Octave 3.8, jcobi/2

Percentage Accurate: 63.1% → 97.7%

Time: 22.6s

Alternatives: 8

Speedup: 4.8×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0) 2.0)))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * i)
    code = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Initial Program: 63.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0) 2.0)))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * i)
    code = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Alternative 1: 97.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 2\right)\right)}, \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)}, 1\right) + 1\\ t_1 := 2 + \left(\beta \cdot 2 + i \cdot 4\right)\\ t_2 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_2}}{2 + t\_2} \leq -0.8:\\ \;\;\;\;\frac{\frac{t\_1 + \left(\beta - \left(\beta + 0.5 \cdot \left(\left(\beta - \beta\right) - t\_1\right)\right)\right)}{\alpha}}{{t\_0}^{2} + \left(t\_0 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0
         (+
          (*
           0.5
           (fma
            (/ (+ alpha beta) (+ alpha (+ beta (fma i 2.0 2.0))))
            (/ (- beta alpha) (+ beta (fma i 2.0 alpha)))
            1.0))
          1.0))
        (t_1 (+ 2.0 (+ (* beta 2.0) (* i 4.0))))
        (t_2 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_2) (+ 2.0 t_2)) -0.8)
     (/
      (/ (+ t_1 (- beta (+ beta (* 0.5 (- (- beta beta) t_1))))) alpha)
      (+ (pow t_0 2.0) (+ t_0 1.0)))
     (/
      (+
       (/
        (* (- beta alpha) (/ (+ alpha beta) (fma 2.0 i (+ alpha beta))))
        (+ alpha (+ beta (fma 2.0 i 2.0))))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (0.5 * fma(((alpha + beta) / (alpha + (beta + fma(i, 2.0, 2.0)))), ((beta - alpha) / (beta + fma(i, 2.0, alpha))), 1.0)) + 1.0;
	double t_1 = 2.0 + ((beta * 2.0) + (i * 4.0));
	double t_2 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_2) / (2.0 + t_2)) <= -0.8) {
		tmp = ((t_1 + (beta - (beta + (0.5 * ((beta - beta) - t_1))))) / alpha) / (pow(t_0, 2.0) + (t_0 + 1.0));
	} else {
		tmp = ((((beta - alpha) * ((alpha + beta) / fma(2.0, i, (alpha + beta)))) / (alpha + (beta + fma(2.0, i, 2.0)))) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(0.5 * fma(Float64(Float64(alpha + beta) / Float64(alpha + Float64(beta + fma(i, 2.0, 2.0)))), Float64(Float64(beta - alpha) / Float64(beta + fma(i, 2.0, alpha))), 1.0)) + 1.0)
	t_1 = Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0)))
	t_2 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_2) / Float64(2.0 + t_2)) <= -0.8)
		tmp = Float64(Float64(Float64(t_1 + Float64(beta - Float64(beta + Float64(0.5 * Float64(Float64(beta - beta) - t_1))))) / alpha) / Float64((t_0 ^ 2.0) + Float64(t_0 + 1.0)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) * Float64(Float64(alpha + beta) / fma(2.0, i, Float64(alpha + beta)))) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(0.5 * N[(N[(N[(alpha + beta), $MachinePrecision] / N[(alpha + N[(beta + N[(i * 2.0 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], -0.8], N[(N[(N[(t$95$1 + N[(beta - N[(beta + N[(0.5 * N[(N[(beta - beta), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $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 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.80000000000000004

   1. Initial program 3.3%

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

   3. Simplified 18.2%

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

      1. clear-num 18.1%

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

      2. inv-pow 18.1%

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

      3. associate-*r/ 3.3%

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

      4. *-commutative 3.3%

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

      5. fma-undefine 3.3%

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

      6. +-commutative 3.3%

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

      7. associate-/l* 18.1%

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

      8. +-commutative 18.1%

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

      9. associate-+r+ 18.1%

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

      10. +-commutative 18.1%

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

      11. fma-define 18.1%

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

   6. Applied egg-rr 18.1%

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

      1. unpow-1 18.1%

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

      2. associate-/r* 18.1%

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

      3. +-commutative 18.1%

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

   8. Simplified 18.1%

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

      1. expm1-log1p-u 18.1%

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

      2. expm1-undefine 18.1%

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

   10. Applied egg-rr 18.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{1}{\frac{\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 2\right)\right)}{\beta + \alpha}}, \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)}, 1\right) \cdot 0.5\right)} - 1} \]
   11. Step-by-step derivation

