Octave 3.8, jcobi/1

Average Error: 15.8 → 0.2

Time: 2.8s

Precision: binary64

\[\alpha > -1 \land \beta > -1\]

Math

\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha} + \left(\mathsf{fma}\left(-2, {\alpha}^{-2}, \frac{\beta}{\alpha}\right) - \frac{\beta}{\alpha} \cdot \left(\frac{\beta}{\alpha} + \frac{3}{\alpha}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{\alpha - \beta}{\mathsf{fma}\left(\beta + \alpha, -2, -4\right)}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))

↓

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ 2.0 (+ beta alpha))) -0.5)
   (+
    (/ 1.0 alpha)
    (-
     (fma -2.0 (pow alpha -2.0) (/ beta alpha))
     (* (/ beta alpha) (+ (/ beta alpha) (/ 3.0 alpha)))))
   (+ 0.5 (/ (- alpha beta) (fma (+ beta alpha) -2.0 -4.0)))))

C

double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}

↓

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / (2.0 + (beta + alpha))) <= -0.5) {
		tmp = (1.0 / alpha) + (fma(-2.0, pow(alpha, -2.0), (beta / alpha)) - ((beta / alpha) * ((beta / alpha) + (3.0 / alpha))));
	} else {
		tmp = 0.5 + ((alpha - beta) / fma((beta + alpha), -2.0, -4.0));
	}
	return tmp;
}

Julia

function code(alpha, beta)
	return Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
end

↓

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha))) <= -0.5)
		tmp = Float64(Float64(1.0 / alpha) + Float64(fma(-2.0, (alpha ^ -2.0), Float64(beta / alpha)) - Float64(Float64(beta / alpha) * Float64(Float64(beta / alpha) + Float64(3.0 / alpha)))));
	else
		tmp = Float64(0.5 + Float64(Float64(alpha - beta) / fma(Float64(beta + alpha), -2.0, -4.0)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]

↓

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(1.0 / alpha), $MachinePrecision] + N[(N[(-2.0 * N[Power[alpha, -2.0], $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision] - N[(N[(beta / alpha), $MachinePrecision] * N[(N[(beta / alpha), $MachinePrecision] + N[(3.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(N[(alpha - beta), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] * -2.0 + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.5

   1. Initial program 58.4

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]

   2. Simplified 58.4

      \[\leadsto \color{blue}{0.5 + \frac{\alpha - \beta}{\mathsf{fma}\left(\beta + \alpha, -2, -4\right)}} \]

   3. Taylor expanded in alpha around inf 4.2

      \[\leadsto \color{blue}{\left(\frac{1}{\alpha} + \frac{\beta}{\alpha}\right) - \left(2 \cdot \frac{1}{{\alpha}^{2}} + \left(3 \cdot \frac{\beta}{{\alpha}^{2}} + \frac{{\beta}^{2}}{{\alpha}^{2}}\right)\right)} \]

   4. Simplified 0.7

      \[\leadsto \color{blue}{\frac{1}{\alpha} + \left(\left(\frac{\beta}{\alpha} + \frac{-2}{\alpha \cdot \alpha}\right) - \frac{\beta}{\alpha} \cdot \left(\frac{\beta}{\alpha} + \frac{3}{\alpha}\right)\right)} \]

   5. Applied egg-rr 0.7

      \[\leadsto \frac{1}{\alpha} + \left(\color{blue}{\mathsf{fma}\left(-2, {\alpha}^{-2}, \frac{\beta}{\alpha}\right)} - \frac{\beta}{\alpha} \cdot \left(\frac{\beta}{\alpha} + \frac{3}{\alpha}\right)\right) \]

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.0

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]

   2. Simplified 0.0

      \[\leadsto \color{blue}{0.5 + \frac{\alpha - \beta}{\mathsf{fma}\left(\beta + \alpha, -2, -4\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha} + \left(\mathsf{fma}\left(-2, {\alpha}^{-2}, \frac{\beta}{\alpha}\right) - \frac{\beta}{\alpha} \cdot \left(\frac{\beta}{\alpha} + \frac{3}{\alpha}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{\alpha - \beta}{\mathsf{fma}\left(\beta + \alpha, -2, -4\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))