Octave 3.8, jcobi/2

Percentage Accurate: 63.0% → 97.9%

Time: 26.7s

Alternatives: 14

Speedup: 9.5×

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: 97.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := 2 + t_1\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{t_2} \leq -0.9999:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha} + \left(\left(t_0 + \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(2, i, \beta\right) - -2}{\alpha}\right) - t_0 \cdot t_0\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\alpha + \beta\right) \cdot \frac{1}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}}{t_2}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (+ 2.0 (fma 2.0 beta (* i 4.0))) alpha))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (+ 2.0 t_1)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) t_2) -0.9999)
     (/
      (+
       (* (/ beta alpha) (/ beta alpha))
       (-
        (+
         t_0
         (* (/ (fma 2.0 i beta) alpha) (/ (- (fma 2.0 i beta) -2.0) alpha)))
        (* t_0 t_0)))
      2.0)
     (/
      (+
       1.0
       (/
        (*
         (+ alpha beta)
         (/ 1.0 (/ (+ beta (fma 2.0 i alpha)) (- beta alpha))))
        t_2))
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (2.0 + fma(2.0, beta, (i * 4.0))) / alpha;
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = 2.0 + t_1;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= -0.9999) {
		tmp = (((beta / alpha) * (beta / alpha)) + ((t_0 + ((fma(2.0, i, beta) / alpha) * ((fma(2.0, i, beta) - -2.0) / alpha))) - (t_0 * t_0))) / 2.0;
	} else {
		tmp = (1.0 + (((alpha + beta) * (1.0 / ((beta + fma(2.0, i, alpha)) / (beta - alpha)))) / t_2)) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(2.0 + fma(2.0, beta, Float64(i * 4.0))) / alpha)
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(2.0 + t_1)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / t_2) <= -0.9999)
		tmp = Float64(Float64(Float64(Float64(beta / alpha) * Float64(beta / alpha)) + Float64(Float64(t_0 + Float64(Float64(fma(2.0, i, beta) / alpha) * Float64(Float64(fma(2.0, i, beta) - -2.0) / alpha))) - Float64(t_0 * t_0))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(alpha + beta) * Float64(1.0 / Float64(Float64(beta + fma(2.0, i, alpha)) / Float64(beta - alpha)))) / t_2)) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(2.0 + N[(2.0 * beta + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], -0.9999], N[(N[(N[(N[(beta / alpha), $MachinePrecision] * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 + N[(N[(N[(2.0 * i + beta), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(N[(2.0 * i + beta), $MachinePrecision] - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + 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] / t$95$2), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.99990000000000001

   1. Initial program 7.4%

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

   3. Simplified 24.4%

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

   4. Taylor expanded in alpha around inf 77.8%

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

   6. Simplified 85.7%

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

   if -0.99990000000000001 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 83.1%

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

      1. *-un-lft-identity 83.1%

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

      2. associate-/l* 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r+ 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-def 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha} + \left(\left(\frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha} + \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(2, i, \beta\right) - -2}{\alpha}\right) - \frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha} \cdot \frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\alpha + \beta\right) \cdot \frac{1}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array} \]

Alternative 2: 97.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{2 + t_1} \leq -0.99999998:\\ \;\;\;\;\frac{\frac{t_0 + \left(2 + t_0\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, \frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i))) (t_1 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ 2.0 t_1))
        -0.99999998)
     (/ (/ (+ t_0 (+ 2.0 t_0)) alpha) 2.0)
     (/
      (fma
       (/ (+ alpha beta) (+ alpha (+ beta (fma 2.0 i 2.0))))
       (/ (- beta alpha) (+ alpha (fma 2.0 i beta)))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.99999998) {
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	} else {
		tmp = fma(((alpha + beta) / (alpha + (beta + fma(2.0, i, 2.0)))), ((beta - alpha) / (alpha + fma(2.0, i, beta))), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(2.0 + t_1)) <= -0.99999998)
		tmp = Float64(Float64(Float64(t_0 + Float64(2.0 + t_0)) / alpha) / 2.0);
	else
		tmp = Float64(fma(Float64(Float64(alpha + beta) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))), Float64(Float64(beta - alpha) / Float64(alpha + fma(2.0, i, beta))), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], -0.99999998], N[(N[(N[(t$95$0 + N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(alpha + beta), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(2.0 * i + beta), $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 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.999999980000000011

