Octave 3.8, jcobi/2

Average Accuracy: 63.4% → 98.2%

Time: 37.7s

Precision: binary64

Cost: 76548

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\\ t_1 := {\left(\mathsf{fma}\left(\beta - \alpha, t_0, 1\right)\right)}^{2}\\ t_2 := 2 + \mathsf{fma}\left(2, i, \beta\right)\\ t_3 := \frac{\beta + t_2}{\alpha}\\ t_4 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_4}}{2 + t_4} \leq -0.99998:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{i}{\alpha} \cdot t_3, t_3 + \left(\mathsf{fma}\left(-4, \frac{i}{\alpha} \cdot \frac{i}{\alpha}, 2 \cdot \frac{i}{\alpha}\right) - t_3 \cdot \frac{t_2}{\alpha}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(t_1 + t_1 \cdot \left(\left(\beta - \alpha\right) \cdot t_0\right)\right)}^{0.3333333333333333}}{2}\\ \end{array} \]

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
         (/
          (/ (+ alpha beta) (+ alpha (fma 2.0 i beta)))
          (+ beta (+ alpha (fma 2.0 i 2.0)))))
        (t_1 (pow (fma (- beta alpha) t_0 1.0) 2.0))
        (t_2 (+ 2.0 (fma 2.0 i beta)))
        (t_3 (/ (+ beta t_2) alpha))
        (t_4 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_4) (+ 2.0 t_4)) -0.99998)
     (/
      (fma
       -2.0
       (* (/ i alpha) t_3)
       (+
        t_3
        (-
         (fma -4.0 (* (/ i alpha) (/ i alpha)) (* 2.0 (/ i alpha)))
         (* t_3 (/ t_2 alpha)))))
      2.0)
     (/ (pow (+ t_1 (* t_1 (* (- beta alpha) t_0))) 0.3333333333333333) 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 = ((alpha + beta) / (alpha + fma(2.0, i, beta))) / (beta + (alpha + fma(2.0, i, 2.0)));
	double t_1 = pow(fma((beta - alpha), t_0, 1.0), 2.0);
	double t_2 = 2.0 + fma(2.0, i, beta);
	double t_3 = (beta + t_2) / alpha;
	double t_4 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_4) / (2.0 + t_4)) <= -0.99998) {
		tmp = fma(-2.0, ((i / alpha) * t_3), (t_3 + (fma(-4.0, ((i / alpha) * (i / alpha)), (2.0 * (i / alpha))) - (t_3 * (t_2 / alpha))))) / 2.0;
	} else {
		tmp = pow((t_1 + (t_1 * ((beta - alpha) * t_0))), 0.3333333333333333) / 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(Float64(Float64(alpha + beta) / Float64(alpha + fma(2.0, i, beta))) / Float64(beta + Float64(alpha + fma(2.0, i, 2.0))))
	t_1 = fma(Float64(beta - alpha), t_0, 1.0) ^ 2.0
	t_2 = Float64(2.0 + fma(2.0, i, beta))
	t_3 = Float64(Float64(beta + t_2) / alpha)
	t_4 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_4) / Float64(2.0 + t_4)) <= -0.99998)
		tmp = Float64(fma(-2.0, Float64(Float64(i / alpha) * t_3), Float64(t_3 + Float64(fma(-4.0, Float64(Float64(i / alpha) * Float64(i / alpha)), Float64(2.0 * Float64(i / alpha))) - Float64(t_3 * Float64(t_2 / alpha))))) / 2.0);
	else
		tmp = Float64((Float64(t_1 + Float64(t_1 * Float64(Float64(beta - alpha) * t_0))) ^ 0.3333333333333333) / 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[(N[(alpha + beta), $MachinePrecision] / N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(beta - alpha), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(beta + t$95$2), $MachinePrecision] / alpha), $MachinePrecision]}, Block[{t$95$4 = 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$4), $MachinePrecision] / N[(2.0 + t$95$4), $MachinePrecision]), $MachinePrecision], -0.99998], N[(N[(-2.0 * N[(N[(i / alpha), $MachinePrecision] * t$95$3), $MachinePrecision] + N[(t$95$3 + N[(N[(-4.0 * N[(N[(i / alpha), $MachinePrecision] * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$3 * N[(t$95$2 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Power[N[(t$95$1 + N[(t$95$1 * N[(N[(beta - alpha), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $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)) < -0.99997999999999998