       1. flip3-- 18.2%

          \[\leadsto \color{blue}{\frac{{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{1}{\frac{\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 2\right)\right)}{\beta + \alpha}}, \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)}, 1\right) \cdot 0.5\right)}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{1}{\frac{\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 2\right)\right)}{\beta + \alpha}}, \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)}, 1\right) \cdot 0.5\right)} \cdot e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{1}{\frac{\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 2\right)\right)}{\beta + \alpha}}, \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)}, 1\right) \cdot 0.5\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{1}{\frac{\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 2\right)\right)}{\beta + \alpha}}, \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)}, 1\right) \cdot 0.5\right)} \cdot 1\right)}} \]

   12. Applied egg-rr 18.2%

       \[\leadsto \color{blue}{\frac{{\left(1 + \mathsf{fma}\left(\frac{1}{\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 2\right)\right)} \cdot \left(\beta + \alpha\right), \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)}, 1\right) \cdot 0.5\right)}^{3} - 1}{{\left(1 + \mathsf{fma}\left(\frac{1}{\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 2\right)\right)} \cdot \left(\beta + \alpha\right), \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)}, 1\right) \cdot 0.5\right)}^{2} + \left(1 + \left(1 + \mathsf{fma}\left(\frac{1}{\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 2\right)\right)} \cdot \left(\beta + \alpha\right), \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)}, 1\right) \cdot 0.5\right) \cdot 1\right)}} \]
   13. Step-by-step derivation

   14. Simplified 18.2%

       \[\leadsto \color{blue}{\frac{{\left(1 + 0.5 \cdot \mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 2\right)\right)}, \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta}, 1\right)\right)}^{3} + -1}{{\left(1 + 0.5 \cdot \mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 2\right)\right)}, \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta}, 1\right)\right)}^{2} + \left(1 + \left(1 + 0.5 \cdot \mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 2\right)\right)}, \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta}, 1\right)\right)\right)}} \]

   15. Taylor expanded in alpha around inf 88.2%

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

   if -0.80000000000000004 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 78.6%

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

   3. Simplified 100.0%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.8:\\ \;\;\;\;\frac{\frac{\left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right) + \left(\beta - \left(\beta + 0.5 \cdot \left(\left(\beta - \beta\right) - \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)\right)\right)\right)}{\alpha}}{{\left(0.5 \cdot \mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 2\right)\right)}, \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)}, 1\right) + 1\right)}^{2} + \left(\left(0.5 \cdot \mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 2\right)\right)}, \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)}, 1\right) + 1\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.8:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.8)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (+
       (/
        (* (- beta alpha) (/ (+ alpha beta) (fma 2.0 i (+ alpha beta))))
        (+ alpha (+ beta (fma 2.0 i 2.0))))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.8) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = ((((beta - alpha) * ((alpha + beta) / fma(2.0, i, (alpha + beta)))) / (alpha + (beta + fma(2.0, i, 2.0)))) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.8)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) * Float64(Float64(alpha + beta) / fma(2.0, i, Float64(alpha + beta)))) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = 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$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.8], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $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 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.80000000000000004

   1. Initial program 3.3%

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

   3. Taylor expanded in i around 0 6.2%

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

      1. associate--l+ 6.2%

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

      2. associate-/l* 9.5%

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

      3. +-commutative 9.5%

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

   5. Simplified 9.5%

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

   6. Taylor expanded in alpha around inf 87.9%

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

   if -0.80000000000000004 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 78.6%

