\sin b \cdot \sin a, -1, \sin b \cdot \left(-\sin a\right)\right)} \]

Alternatives

Alternative 1
Error0.3
Cost39040
\[\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \sin b \cdot \left(-\sin a\right)\right)} \]
Alternative 2
Error0.3
Cost32704
\[r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
Alternative 3
Error0.3
Cost32704
\[\sin b \cdot \frac{r}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
Alternative 4
Error14.3
Cost26432
\[\frac{r \cdot \sin b}{\cos \left(b - a\right) + \left(\sin b \cdot \sin a\right) \cdot -2} \]
Alternative 5
Error15.7
Cost13384
\[\begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -126.89622588844362:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 2.8539410174957196 \cdot 10^{+26}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Error14.9
Cost13248
\[\sin b \cdot \frac{r}{\cos \left(b + a\right)} \]
Alternative 7
Error14.9
Cost13248
\[\frac{r \cdot \sin b}{\cos \left(b + a\right)} \]
Alternative 8
Error15.3
Cost6984
\[\begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -126.89622588844362:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5.3465084723879826 \cdot 10^{-15}:\\ \;\;\;\;\frac{b}{\frac{\cos a}{r}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 9
Error15.2
Cost6984
\[\begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -126.89622588844362:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5.3465084723879826 \cdot 10^{-15}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 10
Error15.2
Cost6984
\[\begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -126.89622588844362:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5.3465084723879826 \cdot 10^{-15}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 11
Error25.4
Cost6592
\[r \cdot \tan b \]
Alternative 12
Error41.7
Cost192
\[r \cdot b \]

Error

Reproduce

herbie shell --seed 2022294 
(FPCore (r a b)
  :name "rsin A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))