Octave 3.8, jcobi/2

Percentage Accurate: 63.0% → 97.9%

Time: 16.9s

Alternatives: 11

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

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

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)
     (/
      (+
       (/
        (/ (+ alpha beta) (+ (+ alpha 2.0) (fma 2.0 i beta)))
        (/ (+ alpha (fma 2.0 i beta)) (- beta alpha)))
       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 = ((((alpha + beta) / ((alpha + 2.0) + fma(2.0, i, beta))) / ((alpha + fma(2.0, i, beta)) / (beta - alpha))) + 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(Float64(Float64(Float64(Float64(alpha + beta) / Float64(Float64(alpha + 2.0) + fma(2.0, i, beta))) / Float64(Float64(alpha + fma(2.0, i, beta)) / Float64(beta - alpha))) + 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[(N[(alpha + beta), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] / N[(beta - 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 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.999999980000000011

   1. Initial program 3.7%

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

   3. Simplified 14.1%

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

   4. Taylor expanded in alpha around inf 91.3%

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

   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 76.8%

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

      1. associate-+r+ 76.8%

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

      2. associate-/l* 99.8%

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

      3. associate-+r+ 99.8%

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

      4. associate-/l/ 99.8%

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

      5. add-sqr-sqrt 73.0%

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

      6. times-frac 73.1%

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

      7. associate-+r+ 73.1%

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

      8. +-commutative 73.1%

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

      9. associate-+r+ 73.1%

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

      10. +-commutative 73.1%

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

      11. fma-udef 73.1%

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

   3. Applied egg-rr 73.1%

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

      1. associate-*r/ 73.1%

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

      2. *-commutative 73.1%

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

      3. associate-*r/ 73.1%

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

      4. rem-square-sqrt 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 98.0%

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

Alternative 2: 97.9% accurate, 0.5× speedup

Math

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

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)
     (/
      (+
       1.0
       (/
        (/ (+ alpha beta) (/ (+ alpha t_0) (- beta alpha)))
        (+ (+ alpha beta) (+ 2.0 (* 2.0 i)))))
      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 = (1.0 + (((alpha + beta) / ((alpha + t_0) / (beta - alpha))) / ((alpha + beta) + (2.0 + (2.0 * i))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = beta + (2.0d0 * i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0d0 + t_1)) <= (-0.99999998d0)) then
        tmp = ((t_0 + (2.0d0 + t_0)) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + (((alpha + beta) / ((alpha + t_0) / (beta - alpha))) / ((alpha + beta) + (2.0d0 + (2.0d0 * i))))) / 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 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 = (1.0 + (((alpha + beta) / ((alpha + t_0) / (beta - alpha))) / ((alpha + beta) + (2.0 + (2.0 * i))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = beta + (2.0 * i)
	t_1 = (alpha + beta) + (2.0 * i)
	tmp = 0
	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 = (1.0 + (((alpha + beta) / ((alpha + t_0) / (beta - alpha))) / ((alpha + beta) + (2.0 + (2.0 * i))))) / 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(Float64(1.0 + Float64(Float64(Float64(alpha + beta) / Float64(Float64(alpha + t_0) / Float64(beta - alpha))) / Float64(Float64(alpha + beta) + Float64(2.0 + Float64(2.0 * i))))) / 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);
	tmp = 0.0;
	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 = (1.0 + (((alpha + beta) / ((alpha + t_0) / (beta - alpha))) / ((alpha + beta) + (2.0 + (2.0 * i))))) / 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]}, 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[(1.0 + N[(N[(N[(alpha + beta), $MachinePrecision] / N[(N[(alpha + t$95$0), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 3.7%

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

   3. Simplified 14.1%

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

   4. Taylor expanded in alpha around inf 91.3%

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

   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 76.8%

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

      1. associate-/l* 99.8%

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

      2. associate-+l+ 99.8%

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

      3. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 98.0%

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

Alternative 3: 97.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := 2 + t_0\\ t_2 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_2}}{2 + t_2} \leq -0.5:\\ \;\;\;\;\frac{\frac{t_0 + t_1}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{t_1 \cdot \left(1 + 2 \cdot \frac{i}{\beta}\right)}}{2}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double t_2 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_2) / (2.0 + t_2)) <= -0.5) {
		tmp = ((t_0 + t_1) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / (t_1 * (1.0 + (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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = beta + (2.0d0 * i)
    t_1 = 2.0d0 + t_0
    t_2 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_2) / (2.0d0 + t_2)) <= (-0.5d0)) then
        tmp = ((t_0 + t_1) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + (beta / (t_1 * (1.0d0 + (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 = beta + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double t_2 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_2) / (2.0 + t_2)) <= -0.5) {
		tmp = ((t_0 + t_1) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / (t_1 * (1.0 + (2.0 * (i / beta)))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = beta + (2.0 * i)
	t_1 = 2.0 + t_0
	t_2 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_2) / (2.0 + t_2)) <= -0.5:
		tmp = ((t_0 + t_1) / alpha) / 2.0
	else:
		tmp = (1.0 + (beta / (t_1 * (1.0 + (2.0 * (i / beta)))))) / 2.0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (2.0 * i);
	t_1 = 2.0 + t_0;
	t_2 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_2) / (2.0 + t_2)) <= -0.5)
		tmp = ((t_0 + t_1) / alpha) / 2.0;
	else
		tmp = (1.0 + (beta / (t_1 * (1.0 + (2.0 * (i / beta)))))) / 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[(2.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(N[(t$95$0 + t$95$1), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(beta / N[(t$95$1 * N[(1.0 + N[(2.0 * N[(i / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.5