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

   3. Simplified 100.0%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.8:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.5:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\frac{\beta}{\beta + 2 \cdot i} \cdot \left(\alpha - \beta\right)}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.5)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (-
       1.0
       (/
        (* (/ beta (+ beta (* 2.0 i))) (- alpha beta))
        (+ alpha (+ beta (fma 2.0 i 2.0)))))
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.5) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 - (((beta / (beta + (2.0 * i))) * (alpha - beta)) / (alpha + (beta + fma(2.0, i, 2.0))))) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.5)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(beta / Float64(beta + Float64(2.0 * i))) * Float64(alpha - beta)) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0))))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = 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$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(beta / N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(alpha - beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $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 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.5

   1. Initial program 4.7%

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

   3. Taylor expanded in i around 0 6.4%

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

      1. associate--l+ 6.4%

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

      2. associate-/l* 9.7%

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

      3. +-commutative 9.7%

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

   5. Simplified 9.7%

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

   6. Taylor expanded in alpha around inf 86.9%

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

   if -0.5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 78.5%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 99.4%

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

4. Final simplification 96.1%

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

Alternative 4: 96.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t\_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_1} \leq -0.5:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t\_1} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))) (t_1 (+ 2.0 t_0)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1) -0.5)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/ (+ (/ beta t_1) 1.0) 2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_1) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = 2.0d0 + t_0
    if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= (-0.5d0)) then
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    else
        tmp = ((beta / t_1) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_1) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = 2.0 + t_0
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	else:
		tmp = ((beta / t_1) + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(2.0 + t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / t_1) <= -0.5)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / t_1) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = 2.0 + t_0;
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5)
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	else
		tmp = ((beta / t_1) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], -0.5], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / t$95$1), $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 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.5

   1. Initial program 4.7%

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

   3. Taylor expanded in i around 0 6.4%

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

      1. associate--l+ 6.4%

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

      2. associate-/l* 9.7%

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

      3. +-commutative 9.7%

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

   5. Simplified 9.7%

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

   6. Taylor expanded in alpha around inf 86.9%

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

   if -0.5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 78.5%

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

   3. Taylor expanded in beta around inf 97.6%

      \[\leadsto \frac{\frac{\color{blue}{\beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.8%

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

Alternative 5: 83.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.3 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.3e+29)
   (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)
   (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.3e+29) {
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.3d+29) then
        tmp = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
    else
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.3e+29) {
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.3e+29:
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0
	else:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.3e+29)
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.3e+29)
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	else
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.3e+29], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.3e29

   1. Initial program 82.1%

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

      1. associate-/l/ 81.6%

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

      2. associate-+l+ 81.6%

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

      3. +-commutative 81.6%

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

      4. associate-+l+ 81.6%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in i around 0 92.0%

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

      1. +-commutative 92.0%

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

   7. Simplified 92.0%

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

   if 1.3e29 < alpha

   1. Initial program 14.6%

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

   3. Taylor expanded in i around 0 18.0%

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

      1. associate--l+ 18.0%

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

      2. associate-/l* 23.6%

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

      3. +-commutative 23.6%

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

   5. Simplified 23.6%

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

   6. Taylor expanded in alpha around inf 65.9%

      \[\leadsto \frac{\color{blue}{\frac{2 + \left(2 \cdot \beta + 4 \cdot i\right)}{\alpha}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.3 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.35 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.35e+29)
   (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.35e+29) {
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.35d+29) then
        tmp = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.35e+29) {
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.35e+29:
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.35e+29)
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.35e+29)
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.35e+29], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.35e29

   1. Initial program 82.1%

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

      1. associate-/l/ 81.6%

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

      2. associate-+l+ 81.6%

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

      3. +-commutative 81.6%

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

      4. associate-+l+ 81.6%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in i around 0 92.0%