   1. Initial program 3.0%

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

   2. Simplified 10.7%

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

      [Start] 3.0	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/l/ [=>] 2.3	\[ \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]
      *-commutative [=>] 2.3	\[ \frac{\frac{\color{blue}{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2} \]
      times-frac [=>] 10.7	\[ \frac{\color{blue}{\frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}} + 1}{2} \]
      fma-def [=>] 10.7	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}, 1\right)}}{2} \]
      associate-+l+ [=>] 10.7	\[ \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}, 1\right)}{2} \]
      fma-def [=>] 10.7	\[ \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \color{blue}{\mathsf{fma}\left(2, i, 2\right)}}, \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}, 1\right)}{2} \]
      associate-+l+ [=>] 10.7	\[ \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}, \frac{\alpha + \beta}{\color{blue}{\alpha + \left(\beta + 2 \cdot i\right)}}, 1\right)}{2} \]
      +-commutative [=>] 10.7	\[ \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}, \frac{\alpha + \beta}{\alpha + \color{blue}{\left(2 \cdot i + \beta\right)}}, 1\right)}{2} \]
      fma-def [=>] 10.7	\[ \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}, \frac{\alpha + \beta}{\alpha + \color{blue}{\mathsf{fma}\left(2, i, \beta\right)}}, 1\right)}{2} \]

   3. Taylor expanded in beta around 0 10.7%

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

   4. Taylor expanded in alpha around inf 73.8%

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

   5. Simplified 96.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-2, \frac{i}{\alpha} \cdot \frac{\beta + \left(2 + \mathsf{fma}\left(2, i, \beta\right)\right)}{\alpha}, \left(\mathsf{fma}\left(-4, \frac{i}{\alpha} \cdot \frac{i}{\alpha}, 2 \cdot \frac{i}{\alpha}\right) - \frac{2 + \mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{\beta + \left(2 + \mathsf{fma}\left(2, i, \beta\right)\right)}{\alpha}\right) + \frac{\beta + \left(2 + \mathsf{fma}\left(2, i, \beta\right)\right)}{\alpha}\right)}}{2} \]
      Step-by-step derivation

      [Start] 73.8	\[ \frac{\left(-2 \cdot \frac{\left(\beta - -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right) \cdot i}{{\alpha}^{2}} + \left(\frac{\beta}{\alpha} + \left(-4 \cdot \frac{{i}^{2}}{{\alpha}^{2}} + \left(2 \cdot \frac{i}{\alpha} + -1 \cdot \frac{\left(\beta - -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\beta + \left(2 + 2 \cdot i\right)}{\alpha}}{2} \]
      associate--l+ [=>] 73.8	\[ \frac{\color{blue}{-2 \cdot \frac{\left(\beta - -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right) \cdot i}{{\alpha}^{2}} + \left(\left(\frac{\beta}{\alpha} + \left(-4 \cdot \frac{{i}^{2}}{{\alpha}^{2}} + \left(2 \cdot \frac{i}{\alpha} + -1 \cdot \frac{\left(\beta - -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}}\right)\right)\right) - -1 \cdot \frac{\beta + \left(2 + 2 \cdot i\right)}{\alpha}\right)}}{2} \]
      fma-def [=>] 73.8	\[ \frac{\color{blue}{\mathsf{fma}\left(-2, \frac{\left(\beta - -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right) \cdot i}{{\alpha}^{2}}, \left(\frac{\beta}{\alpha} + \left(-4 \cdot \frac{{i}^{2}}{{\alpha}^{2}} + \left(2 \cdot \frac{i}{\alpha} + -1 \cdot \frac{\left(\beta - -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}}\right)\right)\right) - -1 \cdot \frac{\beta + \left(2 + 2 \cdot i\right)}{\alpha}\right)}}{2} \]

   if -0.99997999999999998 < (/.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 82.3%

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

   2. Simplified 99.8%

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

      [Start] 82.3	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/l/ [=>] 81.8	\[ \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]
      *-commutative [=>] 81.8	\[ \frac{\frac{\color{blue}{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2} \]
      times-frac [=>] 99.8	\[ \frac{\color{blue}{\frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}} + 1}{2} \]
      associate-+l+ [=>] 99.8	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}} \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} + 1}{2} \]
      fma-def [=>] 99.8	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \color{blue}{\mathsf{fma}\left(2, i, 2\right)}} \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} + 1}{2} \]
      +-commutative [=>] 99.8	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\color{blue}{2 \cdot i + \left(\alpha + \beta\right)}} + 1}{2} \]
      fma-def [=>] 99.8	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\color{blue}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}} + 1}{2} \]

