Octave 3.8, jcobi/1

Average Error: 15.9 → 0.1

Time: 4.9s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.99998:\\ \;\;\;\;\frac{1}{\alpha} + \left(\left(\frac{\beta}{\alpha} + \frac{\frac{\beta}{\alpha}}{\alpha} \cdot \left(-3 - \beta\right)\right) + \frac{-2}{\alpha \cdot \alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{\alpha - \beta}{\mathsf{fma}\left(\beta + \alpha, -2, -4\right)}\\ \end{array} \]

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))) -0.99998)
   (+
    (/ 1.0 alpha)
    (+
     (+ (/ beta alpha) (* (/ (/ beta alpha) alpha) (- -3.0 beta)))
     (/ -2.0 (* alpha 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))) <= -0.99998) {
		tmp = (1.0 / alpha) + (((beta / alpha) + (((beta / alpha) / alpha) * (-3.0 - beta))) + (-2.0 / (alpha * 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))) <= -0.99998)
		tmp = Float64(Float64(1.0 / alpha) + Float64(Float64(Float64(beta / alpha) + Float64(Float64(Float64(beta / alpha) / alpha) * Float64(-3.0 - beta))) + Float64(-2.0 / Float64(alpha * 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], -0.99998], N[(N[(1.0 / alpha), $MachinePrecision] + N[(N[(N[(beta / alpha), $MachinePrecision] + N[(N[(N[(beta / alpha), $MachinePrecision] / alpha), $MachinePrecision] * N[(-3.0 - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(N[(alpha - beta), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] * -2.0 + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99997999999999998

   1. Initial program 59.2

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

   2. Simplified 59.2

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

   3. Taylor expanded in alpha around -inf 2.8

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{\beta}{\alpha} + 0.5 \cdot \frac{\left(0.5 \cdot \beta - 0.25 \cdot \left(-2 \cdot \beta - 4\right)\right) \cdot \left(-2 \cdot \beta - 4\right)}{{\alpha}^{2}}\right) - 0.25 \cdot \frac{-2 \cdot \beta - 4}{\alpha}} \]

   4. Simplified 0.1

      \[\leadsto \color{blue}{0.5 \cdot \left(\frac{\mathsf{fma}\left(\beta, -2, -4\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(0.5, \beta, \mathsf{fma}\left(\beta, -2, -4\right) \cdot -0.25\right)}{\alpha}\right) + \frac{\mathsf{fma}\left(0.5, \beta, \mathsf{fma}\left(\beta, -2, -4\right) \cdot -0.25\right)}{\alpha}} \]

   5. Taylor expanded in beta around 0 2.8

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

   6. Simplified 0.1

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

   if -0.99997999999999998 < (/.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 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.99998:\\ \;\;\;\;\frac{1}{\alpha} + \left(\left(\frac{\beta}{\alpha} + \frac{\frac{\beta}{\alpha}}{\alpha} \cdot \left(-3 - \beta\right)\right) + \frac{-2}{\alpha \cdot \alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{\alpha - \beta}{\mathsf{fma}\left(\beta + \alpha, -2, -4\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022211 
(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))