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

      1. +-commutative 92.0%

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

   7. Simplified 92.0%

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

   if 1.35e29 < alpha

   1. Initial program 14.6%

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

   3. Taylor expanded in i around 0 18.0%

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

      1. associate--l+ 18.0%

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

      2. associate-/l* 23.6%

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

      3. +-commutative 23.6%

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

   5. Simplified 23.6%

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

   6. Taylor expanded in beta around 0 9.7%

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

   7. Taylor expanded in alpha around inf 50.7%

      \[\leadsto \frac{\color{blue}{\frac{2 + 4 \cdot i}{\alpha}}}{2} \]
   8. Step-by-step derivation

      1. *-commutative 50.7%

         \[\leadsto \frac{\frac{2 + \color{blue}{i \cdot 4}}{\alpha}}{2} \]

   9. Simplified 50.7%

      \[\leadsto \frac{\color{blue}{\frac{2 + i \cdot 4}{\alpha}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.35 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.2 \cdot 10^{+16}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 6.2e+16) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.2e+16) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 6.2d+16) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.2e+16) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 6.2e+16:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6.2e+16)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 6.2e+16)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 6.2e+16], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 6.2e16

   1. Initial program 71.7%

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

   3. Simplified 75.8%

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

      1. clear-num 75.7%

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

      2. inv-pow 75.7%

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

      3. associate-*r/ 71.6%

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

      4. *-commutative 71.6%

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

      5. fma-undefine 71.6%

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

      6. +-commutative 71.6%

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

      7. associate-/l* 75.7%

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

      8. +-commutative 75.7%

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

      9. associate-+r+ 75.7%

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

      10. +-commutative 75.7%

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

      11. fma-define 75.7%

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

   6. Applied egg-rr 75.7%

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

      1. unpow-1 75.7%

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

      2. associate-/r* 75.8%

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

      3. +-commutative 75.8%

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

   8. Simplified 75.8%

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

   9. Taylor expanded in i around inf 73.0%

      \[\leadsto \color{blue}{0.5} \]

   if 6.2e16 < beta

   1. Initial program 36.1%

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

   3. Simplified 84.6%

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

      1. clear-num 84.6%

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

      2. inv-pow 84.6%

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

      3. associate-*r/ 36.1%

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

      4. *-commutative 36.1%

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

      5. fma-undefine 36.1%

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

      6. +-commutative 36.1%

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

      7. associate-/l* 84.6%

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

      8. +-commutative 84.6%

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

      9. associate-+r+ 84.6%

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

      10. +-commutative 84.6%

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

      11. fma-define 84.6%

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

   6. Applied egg-rr 84.6%

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

      1. unpow-1 84.6%

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

      2. associate-/r* 84.6%

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

      3. +-commutative 84.6%

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

   8. Simplified 84.6%

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

      1. expm1-log1p-u 84.6%

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

      2. expm1-undefine 84.6%

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

   10. Applied egg-rr 84.6%

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

   11. Taylor expanded in beta around inf 72.2%

       \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 61.1% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 0.5)

C

double code(double alpha, double beta, double i) {
	return 0.5;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.5d0
end function

Java

public static double code(double alpha, double beta, double i) {
	return 0.5;
}

Python

def code(alpha, beta, i):
	return 0.5

Julia

function code(alpha, beta, i)
	return 0.5
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 0.5;
end

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 59.2%

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

3. Simplified 78.9%

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

   1. clear-num 78.9%

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

   2. inv-pow 78.9%

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

   3. associate-*r/ 59.2%

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

   4. *-commutative 59.2%

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

   5. fma-undefine 59.2%

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

   6. +-commutative 59.2%

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

   7. associate-/l* 78.9%

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

   8. +-commutative 78.9%

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

   9. associate-+r+ 78.9%

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

   10. +-commutative 78.9%

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

   11. fma-define 78.9%

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

6. Applied egg-rr 78.9%

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

   1. unpow-1 78.9%

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

   2. associate-/r* 78.9%

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

   3. +-commutative 78.9%

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

8. Simplified 78.9%

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

9. Taylor expanded in i around inf 56.2%

   \[\leadsto \color{blue}{0.5} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(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))