   3. Applied egg-rr 99.8%

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

      [Start] 99.8	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2} \]
      associate-*l/ [=>] 99.8	\[ \frac{\color{blue}{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}} + 1}{2} \]
      div-inv [=>] 99.8	\[ \frac{\color{blue}{\left(\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) \cdot \frac{1}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}} + 1}{2} \]
      fma-def [=>] 99.8	\[ \frac{\color{blue}{\mathsf{fma}\left(\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, \frac{1}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}, 1\right)}}{2} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \frac{\color{blue}{{\left({\left(1 + \frac{\beta - \alpha}{\frac{\alpha + \mathsf{fma}\left(2, i, \beta\right)}{\beta + \alpha} \cdot \left(\mathsf{fma}\left(2, i, 2\right) + \left(\beta + \alpha\right)\right)}\right)}^{3}\right)}^{0.3333333333333333}}}{2} \]
      Step-by-step derivation

      [Start] 99.8	\[ \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\frac{\mathsf{fma}\left(2, i, \beta\right) + \alpha}{\beta + \alpha}}, \frac{1}{\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right)}, 1\right)}{2} \]
      add-cbrt-cube [=>] 99.7	\[ \frac{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(\frac{\beta - \alpha}{\frac{\mathsf{fma}\left(2, i, \beta\right) + \alpha}{\beta + \alpha}}, \frac{1}{\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right)}, 1\right) \cdot \mathsf{fma}\left(\frac{\beta - \alpha}{\frac{\mathsf{fma}\left(2, i, \beta\right) + \alpha}{\beta + \alpha}}, \frac{1}{\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right)}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{\beta - \alpha}{\frac{\mathsf{fma}\left(2, i, \beta\right) + \alpha}{\beta + \alpha}}, \frac{1}{\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right)}, 1\right)}}}{2} \]
      pow1/3 [=>] 99.8	\[ \frac{\color{blue}{{\left(\left(\mathsf{fma}\left(\frac{\beta - \alpha}{\frac{\mathsf{fma}\left(2, i, \beta\right) + \alpha}{\beta + \alpha}}, \frac{1}{\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right)}, 1\right) \cdot \mathsf{fma}\left(\frac{\beta - \alpha}{\frac{\mathsf{fma}\left(2, i, \beta\right) + \alpha}{\beta + \alpha}}, \frac{1}{\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right)}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{\beta - \alpha}{\frac{\mathsf{fma}\left(2, i, \beta\right) + \alpha}{\beta + \alpha}}, \frac{1}{\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right)}, 1\right)\right)}^{0.3333333333333333}}}{2} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \frac{{\color{blue}{\left({\left(\mathsf{fma}\left(\beta - \alpha, \frac{\frac{\beta + \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)\right)}^{2} + \left(\left(\beta - \alpha\right) \cdot \frac{\frac{\beta + \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\right) \cdot {\left(\mathsf{fma}\left(\beta - \alpha, \frac{\frac{\beta + \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)\right)}^{2}\right)}}^{0.3333333333333333}}{2} \]
      Step-by-step derivation

