Octave 3.8, jcobi/2

Percentage Accurate: 62.8% → 97.7%

Time: 28.6s

Precision: binary64

Cost: 55876

• Unsound rule application detected in e-graph. Results may not be sound.

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\mathsf{fma}\left(i, 4, \mathsf{fma}\left(\beta, 2, 2\right)\right)}{\alpha}\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{2 + t_1} \leq -0.99:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha} + \left(t_0 + \left(\frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{\beta + \mathsf{fma}\left(2, i, 2\right)}{\alpha} - t_0 \cdot t_0\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right) + 2\right) \cdot \frac{\alpha + \mathsf{fma}\left(2, i, \beta\right)}{\alpha + \beta}}\right)}}{2}\\ \end{array} \]

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 i 4.0 (fma beta 2.0 2.0)) alpha))
        (t_1 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ 2.0 t_1)) -0.99)
     (/
      (+
       (* (/ beta alpha) (/ beta alpha))
       (+
        t_0
        (-
         (* (/ (fma 2.0 i beta) alpha) (/ (+ beta (fma 2.0 i 2.0)) alpha))
         (* t_0 t_0))))
      2.0)
     (/
      (exp
       (log1p
        (/
         (- beta alpha)
         (*
          (fma 2.0 i (+ (+ alpha beta) 2.0))
          (/ (+ alpha (fma 2.0 i beta)) (+ alpha beta))))))
      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(i, 4.0, fma(beta, 2.0, 2.0)) / alpha;
	double t_1 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.99) {
		tmp = (((beta / alpha) * (beta / alpha)) + (t_0 + (((fma(2.0, i, beta) / alpha) * ((beta + fma(2.0, i, 2.0)) / alpha)) - (t_0 * t_0)))) / 2.0;
	} else {
		tmp = exp(log1p(((beta - alpha) / (fma(2.0, i, ((alpha + beta) + 2.0)) * ((alpha + fma(2.0, i, beta)) / (alpha + beta)))))) / 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(i, 4.0, fma(beta, 2.0, 2.0)) / 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)) <= -0.99)
		tmp = Float64(Float64(Float64(Float64(beta / alpha) * Float64(beta / alpha)) + Float64(t_0 + Float64(Float64(Float64(fma(2.0, i, beta) / alpha) * Float64(Float64(beta + fma(2.0, i, 2.0)) / alpha)) - Float64(t_0 * t_0)))) / 2.0);
	else
		tmp = Float64(exp(log1p(Float64(Float64(beta - alpha) / Float64(fma(2.0, i, Float64(Float64(alpha + beta) + 2.0)) * Float64(Float64(alpha + fma(2.0, i, beta)) / Float64(alpha + beta)))))) / 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[(i * 4.0 + N[(beta * 2.0 + 2.0), $MachinePrecision]), $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], -0.99], N[(N[(N[(N[(beta / alpha), $MachinePrecision] * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + N[(N[(N[(N[(2.0 * i + beta), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Exp[N[Log[1 + N[(N[(beta - alpha), $MachinePrecision] / N[(N[(2.0 * i + N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.98999999999999999

   1. Initial program 2.6%

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

   2. Simplified 1.7%

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

      [Start] 2.6	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/l/ [=>] 1.7	\[ \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]
      associate-+l+ [=>] 1.7	\[ \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2} \]
      associate-+l+ [=>] 1.7	\[ \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \color{blue}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right)}} + 1}{2} \]

   3. Taylor expanded in alpha around inf 81.7%

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

   4. Simplified 97.6%

      \[\leadsto \frac{\color{blue}{\left(0 + \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) - \left(\left(\frac{\mathsf{fma}\left(i, 4, \mathsf{fma}\left(\beta, 2, 2\right)\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(i, 4, \mathsf{fma}\left(\beta, 2, 2\right)\right)}{\alpha} - \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{\beta + \mathsf{fma}\left(2, i, 2\right)}{\alpha}\right) - \frac{\mathsf{fma}\left(i, 4, \mathsf{fma}\left(\beta, 2, 2\right)\right)}{\alpha}\right)}}{2} \]
      Step-by-step derivation

