Octave 3.8, jcobi/2

Average Error: 23.2 → 1.3

Time: 6.9s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha}\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{2 + t_1} \leq -0.9:\\ \;\;\;\;\frac{t_0 + \left(\left(\frac{\beta}{\alpha} - t_0 \cdot \frac{\beta + \mathsf{fma}\left(2, i, \beta\right)}{\alpha}\right) + \left(\frac{\mathsf{fma}\left(2, i, 2\right)}{\alpha} + \frac{\mathsf{fma}\left(i, -2, -2\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(2, \beta + i, 2 + \mathsf{fma}\left(2, i, \beta\right)\right)}{\alpha}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \left(\beta + 2\right)\right)}, \frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)\right)}}{2}\\ \end{array} \]

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 beta) alpha)) (t_1 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ 2.0 t_1)) -0.9)
     (/
      (+
       t_0
       (+
        (- (/ beta alpha) (* t_0 (/ (+ beta (fma 2.0 i beta)) alpha)))
        (+
         (/ (fma 2.0 i 2.0) alpha)
         (*
          (/ (fma i -2.0 -2.0) alpha)
          (/ (fma 2.0 (+ beta i) (+ 2.0 (fma 2.0 i beta))) alpha)))))
      2.0)
     (/
      (exp
       (log
        (fma
         (/ (+ 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, beta) / alpha;
	double t_1 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.9) {
		tmp = (t_0 + (((beta / alpha) - (t_0 * ((beta + fma(2.0, i, beta)) / alpha))) + ((fma(2.0, i, 2.0) / alpha) + ((fma(i, -2.0, -2.0) / alpha) * (fma(2.0, (beta + i), (2.0 + fma(2.0, i, beta))) / alpha))))) / 2.0;
	} else {
		tmp = exp(log(fma(((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, beta) / alpha)
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(2.0 + t_1)) <= -0.9)
		tmp = Float64(Float64(t_0 + Float64(Float64(Float64(beta / alpha) - Float64(t_0 * Float64(Float64(beta + fma(2.0, i, beta)) / alpha))) + Float64(Float64(fma(2.0, i, 2.0) / alpha) + Float64(Float64(fma(i, -2.0, -2.0) / alpha) * Float64(fma(2.0, Float64(beta + i), Float64(2.0 + fma(2.0, i, beta))) / alpha))))) / 2.0);
	else
		tmp = Float64(exp(log(fma(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 + beta), $MachinePrecision] / alpha), $MachinePrecision]}, Block[{t$95$1 = 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$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], -0.9], N[(N[(t$95$0 + N[(N[(N[(beta / alpha), $MachinePrecision] - N[(t$95$0 * N[(N[(beta + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 * i + 2.0), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(N[(i * -2.0 + -2.0), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(2.0 * N[(beta + i), $MachinePrecision] + N[(2.0 + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Exp[N[Log[N[(N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $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)) < -0.900000000000000022

   1. Initial program 61.7

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

   2. Simplified 53.8

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

   3. Taylor expanded in alpha around inf 13.5

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

   4. Simplified 5.6

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

   5. Taylor expanded in alpha around -inf 11.2

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

   6. Simplified 5.6

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

   if -0.900000000000000022 < (/.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 12.0

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

   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

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} + \left(\left(\frac{\beta}{\alpha} - \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{\beta + \mathsf{fma}\left(2, i, \beta\right)}{\alpha}\right) + \left(\frac{\mathsf{fma}\left(2, i, 2\right)}{\alpha} + \frac{\mathsf{fma}\left(i, -2, -2\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(2, \beta + i, 2 + \mathsf{fma}\left(2, i, \beta\right)\right)}{\alpha}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \left(\beta + 2\right)\right)}, \frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)\right)}}{2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022192 
(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))