Octave 3.8, jcobi/2

Average Error: 24.1 → 1.3

Time: 5.9s

Precision: binary64

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 0\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\mathsf{fma}\left(2, i, 2\right)}{\alpha}\\ t_1 := \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha}\\ t_2 := \frac{\beta + \mathsf{fma}\left(2, i, \beta\right)}{\alpha}\\ t_3 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_3}}{2 + t_3} \leq -0.5:\\ \;\;\;\;\frac{t_1 + \left(\left(\frac{\beta}{\alpha} - t_1 \cdot t_2\right) + \left(\mathsf{fma}\left(2, \frac{i}{\alpha}, \frac{2}{\alpha} - t_2 \cdot t_0\right) - t_0 \cdot \frac{\beta + \mathsf{fma}\left(2, i, 2\right)}{\alpha}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \left(\beta + 2\right)\right)}\right)\right), \frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (fma 2.0 i 2.0) alpha))
        (t_1 (/ (fma 2.0 i beta) alpha))
        (t_2 (/ (+ beta (fma 2.0 i beta)) alpha))
        (t_3 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_3) (+ 2.0 t_3)) -0.5)
     (/
      (+
       t_1
       (+
        (- (/ beta alpha) (* t_1 t_2))
        (-
         (fma 2.0 (/ i alpha) (- (/ 2.0 alpha) (* t_2 t_0)))
         (* t_0 (/ (+ beta (fma 2.0 i 2.0)) alpha)))))
      2.0)
     (/
      (fma
       (expm1 (log1p (/ (+ alpha beta) (fma 2.0 i (+ alpha (+ beta 2.0))))))
       (/ (- beta alpha) (+ alpha (fma 2.0 i beta)))
       1.0)
      2.0))))

C

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

↓

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, 2.0) / alpha;
	double t_1 = fma(2.0, i, beta) / alpha;
	double t_2 = (beta + fma(2.0, i, beta)) / alpha;
	double t_3 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_3) / (2.0 + t_3)) <= -0.5) {
		tmp = (t_1 + (((beta / alpha) - (t_1 * t_2)) + (fma(2.0, (i / alpha), ((2.0 / alpha) - (t_2 * t_0))) - (t_0 * ((beta + fma(2.0, i, 2.0)) / alpha))))) / 2.0;
	} else {
		tmp = fma(expm1(log1p(((alpha + beta) / fma(2.0, i, (alpha + (beta + 2.0)))))), ((beta - alpha) / (alpha + fma(2.0, i, beta))), 1.0) / 2.0;
	}
	return tmp;
}

Julia

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

↓

function code(alpha, beta, i)
	t_0 = Float64(fma(2.0, i, 2.0) / alpha)
	t_1 = Float64(fma(2.0, i, beta) / alpha)
	t_2 = Float64(Float64(beta + fma(2.0, i, beta)) / alpha)
	t_3 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_3) / Float64(2.0 + t_3)) <= -0.5)
		tmp = Float64(Float64(t_1 + Float64(Float64(Float64(beta / alpha) - Float64(t_1 * t_2)) + Float64(fma(2.0, Float64(i / alpha), Float64(Float64(2.0 / alpha) - Float64(t_2 * t_0))) - Float64(t_0 * Float64(Float64(beta + fma(2.0, i, 2.0)) / alpha))))) / 2.0);
	else
		tmp = Float64(fma(expm1(log1p(Float64(Float64(alpha + beta) / fma(2.0, i, Float64(alpha + Float64(beta + 2.0)))))), Float64(Float64(beta - alpha) / Float64(alpha + fma(2.0, i, beta))), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

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

↓

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(2.0 * i + 2.0), $MachinePrecision] / alpha), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * i + beta), $MachinePrecision] / alpha), $MachinePrecision]}, Block[{t$95$2 = N[(N[(beta + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]}, Block[{t$95$3 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] / N[(2.0 + t$95$3), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(t$95$1 + N[(N[(N[(beta / alpha), $MachinePrecision] - N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(i / alpha), $MachinePrecision] + N[(N[(2.0 / alpha), $MachinePrecision] - N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[(N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(Exp[N[Log[1 + N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 61.5

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

   2. Simplified 53.7

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

   3. Applied egg-rr 53.8

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

   4. Taylor expanded in alpha around inf 13.7

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

   5. Simplified 5.7

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

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

   1. Initial program 12.5

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

   2. Simplified 0.0

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

   3. Applied egg-rr 0.0

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \left(\beta + 2\right)\right)}\right)\right)}, \frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.3

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

Reproduce

herbie shell --seed 2022210 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))