?

Average Error: 24.15% → 0.52%
Time: 17.6s
Precision: binary64
Cost: 39040

?

\[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
↓
\[\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos a, \cos b, \sin b \cdot \left(-\sin a\right)\right)} \]
(FPCore (r a b) :precision binary64 (/ (* r (sin b)) (cos (+ a b))))
↓
(FPCore (r a b)
 :precision binary64
 (/ (* r (sin b)) (fma (cos a) (cos b) (* (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(a), cos(b), (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)
	return Float64(Float64(r * sin(b)) / fma(cos(a), cos(b), 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_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision] + N[(N[Sin[b], $MachinePrecision] * (-N[Sin[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error?

Derivation?

  1. Initial program 24.15

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

  2. Applied egg-rr0.52

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

  3. Final simplification0.52

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

Alternatives

Alternative 1
Error0.53%
Cost32704
\[\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a} \]
Alternative 2
Error0.59%
Cost32512
\[\frac{r}{\mathsf{fma}\left(\frac{\cos b}{\sin b}, \cos a, -\sin a\right)} \]
Alternative 3
Error23.18%
Cost26432
\[\frac{r \cdot \sin b}{\left(\sin b \cdot \sin a\right) \cdot -2 + \cos \left(a - b\right)} \]
Alternative 4
Error22.81%
Cost13513
\[\begin{array}{l} \mathbf{if}\;a \leq -118 \lor \neg \left(a \leq 0.00031\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{b} - \sin a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]
Alternative 5
Error22.64%
Cost13513
\[\begin{array}{l} \mathbf{if}\;a \leq -118 \lor \neg \left(a \leq 0.003\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{b} - \sin a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b} - a}\\ \end{array} \]
Alternative 6
Error24.62%
Cost13512
\[\begin{array}{l} \mathbf{if}\;a \leq -0.018:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 1500:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot \sin b\right) \cdot \frac{1}{\cos a}\\ \end{array} \]
Alternative 7
Error24.59%
Cost13385
\[\begin{array}{l} \mathbf{if}\;a \leq -0.00094 \lor \neg \left(a \leq 1500\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]
Alternative 8
Error24.22%
Cost13248
\[\frac{r}{\frac{\cos \left(b + a\right)}{\sin b}} \]
Alternative 9
Error24.15%
Cost13248
\[\frac{r \cdot \sin b}{\cos \left(b + a\right)} \]
Alternative 10
Error45.9%
Cost13120
\[\sin b \cdot \frac{r}{\cos a} \]
Alternative 11
Error45.73%
Cost7241
\[\begin{array}{l} \mathbf{if}\;b \leq -1400000 \lor \neg \left(b \leq 250000000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\cos \left(b - a\right)} \cdot \left(r \cdot b\right)\\ \end{array} \]
Alternative 12
Error45.71%
Cost7113
\[\begin{array}{l} \mathbf{if}\;b \leq -1300000 \lor \neg \left(b \leq 250000000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b - a\right)}\\ \end{array} \]
Alternative 13
Error45.7%
Cost6985
\[\begin{array}{l} \mathbf{if}\;b \leq -0.96 \lor \neg \left(b \leq 850000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
Alternative 14
Error45.7%
Cost6985
\[\begin{array}{l} \mathbf{if}\;b \leq -1.66 \lor \neg \left(b \leq 850000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
Alternative 15
Error61.36%
Cost6592
\[r \cdot \sin b \]
Alternative 16
Error65.84%
Cost192
\[r \cdot b \]

Error

Reproduce?

herbie shell --seed 2023090 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))