?

Average Error: 23.52% → 0.49%
Time: 18.8s
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 23.52

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

  2. Simplified23.52

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

    [Start]23.52

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

    +-commutative [=>]23.52

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

  3. Applied egg-rr0.49

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

  4. Final simplification0.49

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

Alternatives

Alternative 1
Error0.5%
Cost32704
\[r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
Alternative 2
Error22.5%
Cost19648
\[r \cdot \frac{\sin b}{\cos b \cdot \cos a} \]
Alternative 3
Error23.6%
Cost13385
\[\begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{-5} \lor \neg \left(a \leq 24000\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \end{array} \]
Alternative 4
Error23.63%
Cost13384
\[\begin{array}{l} \mathbf{if}\;a \leq -0.0003:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 24000:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]
Alternative 5
Error23.72%
Cost13376
\[r \cdot \left(\sin b \cdot \frac{1}{\cos \left(b - a\right)}\right) \]
Alternative 6
Error23.52%
Cost13248
\[r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]
Alternative 7
Error23.54%
Cost13248
\[\sin b \cdot \frac{r}{\cos \left(b + a\right)} \]
Alternative 8
Error25.06%
Cost7369
\[\begin{array}{l} \mathbf{if}\;b \leq -0.0095 \lor \neg \left(b \leq 2.9 \cdot 10^{-8}\right):\\ \;\;\;\;r \cdot \frac{r}{\frac{r}{\tan b} - r \cdot a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(b + a\right)}\\ \end{array} \]
Alternative 9
Error44.72%
Cost7113
\[\begin{array}{l} \mathbf{if}\;b \leq -2.7 \lor \neg \left(b \leq 4.2\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(b + a\right)}\\ \end{array} \]
Alternative 10
Error24.98%
Cost7113
\[\begin{array}{l} \mathbf{if}\;b \leq -0.0095 \lor \neg \left(b \leq 2.9 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{r}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(b + a\right)}\\ \end{array} \]
Alternative 11
Error24.98%
Cost7112
\[\begin{array}{l} t_0 := \frac{1}{\tan b} - a\\ \mathbf{if}\;b \leq -0.0095:\\ \;\;\;\;r \cdot \frac{1}{t_0}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-8}:\\ \;\;\;\;r \cdot \frac{b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{t_0}\\ \end{array} \]
Alternative 12
Error44.74%
Cost6985
\[\begin{array}{l} \mathbf{if}\;b \leq -1.7 \lor \neg \left(b \leq 4.8\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
Alternative 13
Error44.76%
Cost6985
\[\begin{array}{l} \mathbf{if}\;b \leq -1.3 \lor \neg \left(b \leq 4.8\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
Alternative 14
Error61.21%
Cost6592
\[r \cdot \sin b \]
Alternative 15
Error64.82%
Cost704
\[\frac{b}{\frac{1 + -0.5 \cdot \left(b \cdot b\right)}{r}} \]
Alternative 16
Error65.5%
Cost192
\[r \cdot b \]

Error

Reproduce?

herbie shell --seed 2023121 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))