Average Error: 14.6 → 0.3
Time: 13.8s
Precision: binary64
Cost: 39040
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
↓
\[\frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \sin b \cdot \left(-\sin a\right)\right)} \]
(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(Float64(r * sin(b)) / cos(Float64(a + b)))
end
↓
function code(r, a, b)
	return Float64(Float64(r * sin(b)) / fma(cos(b), cos(a), 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[b], $MachinePrecision] * N[Cos[a], $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 b, \cos a, \sin b \cdot \left(-\sin a\right)\right)}

Error

Derivation

  1. Initial program 14.6

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

  2. Simplified14.6

    \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
    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. Final simplification0.3

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

Alternatives

Alternative 1
Error0.3
Cost32704
\[r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
Alternative 2
Error0.3
Cost32704
\[\frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
Alternative 3
Error0.4
Cost32512
\[\frac{r}{\mathsf{fma}\left(\frac{\cos b}{\sin b}, \cos a, -\sin a\right)} \]
Alternative 4
Error14.0
Cost19648
\[\frac{r \cdot \sin b}{\cos b \cdot \cos a} \]
Alternative 5
Error15.5
Cost13384
\[\begin{array}{l} t_0 := \sin b \cdot \frac{r}{\cos a}\\ \mathbf{if}\;a \leq -0.00062:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{+39}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Error15.5
Cost13384
\[\begin{array}{l} \mathbf{if}\;a \leq -0.00055:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{+39}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \end{array} \]
Alternative 7
Error15.5
Cost13384
\[\begin{array}{l} \mathbf{if}\;a \leq -0.00046:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{+39}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \end{array} \]
Alternative 8
Error14.6
Cost13248
\[r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]
Alternative 9
Error14.6
Cost13248
\[\sin b \cdot \frac{r}{\cos \left(b + a\right)} \]
Alternative 10
Error28.8
Cost13120
\[\sin b \cdot \frac{r}{\cos a} \]
Alternative 11
Error28.7
Cost7112
\[\begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -2.4 \cdot 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 3900:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 12
Error28.7
Cost6984
\[\begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -75:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 45000:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 13
Error28.7
Cost6984
\[\begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -80:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 45000:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 14
Error39.0
Cost6592
\[r \cdot \sin b \]
Alternative 15
Error41.9
Cost192
\[r \cdot b \]

Error

Reproduce

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