Octave 3.8, jcobi/1

Average Error: 16.1 → 0.2

Time: 2.9s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.8:\\ \;\;\;\;\frac{\beta}{\alpha} + \left(\left(\mathsf{fma}\left(8, \frac{\beta}{{\alpha}^{3}}, \frac{4}{{\alpha}^{3}}\right) + \mathsf{fma}\left(\frac{\beta}{\alpha \cdot \alpha}, -3, \frac{-2}{\alpha \cdot \alpha}\right)\right) + \frac{1}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\alpha - \beta, \frac{1}{\mathsf{fma}\left(\beta + \alpha, -2, -4\right)}, 0.5\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.8)
   (+
    (/ beta alpha)
    (+
     (+
      (fma 8.0 (/ beta (pow alpha 3.0)) (/ 4.0 (pow alpha 3.0)))
      (fma (/ beta (* alpha alpha)) -3.0 (/ -2.0 (* alpha alpha))))
     (/ 1.0 alpha)))
   (fma (- alpha beta) (/ 1.0 (fma (+ beta alpha) -2.0 -4.0)) 0.5)))

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) / ((beta + alpha) + 2.0)) <= -0.8) {
		tmp = (beta / alpha) + ((fma(8.0, (beta / pow(alpha, 3.0)), (4.0 / pow(alpha, 3.0))) + fma((beta / (alpha * alpha)), -3.0, (-2.0 / (alpha * alpha)))) + (1.0 / alpha));
	} else {
		tmp = fma((alpha - beta), (1.0 / fma((beta + alpha), -2.0, -4.0)), 0.5);
	}
	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(Float64(beta + alpha) + 2.0)) <= -0.8)
		tmp = Float64(Float64(beta / alpha) + Float64(Float64(fma(8.0, Float64(beta / (alpha ^ 3.0)), Float64(4.0 / (alpha ^ 3.0))) + fma(Float64(beta / Float64(alpha * alpha)), -3.0, Float64(-2.0 / Float64(alpha * alpha)))) + Float64(1.0 / alpha)));
	else
		tmp = fma(Float64(alpha - beta), Float64(1.0 / fma(Float64(beta + alpha), -2.0, -4.0)), 0.5);
	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[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.8], N[(N[(beta / alpha), $MachinePrecision] + N[(N[(N[(8.0 * N[(beta / N[Power[alpha, 3.0], $MachinePrecision]), $MachinePrecision] + N[(4.0 / N[Power[alpha, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(beta / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision] * -3.0 + N[(-2.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha - beta), $MachinePrecision] * N[(1.0 / N[(N[(beta + alpha), $MachinePrecision] * -2.0 + -4.0), $MachinePrecision]), $MachinePrecision] + 0.5), $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)) < -0.80000000000000004

   1. Initial program 58.7

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

   2. Simplified 58.7

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

   3. Taylor expanded in alpha around inf 5.7

      \[\leadsto \color{blue}{\left(\frac{{\beta}^{3}}{{\alpha}^{3}} + \left(\frac{1}{\alpha} + \left(4 \cdot \frac{1}{{\alpha}^{3}} + \left(\frac{\beta}{\alpha} + \left(8 \cdot \frac{\beta}{{\alpha}^{3}} + 5 \cdot \frac{{\beta}^{2}}{{\alpha}^{3}}\right)\right)\right)\right)\right) - \left(2 \cdot \frac{1}{{\alpha}^{2}} + \left(3 \cdot \frac{\beta}{{\alpha}^{2}} + \frac{{\beta}^{2}}{{\alpha}^{2}}\right)\right)} \]

   4. Simplified 0.2

      \[\leadsto \color{blue}{\left(\frac{1}{\alpha} + \left(\mathsf{fma}\left(8, \frac{\beta}{{\alpha}^{3}}, \frac{\beta}{\alpha}\right) + \mathsf{fma}\left(5, \frac{\beta}{\frac{{\alpha}^{3}}{\beta}}, \frac{4}{{\alpha}^{3}}\right)\right)\right) + \left({\left(\frac{\beta}{\alpha}\right)}^{3} + \left(\frac{-2}{\alpha \cdot \alpha} - \frac{\beta}{\alpha} \cdot \left(\frac{\beta}{\alpha} + \frac{3}{\alpha}\right)\right)\right)} \]

   5. Taylor expanded in beta around 0 0.6

      \[\leadsto \color{blue}{\left(\frac{1}{\alpha} + \left(\frac{\beta}{\alpha} + \left(8 \cdot \frac{\beta}{{\alpha}^{3}} + 4 \cdot \frac{1}{{\alpha}^{3}}\right)\right)\right) - \left(2 \cdot \frac{1}{{\alpha}^{2}} + 3 \cdot \frac{\beta}{{\alpha}^{2}}\right)} \]

   6. Simplified 0.6

      \[\leadsto \color{blue}{\frac{\beta}{\alpha} + \left(\left(\mathsf{fma}\left(8, \frac{\beta}{{\alpha}^{3}}, \frac{4}{{\alpha}^{3}}\right) + \mathsf{fma}\left(\frac{\beta}{\alpha \cdot \alpha}, -3, \frac{-2}{\alpha \cdot \alpha}\right)\right) + \frac{1}{\alpha}\right)} \]

   if -0.80000000000000004 < (/.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. Applied egg-rr 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha - \beta, \frac{1}{\mathsf{fma}\left(\alpha + \beta, -2, -4\right)}, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.8:\\ \;\;\;\;\frac{\beta}{\alpha} + \left(\left(\mathsf{fma}\left(8, \frac{\beta}{{\alpha}^{3}}, \frac{4}{{\alpha}^{3}}\right) + \mathsf{fma}\left(\frac{\beta}{\alpha \cdot \alpha}, -3, \frac{-2}{\alpha \cdot \alpha}\right)\right) + \frac{1}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\alpha - \beta, \frac{1}{\mathsf{fma}\left(\beta + \alpha, -2, -4\right)}, 0.5\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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))