   1. Initial program 2.6%

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

   3. Simplified 20.9%

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

   4. Taylor expanded in alpha around inf 85.3%

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

   if -0.999999980000000011 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 82.7%

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

   3. Simplified 99.0%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999998:\\ \;\;\;\;\frac{\frac{\left(\beta + 2 \cdot i\right) + \left(2 + \left(\beta + 2 \cdot i\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, \frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \]

Alternative 3: 97.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := 2 + t_1\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{t_2} \leq -0.99999998:\\ \;\;\;\;\frac{\frac{t_0 + \left(2 + t_0\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\alpha + \beta\right) \cdot \frac{1}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}}{t_2}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (+ 2.0 t_1)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) t_2) -0.99999998)
     (/ (/ (+ t_0 (+ 2.0 t_0)) alpha) 2.0)
     (/
      (+
       1.0
       (/
        (*
         (+ alpha beta)
         (/ 1.0 (/ (+ beta (fma 2.0 i alpha)) (- beta alpha))))
        t_2))
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = 2.0 + t_1;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= -0.99999998) {
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((alpha + beta) * (1.0 / ((beta + fma(2.0, i, alpha)) / (beta - alpha)))) / t_2)) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(2.0 + t_1)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / t_2) <= -0.99999998)
		tmp = Float64(Float64(Float64(t_0 + Float64(2.0 + t_0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(alpha + beta) * Float64(1.0 / Float64(Float64(beta + fma(2.0, i, alpha)) / Float64(beta - alpha)))) / t_2)) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], -0.99999998], N[(N[(N[(t$95$0 + N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + 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] / t$95$2), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.999999980000000011

   1. Initial program 2.6%

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

   3. Simplified 20.9%

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

   4. Taylor expanded in alpha around inf 85.3%

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

   if -0.999999980000000011 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 82.7%

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

      1. *-un-lft-identity 82.7%

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

      2. associate-/l* 99.0%

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

      3. +-commutative 99.0%

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

      4. associate-+r+ 99.0%

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

      5. +-commutative 99.0%

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

      6. fma-def 99.0%

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

   3. Applied egg-rr 99.0%

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

      1. div-inv 99.0%

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

   5. Applied egg-rr 99.0%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999998:\\ \;\;\;\;\frac{\frac{\left(\beta + 2 \cdot i\right) + \left(2 + \left(\beta + 2 \cdot i\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\alpha + \beta\right) \cdot \frac{1}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array} \]

Alternative 4: 97.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := 2 + t_1\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{t_2} \leq -0.99999998:\\ \;\;\;\;\frac{\frac{t_0 + \left(2 + t_0\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}}{t_2}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (+ 2.0 t_1)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) t_2) -0.99999998)
     (/ (/ (+ t_0 (+ 2.0 t_0)) alpha) 2.0)
     (/
      (+
       1.0
       (/
        (/ (+ alpha beta) (/ (+ beta (fma 2.0 i alpha)) (- beta alpha)))
        t_2))
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = 2.0 + t_1;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= -0.99999998) {
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((alpha + beta) / ((beta + fma(2.0, i, alpha)) / (beta - alpha))) / t_2)) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(2.0 + t_1)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / t_2) <= -0.99999998)
		tmp = Float64(Float64(Float64(t_0 + Float64(2.0 + t_0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(alpha + beta) / Float64(Float64(beta + fma(2.0, i, alpha)) / Float64(beta - alpha))) / t_2)) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], -0.99999998], N[(N[(N[(t$95$0 + N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(alpha + beta), $MachinePrecision] / N[(N[(beta + N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.999999980000000011

   1. Initial program 2.6%

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

   3. Simplified 20.9%

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

   4. Taylor expanded in alpha around inf 85.3%

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

   if -0.999999980000000011 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 82.7%