      [Start] 81.7	\[ \frac{\left(\frac{\beta}{\alpha} + \left(-1 \cdot \frac{\beta}{\alpha} + \frac{{\beta}^{2}}{{\alpha}^{2}}\right)\right) - \left(-1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha} + \left(-1 \cdot \frac{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}} + \frac{\left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)\right) \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{{\alpha}^{2}}\right)\right)}{2} \]
      associate-+r+ [=>] 81.7	\[ \frac{\color{blue}{\left(\left(\frac{\beta}{\alpha} + -1 \cdot \frac{\beta}{\alpha}\right) + \frac{{\beta}^{2}}{{\alpha}^{2}}\right)} - \left(-1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha} + \left(-1 \cdot \frac{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}} + \frac{\left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)\right) \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{{\alpha}^{2}}\right)\right)}{2} \]
      distribute-rgt1-in [=>] 81.7	\[ \frac{\left(\color{blue}{\left(-1 + 1\right) \cdot \frac{\beta}{\alpha}} + \frac{{\beta}^{2}}{{\alpha}^{2}}\right) - \left(-1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha} + \left(-1 \cdot \frac{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}} + \frac{\left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)\right) \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{{\alpha}^{2}}\right)\right)}{2} \]
      metadata-eval [=>] 81.7	\[ \frac{\left(\color{blue}{0} \cdot \frac{\beta}{\alpha} + \frac{{\beta}^{2}}{{\alpha}^{2}}\right) - \left(-1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha} + \left(-1 \cdot \frac{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}} + \frac{\left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)\right) \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{{\alpha}^{2}}\right)\right)}{2} \]
      mul0-lft [=>] 81.7	\[ \frac{\left(\color{blue}{0} + \frac{{\beta}^{2}}{{\alpha}^{2}}\right) - \left(-1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha} + \left(-1 \cdot \frac{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}} + \frac{\left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)\right) \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{{\alpha}^{2}}\right)\right)}{2} \]
      unpow2 [=>] 81.7	\[ \frac{\left(0 + \frac{\color{blue}{\beta \cdot \beta}}{{\alpha}^{2}}\right) - \left(-1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha} + \left(-1 \cdot \frac{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}} + \frac{\left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)\right) \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{{\alpha}^{2}}\right)\right)}{2} \]
      unpow2 [=>] 81.7	\[ \frac{\left(0 + \frac{\beta \cdot \beta}{\color{blue}{\alpha \cdot \alpha}}\right) - \left(-1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha} + \left(-1 \cdot \frac{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}} + \frac{\left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)\right) \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{{\alpha}^{2}}\right)\right)}{2} \]
      times-frac [=>] 81.7	\[ \frac{\left(0 + \color{blue}{\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}}\right) - \left(-1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha} + \left(-1 \cdot \frac{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}} + \frac{\left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)\right) \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{{\alpha}^{2}}\right)\right)}{2} \]
      +-commutative [=>] 81.7	\[ \frac{\left(0 + \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) - \color{blue}{\left(\left(-1 \cdot \frac{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}} + \frac{\left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)\right) \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{{\alpha}^{2}}\right) + -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}\right)}}{2} \]

   if -0.98999999999999999 < (/.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 79.4%

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

   2. Simplified 100.0%

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

      [Start] 79.4	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/l/ [=>] 78.7	\[ \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]
      *-commutative [=>] 78.7	\[ \frac{\frac{\color{blue}{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2} \]
      times-frac [=>] 100.0	\[ \frac{\color{blue}{\frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}} + 1}{2} \]
      associate-+l+ [=>] 100.0	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}} \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} + 1}{2} \]
      fma-def [=>] 100.0	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \color{blue}{\mathsf{fma}\left(2, i, 2\right)}} \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} + 1}{2} \]
      +-commutative [=>] 100.0	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\color{blue}{2 \cdot i + \left(\alpha + \beta\right)}} + 1}{2} \]
      fma-def [=>] 100.0	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\color{blue}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}} + 1}{2} \]

   3. Applied egg-rr 78.7%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right) \cdot \left(\mathsf{fma}\left(2, i, \beta\right) + \alpha\right)}\right)}}}{2} \]
      Step-by-step derivation

