Octave 3.8, jcobi/2

Percentage Accurate: 63.0% → 97.6%

Time: 22.9s

Alternatives: 12

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: 63.0% 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: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.98:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\mathsf{fma}\left(\beta - \alpha, \frac{\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}, 1\right)}\right)}{2}\\ \end{array} \end{array} \]

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)) -0.98)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (log
       (exp
        (fma
         (- beta alpha)
         (/
          (/ (+ alpha beta) (+ beta (fma 2.0 i alpha)))
          (+ (+ 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)) <= -0.98) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = log(exp(fma((beta - alpha), (((alpha + beta) / (beta + fma(2.0, i, alpha))) / ((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)) <= -0.98)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(log(exp(fma(Float64(beta - alpha), Float64(Float64(Float64(alpha + beta) / Float64(beta + fma(2.0, i, alpha))) / Float64(Float64(alpha + 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], -0.98], 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[Log[N[Exp[N[(N[(beta - alpha), $MachinePrecision] * N[(N[(N[(alpha + beta), $MachinePrecision] / N[(beta + N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.97999999999999998

   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 22.1%

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

   5. Taylor expanded in beta around 0 22.1%

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

   6. Taylor expanded in alpha around inf 85.1%

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

   if -0.97999999999999998 < (/.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 86.6%

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

   3. Simplified 99.9%

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

      1. add-log-exp 99.9%

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

      2. associate-/l* 99.9%

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

      3. fma-define 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.98:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\mathsf{fma}\left(\beta - \alpha, \frac{\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}, 1\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.98:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta \cdot \left(1 + \left(\frac{\alpha}{\beta} + \mathsf{fma}\left(2, \frac{i}{\beta}, \frac{2}{\beta}\right)\right)\right)}, \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

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)) -0.98)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (fma
       (/
        (+ alpha beta)
        (* beta (+ 1.0 (+ (/ alpha beta) (fma 2.0 (/ i beta) (/ 2.0 beta))))))
       (/ (- beta alpha) (+ beta (fma 2.0 i alpha)))
       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)) <= -0.98) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = fma(((alpha + beta) / (beta * (1.0 + ((alpha / beta) + fma(2.0, (i / beta), (2.0 / beta)))))), ((beta - alpha) / (beta + fma(2.0, i, alpha))), 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)) <= -0.98)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(fma(Float64(Float64(alpha + beta) / Float64(beta * Float64(1.0 + Float64(Float64(alpha / beta) + fma(2.0, Float64(i / beta), Float64(2.0 / beta)))))), Float64(Float64(beta - alpha) / Float64(beta + fma(2.0, i, alpha))), 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], -0.98], 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[(alpha + beta), $MachinePrecision] / N[(beta * N[(1.0 + N[(N[(alpha / beta), $MachinePrecision] + N[(2.0 * N[(i / beta), $MachinePrecision] + N[(2.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(2.0 * i + alpha), $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))) < -0.97999999999999998

   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 22.1%

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

   5. Taylor expanded in beta around 0 22.1%

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

   6. Taylor expanded in alpha around inf 85.1%

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

   if -0.97999999999999998 < (/.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 86.6%

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

      1. associate-/l/ 86.2%

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

      2. associate-+l+ 86.2%

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

      3. +-commutative 86.2%

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

      4. associate-+l+ 86.2%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in beta around inf 86.2%

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

      1. times-frac 99.9%

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

      2. +-commutative 99.9%

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

      3. fma-undefine 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-define 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. +-commutative 99.9%

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

      2. +-commutative 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.98:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta \cdot \left(1 + \left(\frac{\alpha}{\beta} + \mathsf{fma}\left(2, \frac{i}{\beta}, \frac{2}{\beta}\right)\right)\right)}, \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.98:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\alpha + \beta\right) \cdot \frac{1}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}, \frac{1}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