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

      1. *-un-lft-identity 82.7%

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

      2. associate-/l* 99.0%

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

      3. +-commutative 99.0%

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

      4. associate-+r+ 99.0%

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

      5. +-commutative 99.0%

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

      6. fma-def 99.0%

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

   3. Applied egg-rr 99.0%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999998:\\ \;\;\;\;\frac{\frac{\left(\beta + 2 \cdot i\right) + \left(2 + \left(\beta + 2 \cdot i\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array} \]

Alternative 5: 97.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := 2 + t_1\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{t_2} \leq -0.999999895:\\ \;\;\;\;\frac{\frac{t_0 + \left(2 + t_0\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\alpha + \beta\right) \cdot \frac{1}{\frac{\alpha + \beta}{\beta - \alpha}}}{t_2}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (+ 2.0 t_1)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) t_2) -0.999999895)
     (/ (/ (+ t_0 (+ 2.0 t_0)) alpha) 2.0)
     (/
      (+
       1.0
       (/ (* (+ alpha beta) (/ 1.0 (/ (+ alpha beta) (- beta alpha)))) t_2))
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = 2.0 + t_1;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= -0.999999895) {
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((alpha + beta) * (1.0 / ((alpha + beta) / (beta - alpha)))) / t_2)) / 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) :: t_2
    real(8) :: tmp
    t_0 = beta + (2.0d0 * i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = 2.0d0 + t_1
    if (((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= (-0.999999895d0)) then
        tmp = ((t_0 + (2.0d0 + t_0)) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + (((alpha + beta) * (1.0d0 / ((alpha + beta) / (beta - alpha)))) / t_2)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = 2.0 + t_1;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= -0.999999895) {
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((alpha + beta) * (1.0 / ((alpha + beta) / (beta - alpha)))) / t_2)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = beta + (2.0 * i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = 2.0 + t_1
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= -0.999999895:
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0
	else:
		tmp = (1.0 + (((alpha + beta) * (1.0 / ((alpha + beta) / (beta - alpha)))) / t_2)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(2.0 + t_1)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / t_2) <= -0.999999895)
		tmp = Float64(Float64(Float64(t_0 + Float64(2.0 + t_0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(alpha + beta) * Float64(1.0 / Float64(Float64(alpha + beta) / Float64(beta - alpha)))) / t_2)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (2.0 * i);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = 2.0 + t_1;
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= -0.999999895)
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	else
		tmp = (1.0 + (((alpha + beta) * (1.0 / ((alpha + beta) / (beta - alpha)))) / t_2)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], -0.999999895], N[(N[(N[(t$95$0 + N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(alpha + beta), $MachinePrecision] * N[(1.0 / N[(N[(alpha + beta), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.99999989499999997

   1. Initial program 5.5%

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

   3. Simplified 23.1%

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

   4. Taylor expanded in alpha around inf 83.9%

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

   if -0.99999989499999997 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 83.0%

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

      1. *-un-lft-identity 83.0%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. +-commutative 99.5%

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

      6. fma-def 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in i around 0 98.4%

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

      1. +-commutative 98.4%

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

   8. Simplified 98.4%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999999895:\\ \;\;\;\;\frac{\frac{\left(\beta + 2 \cdot i\right) + \left(2 + \left(\beta + 2 \cdot i\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\alpha + \beta\right) \cdot \frac{1}{\frac{\alpha + \beta}{\beta - \alpha}}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array} \]

