Octave 3.8, jcobi/1

Average Error: 16.0 → 0.1

Time: 4.7s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999995:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(0.5, \beta, \mathsf{fma}\left(\beta, -2, -4\right) \cdot -0.25\right)}{\alpha} \cdot \left(0.5 \cdot \frac{\mathsf{fma}\left(\beta, -2, -4\right)}{\alpha} + 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha - \beta}{\mathsf{fma}\left(\beta + \alpha, -2, -4\right)}\right)\right)\\ \end{array} \]

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)) -0.999995)
   (log1p
    (expm1
     (*
      (/ (fma 0.5 beta (* (fma beta -2.0 -4.0) -0.25)) alpha)
      (+ (* 0.5 (/ (fma beta -2.0 -4.0) alpha)) 1.0))))
   (+ 0.5 (expm1 (log1p (/ (- 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)) <= -0.999995) {
		tmp = log1p(expm1(((fma(0.5, beta, (fma(beta, -2.0, -4.0) * -0.25)) / alpha) * ((0.5 * (fma(beta, -2.0, -4.0) / alpha)) + 1.0))));
	} else {
		tmp = 0.5 + expm1(log1p(((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)) <= -0.999995)
		tmp = log1p(expm1(Float64(Float64(fma(0.5, beta, Float64(fma(beta, -2.0, -4.0) * -0.25)) / alpha) * Float64(Float64(0.5 * Float64(fma(beta, -2.0, -4.0) / alpha)) + 1.0))));
	else
		tmp = Float64(0.5 + expm1(log1p(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], -0.999995], N[Log[1 + N[(Exp[N[(N[(N[(0.5 * beta + N[(N[(beta * -2.0 + -4.0), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(0.5 * N[(N[(beta * -2.0 + -4.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], N[(0.5 + N[(Exp[N[Log[1 + N[(N[(alpha - beta), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] * -2.0 + -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99999499999999997

   1. Initial program 59.5

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

   2. Simplified 59.5

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

   3. Taylor expanded in alpha around -inf 3.1

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

   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. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(0.5, \beta, \mathsf{fma}\left(\beta, -2, -4\right) \cdot -0.25\right)}{\alpha} \cdot \left(0.5 \cdot \frac{\mathsf{fma}\left(\beta, -2, -4\right)}{\alpha} + 1\right)\right)\right)} \]

   if -0.99999499999999997 < (/.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. Applied egg-rr 0.1

      \[\leadsto 0.5 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha - \beta}{\mathsf{fma}\left(\alpha + \beta, -2, -4\right)}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999995:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(0.5, \beta, \mathsf{fma}\left(\beta, -2, -4\right) \cdot -0.25\right)}{\alpha} \cdot \left(0.5 \cdot \frac{\mathsf{fma}\left(\beta, -2, -4\right)}{\alpha} + 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha - \beta}{\mathsf{fma}\left(\beta + \alpha, -2, -4\right)}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022181 
(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))