Average Error: 14.4 → 0.3
Time: 15.7s
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 14.4

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

  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
Cost39040
\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \sin b \cdot \left(-\sin a\right)\right)} \]
Alternative 2
Error0.3
Cost32704
\[r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
Alternative 3
Error0.4
Cost32512
\[\frac{r}{\mathsf{fma}\left(\frac{\cos b}{\sin b}, \cos a, -\sin a\right)} \]
Alternative 4
Error14.5
Cost13385
\[\begin{array}{l} \mathbf{if}\;b \leq -0.0039 \lor \neg \left(b \leq 0.0012\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
Alternative 5
Error14.6
Cost13385
\[\begin{array}{l} \mathbf{if}\;b \leq -0.00032 \lor \neg \left(b \leq 112\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]
Alternative 6
Error14.6
Cost13384
\[\begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;b \leq 112:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \end{array} \]
Alternative 7
Error14.6
Cost13384
\[\begin{array}{l} \mathbf{if}\;b \leq -0.00032:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \mathbf{elif}\;b \leq 112:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \end{array} \]
Alternative 8
Error14.6
Cost13384
\[\begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{elif}\;b \leq 112:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \end{array} \]
Alternative 9
Error14.4
Cost13248
\[r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]
Alternative 10
Error14.4
Cost13248
\[\frac{r \cdot \sin b}{\cos \left(b + a\right)} \]
Alternative 11
Error14.4
Cost13248
\[\frac{r \cdot \sin b}{\cos \left(b - a\right)} \]
Alternative 12
Error29.9
Cost6852
\[\begin{array}{l} \mathbf{if}\;b \leq 1.32:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \sin b\\ \end{array} \]
Alternative 13
Error29.9
Cost6852
\[\begin{array}{l} \mathbf{if}\;b \leq 0.69:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \sin b\\ \end{array} \]
Alternative 14
Error39.0
Cost6592
\[r \cdot \sin b \]
Alternative 15
Error41.3
Cost576
\[\frac{r}{b \cdot -0.3333333333333333 + \frac{1}{b}} \]
Alternative 16
Error41.8
Cost192
\[r \cdot b \]

Error

Reproduce

herbie shell --seed 2023012 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))