?

Average Error: 15.3 → 0.3
Time: 19.8s
Precision: binary64
Cost: 39040

?

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

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

Error?

Derivation?

  1. Initial program 15.3

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

  2. Applied egg-rr0.3

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

  3. Simplified0.3

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

    [Start]0.3

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

    +-commutative [=>]0.3

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

    *-commutative [=>]0.3

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

    fma-def [=>]0.3

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

    *-commutative [=>]0.3

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

  4. Final simplification0.3

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

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
Error15.5
Cost13385
\[\begin{array}{l} \mathbf{if}\;b \leq -0.00085 \lor \neg \left(b \leq 0.00062\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
Alternative 5
Error15.5
Cost13385
\[\begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{-5} \lor \neg \left(a \leq 29\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
Alternative 6
Error15.5
Cost13384
\[\begin{array}{l} \mathbf{if}\;b \leq -0.00085:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;b \leq 0.0007:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
Alternative 7
Error15.5
Cost13384
\[\begin{array}{l} \mathbf{if}\;a \leq -0.000155:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{elif}\;a \leq 29:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \end{array} \]
Alternative 8
Error15.5
Cost13384
\[\begin{array}{l} \mathbf{if}\;a \leq -4.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{elif}\;a \leq 29:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \end{array} \]
Alternative 9
Error15.5
Cost13384
\[\begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;a \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_0}{\cos a}\\ \mathbf{elif}\;a \leq 29:\\ \;\;\;\;\frac{t_0}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \end{array} \]
Alternative 10
Error15.3
Cost13248
\[\sin b \cdot \frac{r}{\cos \left(b + a\right)} \]
Alternative 11
Error15.3
Cost13248
\[r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]
Alternative 12
Error15.3
Cost13248
\[\frac{r \cdot \sin b}{\cos \left(b + a\right)} \]
Alternative 13
Error30.5
Cost6980
\[\begin{array}{l} \mathbf{if}\;b \leq -0.00182:\\ \;\;\;\;\frac{r}{\frac{1}{b} - \sin a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
Alternative 14
Error30.5
Cost6852
\[\begin{array}{l} \mathbf{if}\;b \leq -550000:\\ \;\;\;\;\frac{-r}{\sin a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]
Alternative 15
Error40.9
Cost6788
\[\begin{array}{l} \mathbf{if}\;b \leq -640000:\\ \;\;\;\;\frac{-r}{\sin a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot b\\ \end{array} \]
Alternative 16
Error41.9
Cost192
\[r \cdot b \]

Error

Reproduce?

herbie shell --seed 2023060 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))