Alternative 6: 89.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ \mathbf{if}\;\alpha \leq 1.56 \cdot 10^{+69}:\\ \;\;\;\;\frac{1 + \frac{\beta}{t_0} \cdot \frac{\beta}{2 \cdot i + \left(\beta + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0 + \left(2 + t_0\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i))))
   (if (<= alpha 1.56e+69)
     (/ (+ 1.0 (* (/ beta t_0) (/ beta (+ (* 2.0 i) (+ beta 2.0))))) 2.0)
     (/ (/ (+ t_0 (+ 2.0 t_0)) alpha) 2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double tmp;
	if (alpha <= 1.56e+69) {
		tmp = (1.0 + ((beta / t_0) * (beta / ((2.0 * i) + (beta + 2.0))))) / 2.0;
	} else {
		tmp = ((t_0 + (2.0 + t_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) :: t_0
    real(8) :: tmp
    t_0 = beta + (2.0d0 * i)
    if (alpha <= 1.56d+69) then
        tmp = (1.0d0 + ((beta / t_0) * (beta / ((2.0d0 * i) + (beta + 2.0d0))))) / 2.0d0
    else
        tmp = ((t_0 + (2.0d0 + t_0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double tmp;
	if (alpha <= 1.56e+69) {
		tmp = (1.0 + ((beta / t_0) * (beta / ((2.0 * i) + (beta + 2.0))))) / 2.0;
	} else {
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = beta + (2.0 * i)
	tmp = 0
	if alpha <= 1.56e+69:
		tmp = (1.0 + ((beta / t_0) * (beta / ((2.0 * i) + (beta + 2.0))))) / 2.0
	else:
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	tmp = 0.0
	if (alpha <= 1.56e+69)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta / t_0) * Float64(beta / Float64(Float64(2.0 * i) + Float64(beta + 2.0))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(t_0 + Float64(2.0 + t_0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (2.0 * i);
	tmp = 0.0;
	if (alpha <= 1.56e+69)
		tmp = (1.0 + ((beta / t_0) * (beta / ((2.0 * i) + (beta + 2.0))))) / 2.0;
	else
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 1.56e+69], N[(N[(1.0 + N[(N[(beta / t$95$0), $MachinePrecision] * N[(beta / N[(N[(2.0 * i), $MachinePrecision] + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(t$95$0 + N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.56000000000000007e69

   1. Initial program 83.2%

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

      1. *-un-lft-identity 83.2%

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

      2. associate-/l* 97.7%

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

      3. +-commutative 97.7%

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

      4. associate-+r+ 97.7%

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

      5. +-commutative 97.7%

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

      6. fma-def 97.7%

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

   3. Applied egg-rr 97.7%

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

   4. Taylor expanded in alpha around 0 80.9%

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

      1. unpow2 80.9%

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

      2. *-commutative 80.9%

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

      3. times-frac 95.9%

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

      4. associate-+r+ 95.9%

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

      5. +-commutative 95.9%

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

   6. Simplified 95.9%

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

   if 1.56000000000000007e69 < alpha

   1. Initial program 11.6%

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

   3. Simplified 34.6%

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

   4. Taylor expanded in alpha around inf 72.2%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.56 \cdot 10^{+69}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2 \cdot i} \cdot \frac{\beta}{2 \cdot i + \left(\beta + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + 2 \cdot i\right) + \left(2 + \left(\beta + 2 \cdot i\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 88.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ \mathbf{if}\;\alpha \leq 4.5 \cdot 10^{+67}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0 + \left(2 + t_0\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i))))
   (if (<= alpha 4.5e+67)
     (/ (+ 1.0 (/ beta (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
     (/ (/ (+ t_0 (+ 2.0 t_0)) alpha) 2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double tmp;
	if (alpha <= 4.5e+67) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((t_0 + (2.0 + t_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) :: t_0
    real(8) :: tmp
    t_0 = beta + (2.0d0 * i)
    if (alpha <= 4.5d+67) then
        tmp = (1.0d0 + (beta / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = ((t_0 + (2.0d0 + t_0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double tmp;
	if (alpha <= 4.5e+67) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = beta + (2.0 * i)
	tmp = 0
	if alpha <= 4.5e+67:
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	tmp = 0.0
	if (alpha <= 4.5e+67)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(t_0 + Float64(2.0 + t_0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (2.0 * i);
	tmp = 0.0;
	if (alpha <= 4.5e+67)
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 4.5e+67], N[(N[(1.0 + N[(beta / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(t$95$0 + N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.4999999999999998e67