      [Start] 100.0	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2} \]
      add-exp-log [=>] 100.0	\[ \frac{\color{blue}{e^{\log \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1\right)}}}{2} \]
      +-commutative [=>] 100.0	\[ \frac{e^{\log \color{blue}{\left(1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)}}}{2} \]
      log1p-udef [<=] 100.0	\[ \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)}}}{2} \]
      frac-times [=>] 78.7	\[ \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}{\left(\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)\right) \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right)}}\right)}}{2} \]
      *-commutative [=>] 78.7	\[ \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}}{\left(\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)\right) \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)}}{2} \]
      +-commutative [=>] 78.7	\[ \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(\beta + \alpha\right)} \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)\right) \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)}}{2} \]
      difference-of-squares [<=] 78.7	\[ \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\beta \cdot \beta - \alpha \cdot \alpha}}{\left(\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)\right) \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)}}{2} \]
      +-commutative [=>] 78.7	\[ \frac{e^{\mathsf{log1p}\left(\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\color{blue}{\left(\mathsf{fma}\left(2, i, 2\right) + \left(\alpha + \beta\right)\right)} \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)}}{2} \]
      fma-udef [=>] 78.7	\[ \frac{e^{\mathsf{log1p}\left(\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\left(\color{blue}{\left(2 \cdot i + 2\right)} + \left(\alpha + \beta\right)\right) \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)}}{2} \]
      associate-+l+ [=>] 78.7	\[ \frac{e^{\mathsf{log1p}\left(\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\color{blue}{\left(2 \cdot i + \left(2 + \left(\alpha + \beta\right)\right)\right)} \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)}}{2} \]
      fma-def [=>] 78.7	\[ \frac{e^{\mathsf{log1p}\left(\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\color{blue}{\mathsf{fma}\left(2, i, 2 + \left(\alpha + \beta\right)\right)} \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)}}{2} \]
      +-commutative [=>] 78.7	\[ \frac{e^{\mathsf{log1p}\left(\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\mathsf{fma}\left(2, i, 2 + \color{blue}{\left(\beta + \alpha\right)}\right) \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)}}{2} \]
      fma-udef [=>] 78.7	\[ \frac{e^{\mathsf{log1p}\left(\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right) \cdot \color{blue}{\left(2 \cdot i + \left(\alpha + \beta\right)\right)}}\right)}}{2} \]
      +-commutative [=>] 78.7	\[ \frac{e^{\mathsf{log1p}\left(\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right) \cdot \left(2 \cdot i + \color{blue}{\left(\beta + \alpha\right)}\right)}\right)}}{2} \]
      associate-+r+ [=>] 78.7	\[ \frac{e^{\mathsf{log1p}\left(\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right) \cdot \color{blue}{\left(\left(2 \cdot i + \beta\right) + \alpha\right)}}\right)}}{2} \]
      fma-def [=>] 78.7	\[ \frac{e^{\mathsf{log1p}\left(\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left(2, i, \beta\right)} + \alpha\right)}\right)}}{2} \]

   4. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, 2 + \left(\alpha + \beta\right)\right) \cdot \frac{\alpha + \mathsf{fma}\left(2, i, \beta\right)}{\alpha + \beta}}\right)}}}{2} \]
      Step-by-step derivation

