?

Average Error: 23.16% → 0.5%
Time: 30.9s
Precision: binary64

?

\[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
↓
\[\frac{r}{1} \cdot \frac{-\sin b}{\mathsf{fma}\left(\sin a, \sin b, -\cos b \cdot \cos a\right)} \]
(FPCore (r a b) :precision binary64 (* r (/ (sin b) (cos (+ a b)))))
↓
(FPCore (r a b)
 :precision binary64
 (* (/ r 1.0) (/ (- (sin b)) (fma (sin a) (sin b) (- (* (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 / 1.0) * (-sin(b) / fma(sin(a), sin(b), -(cos(b) * cos(a))));
}

function code(r, a, b)
	return Float64(r * Float64(sin(b) / cos(Float64(a + b))))
end
↓
function code(r, a, b)
	return Float64(Float64(r / 1.0) * Float64(Float64(-sin(b)) / fma(sin(a), sin(b), Float64(-Float64(cos(b) * cos(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[(N[(r / 1.0), $MachinePrecision] * N[((-N[Sin[b], $MachinePrecision]) / N[(N[Sin[a], $MachinePrecision] * N[Sin[b], $MachinePrecision] + (-N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r}{1} \cdot \frac{-\sin b}{\mathsf{fma}\left(\sin a, \sin b, -\cos b \cdot \cos a\right)}

Error?

Derivation?

  1. Initial program 23.16

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

  2. Simplified23.16

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

  3. Applied egg-rr0.52

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

  4. Simplified0.5

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

  5. Applied egg-rr0.5

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

  6. Applied egg-rr0.53

    \[\leadsto \frac{\left(-\sin b\right) \cdot r}{\color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\mathsf{fma}\left(\sin a, \sin b, -\cos b \cdot \cos a\right) > 0:\\ \;\;\;\;e^{\log \left(\mathsf{fma}\left(\sin a, \sin b, -\cos b \cdot \cos a\right)\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin a, \sin b, -\cos b \cdot \cos a\right)\\ } \end{array}}} \]

  7. Applied egg-rr0.5

    \[\leadsto \color{blue}{\frac{r}{1} \cdot \frac{-\sin b}{\mathsf{fma}\left(\sin a, \sin b, -\cos b \cdot \cos a\right)}} \]

Reproduce?

herbie shell --seed 2023136 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))