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)) -0.98)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (fma
       (* (+ alpha beta) (/ 1.0 (/ (+ beta (fma 2.0 i alpha)) (- beta alpha))))
       (/ 1.0 (+ (+ 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)) <= -0.98) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = fma(((alpha + beta) * (1.0 / ((beta + fma(2.0, i, alpha)) / (beta - alpha)))), (1.0 / ((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)) <= -0.98)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(fma(Float64(Float64(alpha + beta) * Float64(1.0 / Float64(Float64(beta + fma(2.0, i, alpha)) / Float64(beta - alpha)))), Float64(1.0 / Float64(Float64(alpha + 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], -0.98], 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[(alpha + beta), $MachinePrecision] * N[(1.0 / N[(N[(beta + N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $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))) < -0.97999999999999998

   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 22.1%

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

   5. Taylor expanded in beta around 0 22.1%

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

   6. Taylor expanded in alpha around inf 85.1%

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

   if -0.97999999999999998 < (/.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 86.6%

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

   3. Simplified 99.9%

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

      1. div-inv 99.9%

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

      2. associate-*r/ 86.6%

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

      3. *-commutative 86.6%

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

      4. fma-undefine 86.6%

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

      5. +-commutative 86.6%

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

      6. fma-define 86.6%

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

   6. Applied egg-rr 99.9%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

      2. +-commutative 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.98:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\alpha + \beta\right) \cdot \frac{1}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}, \frac{1}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.98:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}}}{2}\\ \end{array} \end{array} \]

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)) -0.98)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (+
       1.0
       (/
        1.0
        (/
         (+ (+ alpha beta) (fma 2.0 i 2.0))
         (* (+ alpha beta) (/ (- beta alpha) (+ beta (fma 2.0 i alpha)))))))
      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)) <= -0.98) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (1.0 / (((alpha + beta) + fma(2.0, i, 2.0)) / ((alpha + beta) * ((beta - alpha) / (beta + fma(2.0, i, alpha))))))) / 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)) <= -0.98)
		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(1.0 / Float64(Float64(Float64(alpha + beta) + fma(2.0, i, 2.0)) / Float64(Float64(alpha + beta) * Float64(Float64(beta - alpha) / Float64(beta + fma(2.0, i, alpha))))))) / 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], -0.98], 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[(1.0 / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(2.0 * i + alpha), $MachinePrecision]), $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 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.97999999999999998

   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 22.1%

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

   5. Taylor expanded in beta around 0 22.1%

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

   6. Taylor expanded in alpha around inf 85.1%

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

   if -0.97999999999999998 < (/.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 86.6%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. fma-define 99.9%

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

      3. associate-+r+ 99.9%

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

      4. associate-+r+ 99.9%

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

      5. associate-*r/ 86.6%

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

      6. *-commutative 86.6%

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

      7. fma-undefine 86.6%

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

      8. +-commutative 86.6%

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

      9. inv-pow 86.6%

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

   6. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

      2. +-commutative 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.98:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.98:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \end{array} \]

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)) -0.98)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (+
       1.0
       (/
        (* (- beta alpha) (/ (+ alpha beta) (fma 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)) <= -0.98) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((beta - alpha) * ((alpha + beta) / fma(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)) <= -0.98)
		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 - alpha) * Float64(Float64(alpha + beta) / fma(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], -0.98], 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 - 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]), $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))) < -0.97999999999999998

   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 22.1%

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

   5. Taylor expanded in beta around 0 22.1%

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

   6. Taylor expanded in alpha around inf 85.1%

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

   if -0.97999999999999998 < (/.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 86.6%

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

   3. Simplified 99.9%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.98:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.6:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \end{array} \]

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)) -0.6)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (+
       1.0
       (/
        (* (- beta alpha) (/ beta (+ beta (* 2.0 i))))
        (+ 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)) <= -0.6) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((beta - alpha) * (beta / (beta + (2.0 * i)))) / (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)) <= -0.6)
		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 - alpha) * Float64(beta / Float64(beta + Float64(2.0 * i)))) / 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], -0.6], 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 - alpha), $MachinePrecision] * N[(beta / N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $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))) < -0.599999999999999978