      [Start] 78.7	\[ \frac{e^{\mathsf{log1p}\left(\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right) \cdot \left(\mathsf{fma}\left(2, i, \beta\right) + \alpha\right)}\right)}}{2} \]
      *-lft-identity [<=] 78.7	\[ \frac{e^{\mathsf{log1p}\left(\color{blue}{1 \cdot \frac{\beta \cdot \beta - \alpha \cdot \alpha}{\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right) \cdot \left(\mathsf{fma}\left(2, i, \beta\right) + \alpha\right)}}\right)}}{2} \]
      metadata-eval [<=] 78.7	\[ \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{-1}{-1}} \cdot \frac{\beta \cdot \beta - \alpha \cdot \alpha}{\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right) \cdot \left(\mathsf{fma}\left(2, i, \beta\right) + \alpha\right)}\right)}}{2} \]
      times-frac [<=] 78.7	\[ \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{-1 \cdot \left(\beta \cdot \beta - \alpha \cdot \alpha\right)}{-1 \cdot \left(\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right) \cdot \left(\mathsf{fma}\left(2, i, \beta\right) + \alpha\right)\right)}}\right)}}{2} \]
      neg-mul-1 [<=] 78.7	\[ \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{-\left(\beta \cdot \beta - \alpha \cdot \alpha\right)}}{-1 \cdot \left(\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right) \cdot \left(\mathsf{fma}\left(2, i, \beta\right) + \alpha\right)\right)}\right)}}{2} \]
      difference-of-squares [=>] 78.7	\[ \frac{e^{\mathsf{log1p}\left(\frac{-\color{blue}{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}}{-1 \cdot \left(\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right) \cdot \left(\mathsf{fma}\left(2, i, \beta\right) + \alpha\right)\right)}\right)}}{2} \]
      distribute-lft-neg-out [<=] 78.7	\[ \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(-\left(\beta + \alpha\right)\right) \cdot \left(\beta - \alpha\right)}}{-1 \cdot \left(\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right) \cdot \left(\mathsf{fma}\left(2, i, \beta\right) + \alpha\right)\right)}\right)}}{2} \]
      *-commutative [=>] 78.7	\[ \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(\beta - \alpha\right) \cdot \left(-\left(\beta + \alpha\right)\right)}}{-1 \cdot \left(\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right) \cdot \left(\mathsf{fma}\left(2, i, \beta\right) + \alpha\right)\right)}\right)}}{2} \]
      neg-mul-1 [<=] 78.7	\[ \frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(-\left(\beta + \alpha\right)\right)}{\color{blue}{-\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right) \cdot \left(\mathsf{fma}\left(2, i, \beta\right) + \alpha\right)}}\right)}}{2} \]
      associate-/l* [=>] 85.5	\[ \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta - \alpha}{\frac{-\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right) \cdot \left(\mathsf{fma}\left(2, i, \beta\right) + \alpha\right)}{-\left(\beta + \alpha\right)}}}\right)}}{2} \]
      distribute-rgt-neg-in [=>] 85.5	\[ \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\frac{\color{blue}{\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right) \cdot \left(-\left(\mathsf{fma}\left(2, i, \beta\right) + \alpha\right)\right)}}{-\left(\beta + \alpha\right)}}\right)}}{2} \]
      *-lft-identity [<=] 85.5	\[ \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\frac{\mathsf{fma}\left(2, i, 2 + \left(\beta + \alpha\right)\right) \cdot \left(-\left(\mathsf{fma}\left(2, i, \beta\right) + \alpha\right)\right)}{\color{blue}{1 \cdot \left(-\left(\beta + \alpha\right)\right)}}}\right)}}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha} + \left(\frac{\mathsf{fma}\left(i, 4, \mathsf{fma}\left(\beta, 2, 2\right)\right)}{\alpha} + \left(\frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{\beta + \mathsf{fma}\left(2, i, 2\right)}{\alpha} - \frac{\mathsf{fma}\left(i, 4, \mathsf{fma}\left(\beta, 2, 2\right)\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(i, 4, \mathsf{fma}\left(\beta, 2, 2\right)\right)}{\alpha}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right) + 2\right) \cdot \frac{\alpha + \mathsf{fma}\left(2, i, \beta\right)}{\alpha + \beta}}\right)}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.5%
Cost	28740

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.99:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right) + 2\right) \cdot \frac{\alpha + \mathsf{fma}\left(2, i, \beta\right)}{\alpha + \beta}}\right)}}{2}\\ \end{array} \]

Alternative 2
Accuracy	97.5%
Cost	16068

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.99:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}\\ \end{array} \]

Alternative 3
Accuracy	97.5%
Cost	9796

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.99:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{t_1} + 1}{2}\\ \end{array} \]

Alternative 4
Accuracy	97.1%
Cost	9668

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

Alternative 5
Accuracy	97.1%
Cost	9540

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

Alternative 6
Accuracy	96.6%
Cost	3140

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

Alternative 7
Accuracy	89.0%
Cost	1476

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.7 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2 \cdot i} \cdot \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \end{array} \]

Alternative 8
Accuracy	88.4%
Cost	1348

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.9 \cdot 10^{+146}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \end{array} \]

Alternative 9
Accuracy	88.5%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 5.8 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 10
Accuracy	75.1%
Cost	973

\[\begin{array}{l} \mathbf{if}\;i \leq 7 \cdot 10^{+81} \lor \neg \left(i \leq 1.95 \cdot 10^{+148}\right) \land i \leq 7.2 \cdot 10^{+204}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 11
Accuracy	75.5%
Cost	972

\[\begin{array}{l} t_0 := \frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{if}\;\alpha \leq 7.8 \cdot 10^{-161}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 1120:\\ \;\;\;\;\frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 3.1 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 12
Accuracy	77.7%
Cost	972

\[\begin{array}{l} t_0 := \frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{if}\;\alpha \leq 1.46 \cdot 10^{-161}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 0.00052:\\ \;\;\;\;\frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 1.25 \cdot 10^{+142}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 13
Accuracy	79.8%
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 6.1 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \]

Alternative 14
Accuracy	82.6%
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4 \cdot 10^{+147}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 15
Accuracy	71.7%
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+124}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16
Accuracy	61.6%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023159 
(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))