VandenBroeck and Keller, Equation (24)

Average Error: 0.2 → 0.2

Time: 5.9s

Precision: binary64

Math

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

↓

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

FPCore

(FPCore (B x)
 :precision binary64
 (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))

↓

(FPCore (B x) :precision binary64 (fma 1.0 (/ 1.0 (sin B)) (/ (- x) (tan B))))

C

double code(double B, double x) {
	return -(x * (1.0 / tan(B))) + (1.0 / sin(B));
}

↓

double code(double B, double x) {
	return fma(1.0, (1.0 / sin(B)), (-x / tan(B)));
}

Julia

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

↓

function code(B, x)
	return fma(1.0, Float64(1.0 / sin(B)), Float64(Float64(-x) / tan(B)))
end

Wolfram

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_] := N[(1.0 * N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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. Simplified 0.2

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

3. Applied egg-rr 0.2

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

4. Applied egg-rr 0.2

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

5. Applied egg-rr 0.2

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

6. Final simplification 0.2

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

Reproduce

herbie shell --seed 2022178 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))