   1. Initial program 6.2%

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

   3. Simplified 24.4%

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

   5. Taylor expanded in beta around 0 24.4%

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

   6. Taylor expanded in alpha around inf 83.2%

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

   if -0.599999999999999978 < (/.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 86.6%

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

   5. Taylor expanded in alpha around 0 99.6%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.6:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.6:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\beta + 2 \cdot i\right) \cdot \left(1 + \left(2 \cdot \frac{1}{\beta} + 2 \cdot \frac{i}{\beta}\right)\right)}}{2}\\ \end{array} \end{array} \]

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)) -0.6)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (+
       1.0
       (/
        beta
        (*
         (+ beta (* 2.0 i))
         (+ 1.0 (+ (* 2.0 (/ 1.0 beta)) (* 2.0 (/ i beta)))))))
      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)) <= -0.6) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / ((beta + (2.0 * i)) * (1.0 + ((2.0 * (1.0 / beta)) + (2.0 * (i / beta))))))) / 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) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)) <= (-0.6d0)) then
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + (beta / ((beta + (2.0d0 * i)) * (1.0d0 + ((2.0d0 * (1.0d0 / beta)) + (2.0d0 * (i / beta))))))) / 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 tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.6) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / ((beta + (2.0 * i)) * (1.0 + ((2.0 * (1.0 / beta)) + (2.0 * (i / beta))))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.6:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	else:
		tmp = (1.0 + (beta / ((beta + (2.0 * i)) * (1.0 + ((2.0 * (1.0 / beta)) + (2.0 * (i / beta))))))) / 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)) <= -0.6)
		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(beta / Float64(Float64(beta + Float64(2.0 * i)) * Float64(1.0 + Float64(Float64(2.0 * Float64(1.0 / beta)) + Float64(2.0 * Float64(i / beta))))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.6)
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	else
		tmp = (1.0 + (beta / ((beta + (2.0 * i)) * (1.0 + ((2.0 * (1.0 / beta)) + (2.0 * (i / beta))))))) / 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]}, 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], -0.6], 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[(beta / N[(N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(2.0 * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(i / beta), $MachinePrecision]), $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 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.599999999999999978

   1. Initial program 6.2%

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

   3. Simplified 24.4%

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

   5. Taylor expanded in beta around 0 24.4%

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

   6. Taylor expanded in alpha around inf 83.2%

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

   if -0.599999999999999978 < (/.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 86.6%

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

      1. associate-/l/ 86.1%

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

      2. associate-+l+ 86.1%

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

      3. +-commutative 86.1%

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

      4. associate-+l+ 86.1%

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

   3. Simplified 86.1%

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

   5. Taylor expanded in beta around inf 86.1%

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

   6. Taylor expanded in alpha around 0 99.3%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.6:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\beta + 2 \cdot i\right) \cdot \left(1 + \left(2 \cdot \frac{1}{\beta} + 2 \cdot \frac{i}{\beta}\right)\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.7 \cdot 10^{+119}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\beta + 2 \cdot i\right) \cdot \left(1 + \frac{2}{\beta}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.7e+119)
   (/ (+ 1.0 (/ beta (* (+ beta (* 2.0 i)) (+ 1.0 (/ 2.0 beta))))) 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 <= 2.7e+119) {
		tmp = (1.0 + (beta / ((beta + (2.0 * i)) * (1.0 + (2.0 / beta))))) / 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 <= 2.7d+119) then
        tmp = (1.0d0 + (beta / ((beta + (2.0d0 * i)) * (1.0d0 + (2.0d0 / beta))))) / 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 <= 2.7e+119) {
		tmp = (1.0 + (beta / ((beta + (2.0 * i)) * (1.0 + (2.0 / beta))))) / 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 <= 2.7e+119:
		tmp = (1.0 + (beta / ((beta + (2.0 * i)) * (1.0 + (2.0 / beta))))) / 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 <= 2.7e+119)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(Float64(beta + Float64(2.0 * i)) * Float64(1.0 + Float64(2.0 / beta))))) / 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 <= 2.7e+119)
		tmp = (1.0 + (beta / ((beta + (2.0 * i)) * (1.0 + (2.0 / beta))))) / 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, 2.7e+119], N[(N[(1.0 + N[(beta / N[(N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(2.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 2.6999999999999998e119