   1. Initial program 83.2%

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

   2. Taylor expanded in beta around inf 95.1%

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

   if 4.4999999999999998e67 < alpha

   1. Initial program 11.6%

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

   3. Simplified 34.6%

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

   4. Taylor expanded in alpha around inf 72.2%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.5 \cdot 10^{+67}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + 2 \cdot i\right) + \left(2 + \left(\beta + 2 \cdot i\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 85.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.8 \cdot 10^{+110}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 4.8e+110)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.8e+110) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 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 <= 4.8d+110) then
        tmp = (1.0d0 + (beta / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 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 <= 4.8e+110) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 4.8e+110:
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 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 <= 4.8e+110)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 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 <= 4.8e+110)
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 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, 4.8e+110], N[(N[(1.0 + N[(beta / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $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 < 4.80000000000000025e110

   1. Initial program 80.4%

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

   2. Taylor expanded in beta around inf 92.1%

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

   if 4.80000000000000025e110 < alpha

   1. Initial program 7.7%

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

      1. associate-/l/ 6.8%

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

      2. +-commutative 6.8%

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

      3. +-commutative 6.8%

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

      4. associate-+l+ 6.8%

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

      5. +-commutative 6.8%

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

      6. associate-+l+ 6.8%

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

   3. Simplified 6.8%

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

   4. Taylor expanded in beta around 0 5.3%

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

   5. Taylor expanded in alpha around inf 58.5%

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

      1. distribute-rgt1-in 58.5%

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

      2. metadata-eval 58.5%

         \[\leadsto \frac{\frac{\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + 4 \cdot i\right)}{\alpha}}{2} \]

      3. mul0-lft 58.5%

         \[\leadsto \frac{\frac{\color{blue}{0} - -1 \cdot \left(2 + 4 \cdot i\right)}{\alpha}}{2} \]

      4. neg-sub0 58.5%

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

      5. mul-1-neg 58.5%

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

      6. remove-double-neg 58.5%

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

      7. *-commutative 58.5%

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

   7. Simplified 58.5%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.8 \cdot 10^{+110}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 9: 85.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.02 \cdot 10^{+99}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\beta + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.02e+99)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ beta (* 2.0 i))))) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.02e+99) {
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 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 <= 1.02d+99) then
        tmp = (1.0d0 + (beta / (2.0d0 + (beta + (2.0d0 * i))))) / 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 <= 1.02e+99) {
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.02e+99:
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 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 <= 1.02e+99)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + Float64(beta + Float64(2.0 * i))))) / 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 <= 1.02e+99)
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 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, 1.02e+99], N[(N[(1.0 + N[(beta / N[(2.0 + N[(beta + N[(2.0 * i), $MachinePrecision]), $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 < 1.01999999999999998e99

   1. Initial program 80.4%

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

   2. Taylor expanded in beta around inf 92.1%

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

   3. Taylor expanded in alpha around 0 92.1%

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

   if 1.01999999999999998e99 < alpha

   1. Initial program 7.7%

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

      1. associate-/l/ 6.8%

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

      2. +-commutative 6.8%

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

      3. +-commutative 6.8%

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

      4. associate-+l+ 6.8%

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

      5. +-commutative 6.8%

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

      6. associate-+l+ 6.8%

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

   3. Simplified 6.8%

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

   4. Taylor expanded in beta around 0 5.3%

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

   5. Taylor expanded in alpha around inf 58.5%

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

      1. distribute-rgt1-in 58.5%

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

      2. metadata-eval 58.5%

         \[\leadsto \frac{\frac{\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + 4 \cdot i\right)}{\alpha}}{2} \]

      3. mul0-lft 58.5%

         \[\leadsto \frac{\frac{\color{blue}{0} - -1 \cdot \left(2 + 4 \cdot i\right)}{\alpha}}{2} \]

      4. neg-sub0 58.5%

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

      5. mul-1-neg 58.5%

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

      6. remove-double-neg 58.5%

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

      7. *-commutative 58.5%

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

   7. Simplified 58.5%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.02 \cdot 10^{+99}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\beta + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 10: 75.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 3.1 \cdot 10^{+159}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 3.1e+159) (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 3.1e+159) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	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 (i <= 3.1d+159) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 3.1e+159) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if i <= 3.1e+159:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 3.1e+159)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 3.1e+159)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 3.1e+159], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if i < 3.0999999999999998e159

