Octave 3.8, jcobi/1

Average Error: 15.7 → 0.1

Time: 2.5s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999496553756:\\ \;\;\;\;\left(\frac{1}{\alpha} + \left(\mathsf{fma}\left(8, \frac{\beta}{{\alpha}^{3}}, \frac{\beta}{\alpha}\right) + \mathsf{fma}\left(5, \frac{\beta}{\frac{{\alpha}^{3}}{\beta}}, \frac{4}{{\alpha}^{3}}\right)\right)\right) + \left({\left(\frac{\beta}{\alpha}\right)}^{3} + \left(\frac{-2}{\alpha \cdot \alpha} - \frac{\beta}{\alpha} \cdot \left(\frac{\beta}{\alpha} + \frac{3}{\alpha}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{\alpha - \beta}{\mathsf{fma}\left(\beta + \alpha, -2, -4\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9999999496553756)
   (+
    (+
     (/ 1.0 alpha)
     (+
      (fma 8.0 (/ beta (pow alpha 3.0)) (/ beta alpha))
      (fma 5.0 (/ beta (/ (pow alpha 3.0) beta)) (/ 4.0 (pow alpha 3.0)))))
    (+
     (pow (/ beta alpha) 3.0)
     (-
      (/ -2.0 (* alpha 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) / ((beta + alpha) + 2.0)) <= -0.9999999496553756) {
		tmp = ((1.0 / alpha) + (fma(8.0, (beta / pow(alpha, 3.0)), (beta / alpha)) + fma(5.0, (beta / (pow(alpha, 3.0) / beta)), (4.0 / pow(alpha, 3.0))))) + (pow((beta / alpha), 3.0) + ((-2.0 / (alpha * 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(Float64(beta + alpha) + 2.0)) <= -0.9999999496553756)
		tmp = Float64(Float64(Float64(1.0 / alpha) + Float64(fma(8.0, Float64(beta / (alpha ^ 3.0)), Float64(beta / alpha)) + fma(5.0, Float64(beta / Float64((alpha ^ 3.0) / beta)), Float64(4.0 / (alpha ^ 3.0))))) + Float64((Float64(beta / alpha) ^ 3.0) + Float64(Float64(-2.0 / Float64(alpha * 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[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999999496553756], N[(N[(N[(1.0 / alpha), $MachinePrecision] + N[(N[(8.0 * N[(beta / N[Power[alpha, 3.0], $MachinePrecision]), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(5.0 * N[(beta / N[(N[Power[alpha, 3.0], $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] + N[(4.0 / N[Power[alpha, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(beta / alpha), $MachinePrecision], 3.0], $MachinePrecision] + N[(N[(-2.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision] - N[(N[(beta / alpha), $MachinePrecision] * N[(N[(beta / alpha), $MachinePrecision] + N[(3.0 / alpha), $MachinePrecision]), $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)) < -0.999999949655375642

   1. Initial program 59.7

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

   2. Simplified 59.7

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

   3. Taylor expanded in alpha around inf 4.9

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

   4. Simplified 0.0

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

   if -0.999999949655375642 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.1

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

   2. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999496553756:\\ \;\;\;\;\left(\frac{1}{\alpha} + \left(\mathsf{fma}\left(8, \frac{\beta}{{\alpha}^{3}}, \frac{\beta}{\alpha}\right) + \mathsf{fma}\left(5, \frac{\beta}{\frac{{\alpha}^{3}}{\beta}}, \frac{4}{{\alpha}^{3}}\right)\right)\right) + \left({\left(\frac{\beta}{\alpha}\right)}^{3} + \left(\frac{-2}{\alpha \cdot \alpha} - \frac{\beta}{\alpha} \cdot \left(\frac{\beta}{\alpha} + \frac{3}{\alpha}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{\alpha - \beta}{\mathsf{fma}\left(\beta + \alpha, -2, -4\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022148 
(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))