   1. Initial program 84.7%

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

      1. associate-/l/ 84.4%

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

      2. associate-+l+ 84.4%

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

      3. +-commutative 84.4%

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

      4. associate-+l+ 84.4%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in beta around inf 84.4%

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

   6. Taylor expanded in alpha around 0 95.2%

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

   7. Taylor expanded in i around 0 94.3%

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

   if 2.6999999999999998e119 < alpha

   1. Initial program 4.1%

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

   3. Simplified 29.6%

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

   5. Taylor expanded in beta around 0 28.1%

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

   6. Taylor expanded in alpha around inf 77.8%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.7 \cdot 10^{+119}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\beta + 2 \cdot i\right) \cdot \left(1 + \frac{2}{\beta}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3.8 \cdot 10^{+118}:\\ \;\;\;\;\frac{1 + \frac{1}{1 + \frac{2}{\beta}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 3.8e+118)
   (/ (+ 1.0 (/ 1.0 (+ 1.0 (/ 2.0 beta)))) 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 <= 3.8e+118) {
		tmp = (1.0 + (1.0 / (1.0 + (2.0 / beta)))) / 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 <= 3.8d+118) then
        tmp = (1.0d0 + (1.0d0 / (1.0d0 + (2.0d0 / beta)))) / 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 <= 3.8e+118) {
		tmp = (1.0 + (1.0 / (1.0 + (2.0 / beta)))) / 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 <= 3.8e+118:
		tmp = (1.0 + (1.0 / (1.0 + (2.0 / beta)))) / 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 <= 3.8e+118)
		tmp = Float64(Float64(1.0 + Float64(1.0 / Float64(1.0 + Float64(2.0 / beta)))) / 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 <= 3.8e+118)
		tmp = (1.0 + (1.0 / (1.0 + (2.0 / beta)))) / 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, 3.8e+118], N[(N[(1.0 + N[(1.0 / N[(1.0 + N[(2.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 3.80000000000000016e118

   1. Initial program 84.7%

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

      1. associate-/l/ 84.4%

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

      2. associate-+l+ 84.4%

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

      3. +-commutative 84.4%

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

      4. associate-+l+ 84.4%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in beta around inf 84.4%

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

   6. Taylor expanded in i around 0 83.4%

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

      1. +-commutative 83.4%

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

      2. associate-*r/ 83.4%

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

      3. metadata-eval 83.4%

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

   8. Simplified 83.4%

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

   9. Taylor expanded in alpha around 0 88.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{1 + 2 \cdot \frac{1}{\beta}}} + 1}{2} \]
   10. Step-by-step derivation

       1. associate-*r/ 88.2%

          \[\leadsto \frac{\frac{1}{1 + \color{blue}{\frac{2 \cdot 1}{\beta}}} + 1}{2} \]

       2. metadata-eval 88.2%

          \[\leadsto \frac{\frac{1}{1 + \frac{\color{blue}{2}}{\beta}} + 1}{2} \]

