Average Error: 0.2 → 0.2
Time: 6.3s
Precision: binary64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
↓
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathsf{fma}\left(\frac{1}{\sin B}, 1, \frac{-x}{\tan B}\right) + \mathsf{fma}\left(-1, t_0, t_0\right) \end{array} \]
(FPCore (B x)
 :precision binary64
 (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))
↓
(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (+ (fma (/ 1.0 (sin B)) 1.0 (/ (- x) (tan B))) (fma -1.0 t_0 t_0))))
double code(double B, double x) {
	return -(x * (1.0 / tan(B))) + (1.0 / sin(B));
}
↓
double code(double B, double x) {
	double t_0 = x / tan(B);
	return fma((1.0 / sin(B)), 1.0, (-x / tan(B))) + fma(-1.0, t_0, t_0);
}

function code(B, x)
	return Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(1.0 / sin(B)))
end
↓
function code(B, x)
	t_0 = Float64(x / tan(B))
	return Float64(fma(Float64(1.0 / sin(B)), 1.0, Float64(Float64(-x) / tan(B))) + fma(-1.0, t_0, t_0))
end

code[B_, x_] := N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * 1.0 + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * t$95$0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\begin{array}{l}
t_0 := \frac{x}{\tan B}\\
\mathsf{fma}\left(\frac{1}{\sin B}, 1, \frac{-x}{\tan B}\right) + \mathsf{fma}\left(-1, t_0, t_0\right)
\end{array}

Error

Bits error versus B

Bits error versus x

Derivation

  1. Initial program 0.2

    \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]

  2. Simplified0.2

    \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]

  3. Taylor expanded in x around 0 0.2

    \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

  4. Applied egg-rr0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, 1, \frac{-x}{\tan B}\right) + \mathsf{fma}\left(-1, \frac{x}{\tan B}, \frac{x}{\tan B}\right)} \]

  5. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, 1, \frac{-x}{\tan B}\right) + \mathsf{fma}\left(-1, \frac{x}{\tan B}, \frac{x}{\tan B}\right) \]

Reproduce

herbie shell --seed 2022160 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))