   1. Initial program 62.4%

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

   3. Simplified 76.0%

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

   4. Taylor expanded in i around 0 71.0%

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

   5. Taylor expanded in alpha around 0 71.2%

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

      1. +-commutative 71.2%

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

   7. Simplified 71.2%

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

   if 3.0999999999999998e159 < i

   1. Initial program 68.8%

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

   3. Simplified 96.0%

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

   4. Taylor expanded in i around inf 87.4%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 3.1 \cdot 10^{+159}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 11: 77.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3.3 \cdot 10^{+78}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 3.3e+78)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.3e+78) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / 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 <= 3.3d+78) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.3e+78) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 3.3e+78:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 3.3e+78)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 3.3e+78)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 3.3e+78], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3.3e78

   1. Initial program 81.9%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in i around 0 85.0%

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

   5. Taylor expanded in alpha around 0 87.4%

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

      1. +-commutative 87.4%

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

   7. Simplified 87.4%

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

   if 3.3e78 < alpha

   1. Initial program 9.8%

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

   3. Simplified 33.0%

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

   4. Taylor expanded in i around 0 13.4%

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

   5. Taylor expanded in alpha around inf 54.2%

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

      1. *-commutative 54.2%

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

   7. Simplified 54.2%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.3 \cdot 10^{+78}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 12: 79.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.75 \cdot 10^{+108}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.75e+108)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.75e+108) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 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 <= 1.75d+108) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 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 <= 1.75e+108) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.75e+108:
		tmp = (1.0 + (beta / (beta + 2.0))) / 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 <= 1.75e+108)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 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 <= 1.75e+108)
		tmp = (1.0 + (beta / (beta + 2.0))) / 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, 1.75e+108], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $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 < 1.7500000000000001e108

   1. Initial program 80.4%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in i around 0 83.3%

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

   5. Taylor expanded in alpha around 0 85.9%

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

      1. +-commutative 85.9%

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

   7. Simplified 85.9%

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

   if 1.7500000000000001e108 < alpha

   1. Initial program 7.7%

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

      1. associate-/l/ 6.8%

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

      2. +-commutative 6.8%

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

      3. +-commutative 6.8%

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

      4. associate-+l+ 6.8%

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

      5. +-commutative 6.8%

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

      6. associate-+l+ 6.8%

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

   3. Simplified 6.8%

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

   4. Taylor expanded in beta around 0 5.3%

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

   5. Taylor expanded in alpha around inf 58.5%

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

      1. distribute-rgt1-in 58.5%

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

      2. metadata-eval 58.5%

         \[\leadsto \frac{\frac{\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + 4 \cdot i\right)}{\alpha}}{2} \]

      3. mul0-lft 58.5%

         \[\leadsto \frac{\frac{\color{blue}{0} - -1 \cdot \left(2 + 4 \cdot i\right)}{\alpha}}{2} \]

      4. neg-sub0 58.5%

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

      5. mul-1-neg 58.5%

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

      6. remove-double-neg 58.5%

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

      7. *-commutative 58.5%

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

   7. Simplified 58.5%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.75 \cdot 10^{+108}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 13: 72.2% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+29}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 1.15e+29) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.15e+29) {
		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 <= 1.15d+29) 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 <= 1.15e+29) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.15e+29:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.15e+29)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.15e+29)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.15e+29], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 1.1500000000000001e29

   1. Initial program 73.1%

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

   3. Simplified 78.0%

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

   4. Taylor expanded in i around inf 73.3%

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

   if 1.1500000000000001e29 < beta

   1. Initial program 42.9%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in beta around inf 69.1%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+29}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14: 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 63.9%

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

3. Simplified 80.7%

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

4. Taylor expanded in i around inf 59.9%

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

5. Final simplification 59.9%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023279 
(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))