Average Error: 14.6 → 0.3
Time: 14.3s
Precision: binary64
Cost: 39040
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
↓
\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)} \]
(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(Float64(-sin(b)) * 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, \left(-\sin b\right) \cdot \sin a\right)}

Error

Derivation

  1. Initial program 14.6

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

  2. Simplified14.6

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

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

  4. Final simplification0.3

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

Alternatives

Alternative 1
Error0.4
Cost19648
\[\frac{r}{\frac{\cos a}{\tan b} - \sin a} \]
Alternative 2
Error15.5
Cost13384
\[\begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos a}\\ \mathbf{if}\;a \leq -0.00062:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{+39}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 3
Error15.5
Cost13384
\[\begin{array}{l} t_0 := \frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{if}\;a \leq -0.00055:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{+39}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 4
Error15.5
Cost13384
\[\begin{array}{l} \mathbf{if}\;a \leq -0.00046:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{+39}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \end{array} \]
Alternative 5
Error14.6
Cost13248
\[r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]
Alternative 6
Error14.6
Cost13248
\[\sin b \cdot \frac{r}{\cos \left(b + a\right)} \]
Alternative 7
Error14.7
Cost6984
\[\begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -1.55 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 6.3 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 8
Error14.7
Cost6984
\[\begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -1.55 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-6}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 9
Error14.7
Cost6984
\[\begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{-7}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b}}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-6}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
Alternative 10
Error25.0
Cost6592
\[r \cdot \tan b \]
Alternative 11
Error41.9
Cost192
\[r \cdot b \]

Error

Reproduce

herbie shell --seed 2022329 
(FPCore (r a b)
  :name "rsin B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))