      [Start] 99.8	\[ \frac{{\left({\left(1 + \frac{\beta - \alpha}{\frac{\alpha + \mathsf{fma}\left(2, i, \beta\right)}{\beta + \alpha} \cdot \left(\mathsf{fma}\left(2, i, 2\right) + \left(\beta + \alpha\right)\right)}\right)}^{3}\right)}^{0.3333333333333333}}{2} \]
      unpow3 [=>] 99.8	\[ \frac{{\color{blue}{\left(\left(\left(1 + \frac{\beta - \alpha}{\frac{\alpha + \mathsf{fma}\left(2, i, \beta\right)}{\beta + \alpha} \cdot \left(\mathsf{fma}\left(2, i, 2\right) + \left(\beta + \alpha\right)\right)}\right) \cdot \left(1 + \frac{\beta - \alpha}{\frac{\alpha + \mathsf{fma}\left(2, i, \beta\right)}{\beta + \alpha} \cdot \left(\mathsf{fma}\left(2, i, 2\right) + \left(\beta + \alpha\right)\right)}\right)\right) \cdot \left(1 + \frac{\beta - \alpha}{\frac{\alpha + \mathsf{fma}\left(2, i, \beta\right)}{\beta + \alpha} \cdot \left(\mathsf{fma}\left(2, i, 2\right) + \left(\beta + \alpha\right)\right)}\right)\right)}}^{0.3333333333333333}}{2} \]
      distribute-rgt-in [=>] 99.8	\[ \frac{{\color{blue}{\left(1 \cdot \left(\left(1 + \frac{\beta - \alpha}{\frac{\alpha + \mathsf{fma}\left(2, i, \beta\right)}{\beta + \alpha} \cdot \left(\mathsf{fma}\left(2, i, 2\right) + \left(\beta + \alpha\right)\right)}\right) \cdot \left(1 + \frac{\beta - \alpha}{\frac{\alpha + \mathsf{fma}\left(2, i, \beta\right)}{\beta + \alpha} \cdot \left(\mathsf{fma}\left(2, i, 2\right) + \left(\beta + \alpha\right)\right)}\right)\right) + \frac{\beta - \alpha}{\frac{\alpha + \mathsf{fma}\left(2, i, \beta\right)}{\beta + \alpha} \cdot \left(\mathsf{fma}\left(2, i, 2\right) + \left(\beta + \alpha\right)\right)} \cdot \left(\left(1 + \frac{\beta - \alpha}{\frac{\alpha + \mathsf{fma}\left(2, i, \beta\right)}{\beta + \alpha} \cdot \left(\mathsf{fma}\left(2, i, 2\right) + \left(\beta + \alpha\right)\right)}\right) \cdot \left(1 + \frac{\beta - \alpha}{\frac{\alpha + \mathsf{fma}\left(2, i, \beta\right)}{\beta + \alpha} \cdot \left(\mathsf{fma}\left(2, i, 2\right) + \left(\beta + \alpha\right)\right)}\right)\right)\right)}}^{0.3333333333333333}}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99998:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{i}{\alpha} \cdot \frac{\beta + \left(2 + \mathsf{fma}\left(2, i, \beta\right)\right)}{\alpha}, \frac{\beta + \left(2 + \mathsf{fma}\left(2, i, \beta\right)\right)}{\alpha} + \left(\mathsf{fma}\left(-4, \frac{i}{\alpha} \cdot \frac{i}{\alpha}, 2 \cdot \frac{i}{\alpha}\right) - \frac{\beta + \left(2 + \mathsf{fma}\left(2, i, \beta\right)\right)}{\alpha} \cdot \frac{2 + \mathsf{fma}\left(2, i, \beta\right)}{\alpha}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(\mathsf{fma}\left(\beta - \alpha, \frac{\frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)\right)}^{2} + {\left(\mathsf{fma}\left(\beta - \alpha, \frac{\frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)\right)}^{2} \cdot \left(\left(\beta - \alpha\right) \cdot \frac{\frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\right)\right)}^{0.3333333333333333}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.2%
Cost	43716

\[\begin{array}{l} t_0 := 2 + \mathsf{fma}\left(2, i, \beta\right)\\ t_1 := \frac{\beta + t_0}{\alpha}\\ t_2 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_2}}{2 + t_2} \leq -0.99998:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{i}{\alpha} \cdot t_1, t_1 + \left(\mathsf{fma}\left(-4, \frac{i}{\alpha} \cdot \frac{i}{\alpha}, 2 \cdot \frac{i}{\alpha}\right) - t_1 \cdot \frac{t_0}{\alpha}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{2}\\ \end{array} \]

Alternative 2
Accuracy	97.6%
Cost	16068

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -1:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{2}\\ \end{array} \]

Alternative 3
Accuracy	97.6%
Cost	9796

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -1:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{t_1}}{2}\\ \end{array} \]

Alternative 4
Accuracy	97.5%
Cost	3268

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.5:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{t_1}}{2}\\ \end{array} \]

Alternative 5
Accuracy	89.0%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.5 \cdot 10^{+98}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Accuracy	83.8%
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Accuracy	75.0%
Cost	845

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 9.5 \cdot 10^{+98}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 1.75 \cdot 10^{+200} \lor \neg \left(\alpha \leq 4.2 \cdot 10^{+281}\right):\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4}{\frac{\alpha}{i}}}{2}\\ \end{array} \]

Alternative 8
Accuracy	78.1%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 7.2 \cdot 10^{+90}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Accuracy	80.5%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.8 \cdot 10^{+92}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 10
Accuracy	72.6%
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+26}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Accuracy	60.9%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023158 
(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))