   11. Simplified 88.2%

       \[\leadsto \frac{\color{blue}{\frac{1}{1 + \frac{2}{\beta}}} + 1}{2} \]

   if 3.80000000000000016e118 < alpha

   1. Initial program 4.1%

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

   3. Simplified 29.6%

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

   5. Taylor expanded in beta around 0 28.1%

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

   6. Taylor expanded in alpha around inf 77.8%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.8 \cdot 10^{+118}:\\ \;\;\;\;\frac{1 + \frac{1}{1 + \frac{2}{\beta}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3.5 \cdot 10^{+119}:\\ \;\;\;\;\frac{1 + \frac{1}{1 + \frac{2}{\beta}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 3.5e+119)
   (/ (+ 1.0 (/ 1.0 (+ 1.0 (/ 2.0 beta)))) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.5e+119) {
		tmp = (1.0 + (1.0 / (1.0 + (2.0 / beta)))) / 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 <= 3.5d+119) then
        tmp = (1.0d0 + (1.0d0 / (1.0d0 + (2.0d0 / beta)))) / 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 <= 3.5e+119) {
		tmp = (1.0 + (1.0 / (1.0 + (2.0 / beta)))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 3.5e+119:
		tmp = (1.0 + (1.0 / (1.0 + (2.0 / beta)))) / 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 <= 3.5e+119)
		tmp = Float64(Float64(1.0 + Float64(1.0 / Float64(1.0 + Float64(2.0 / beta)))) / 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 <= 3.5e+119)
		tmp = (1.0 + (1.0 / (1.0 + (2.0 / beta)))) / 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, 3.5e+119], N[(N[(1.0 + N[(1.0 / N[(1.0 + N[(2.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 3.5000000000000001e119

   1. Initial program 84.7%

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

      1. associate-/l/ 84.4%

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

      2. associate-+l+ 84.4%

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

      3. +-commutative 84.4%

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

      4. associate-+l+ 84.4%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in beta around inf 84.4%

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

   6. Taylor expanded in i around 0 83.4%

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

      1. +-commutative 83.4%

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

      2. associate-*r/ 83.4%

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

      3. metadata-eval 83.4%

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

   8. Simplified 83.4%

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

   9. Taylor expanded in alpha around 0 88.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{1 + 2 \cdot \frac{1}{\beta}}} + 1}{2} \]
   10. Step-by-step derivation

       1. associate-*r/ 88.2%

          \[\leadsto \frac{\frac{1}{1 + \color{blue}{\frac{2 \cdot 1}{\beta}}} + 1}{2} \]

       2. metadata-eval 88.2%

          \[\leadsto \frac{\frac{1}{1 + \frac{\color{blue}{2}}{\beta}} + 1}{2} \]

   11. Simplified 88.2%

       \[\leadsto \frac{\color{blue}{\frac{1}{1 + \frac{2}{\beta}}} + 1}{2} \]

   if 3.5000000000000001e119 < alpha

   1. Initial program 4.1%

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

      1. associate-/l/ 3.1%

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

      2. associate-+l+ 3.1%

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

      3. +-commutative 3.1%

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

      4. associate-+l+ 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in beta around inf 2.4%

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

   6. Taylor expanded in alpha around inf 62.8%

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

   7. Taylor expanded in beta around 0 60.6%

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

      1. *-commutative 60.6%

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

   9. Simplified 60.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.5 \cdot 10^{+119}:\\ \;\;\;\;\frac{1 + \frac{1}{1 + \frac{2}{\beta}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3800:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 3800.0) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3800.0) {
		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 <= 3800.0d0) 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 <= 3800.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 3800.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3800.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3800.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 3800.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 3800

   1. Initial program 75.5%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in i around inf 77.2%

      \[\leadsto \frac{\color{blue}{1}}{2} \]

   if 3800 < beta

   1. Initial program 47.9%

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

   3. Simplified 85.1%

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

   5. Taylor expanded in beta around inf 63.9%

      \[\leadsto \frac{\color{blue}{2}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3800:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.4% 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 67.4%

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

3. Simplified 82.0%

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

5. Taylor expanded in i around inf 64.1%

   \[\leadsto \frac{\color{blue}{1}}{2} \]

6. Final simplification 64.1%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024090 
(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))