   1. Initial program 4.8%

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

   3. Simplified 15.0%

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

   4. Taylor expanded in alpha around inf 90.7%

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

   if -0.5 < (/.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 76.9%

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

      1. associate-/l* 100.0%

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

      2. associate-+l+ 100.0%

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

      3. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in beta around inf 99.5%

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

   5. Taylor expanded in alpha around 0 99.3%

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

4. Final simplification 97.4%

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

Alternative 4: 89.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ \mathbf{if}\;\alpha \leq 2.7 \cdot 10^{+74} \lor \neg \left(\alpha \leq 1.3 \cdot 10^{+117}\right) \land \alpha \leq 3.6 \cdot 10^{+144}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\beta + 2\right) + i \cdot 4}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0 + \left(2 + t_0\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i))))
   (if (or (<= alpha 2.7e+74)
           (and (not (<= alpha 1.3e+117)) (<= alpha 3.6e+144)))
     (/ (+ 1.0 (/ beta (+ (+ beta 2.0) (* i 4.0)))) 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 <= 2.7e+74) || (!(alpha <= 1.3e+117) && (alpha <= 3.6e+144))) {
		tmp = (1.0 + (beta / ((beta + 2.0) + (i * 4.0)))) / 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 <= 2.7d+74) .or. (.not. (alpha <= 1.3d+117)) .and. (alpha <= 3.6d+144)) then
        tmp = (1.0d0 + (beta / ((beta + 2.0d0) + (i * 4.0d0)))) / 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 <= 2.7e+74) || (!(alpha <= 1.3e+117) && (alpha <= 3.6e+144))) {
		tmp = (1.0 + (beta / ((beta + 2.0) + (i * 4.0)))) / 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 <= 2.7e+74) or (not (alpha <= 1.3e+117) and (alpha <= 3.6e+144)):
		tmp = (1.0 + (beta / ((beta + 2.0) + (i * 4.0)))) / 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 <= 2.7e+74) || (!(alpha <= 1.3e+117) && (alpha <= 3.6e+144)))
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(Float64(beta + 2.0) + Float64(i * 4.0)))) / 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 <= 2.7e+74) || (~((alpha <= 1.3e+117)) && (alpha <= 3.6e+144)))
		tmp = (1.0 + (beta / ((beta + 2.0) + (i * 4.0)))) / 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[Or[LessEqual[alpha, 2.7e+74], And[N[Not[LessEqual[alpha, 1.3e+117]], $MachinePrecision], LessEqual[alpha, 3.6e+144]]], N[(N[(1.0 + N[(beta / N[(N[(beta + 2.0), $MachinePrecision] + N[(i * 4.0), $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 < 2.6999999999999998e74 or 1.3e117 < alpha < 3.5999999999999997e144

   1. Initial program 74.6%

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

      1. associate-/l* 95.5%

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

      2. associate-+l+ 95.5%

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

      3. associate-+l+ 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in beta around inf 94.6%

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

   5. Taylor expanded in alpha around 0 94.6%

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

   6. Taylor expanded in beta around inf 94.2%

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

      1. associate-+r+ 94.2%

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

      2. +-commutative 94.2%

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

      3. *-commutative 94.2%

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

   8. Simplified 94.2%

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

   if 2.6999999999999998e74 < alpha < 1.3e117 or 3.5999999999999997e144 < alpha

   1. Initial program 3.9%

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

   3. Simplified 21.8%

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

   4. Taylor expanded in alpha around inf 83.6%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.7 \cdot 10^{+74} \lor \neg \left(\alpha \leq 1.3 \cdot 10^{+117}\right) \land \alpha \leq 3.6 \cdot 10^{+144}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\beta + 2\right) + i \cdot 4}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + 2 \cdot i\right) + \left(2 + \left(\beta + 2 \cdot i\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 5: 86.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.15 \cdot 10^{+75} \lor \neg \left(\alpha \leq 1.05 \cdot 10^{+112}\right) \land \alpha \leq 3 \cdot 10^{+144}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\beta + 2\right) + i \cdot 4}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (or (<= alpha 2.15e+75)
         (and (not (<= alpha 1.05e+112)) (<= alpha 3e+144)))
   (/ (+ 1.0 (/ beta (+ (+ beta 2.0) (* i 4.0)))) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if ((alpha <= 2.15e+75) || (!(alpha <= 1.05e+112) && (alpha <= 3e+144))) {
		tmp = (1.0 + (beta / ((beta + 2.0) + (i * 4.0)))) / 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 <= 2.15d+75) .or. (.not. (alpha <= 1.05d+112)) .and. (alpha <= 3d+144)) then
        tmp = (1.0d0 + (beta / ((beta + 2.0d0) + (i * 4.0d0)))) / 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 <= 2.15e+75) || (!(alpha <= 1.05e+112) && (alpha <= 3e+144))) {
		tmp = (1.0 + (beta / ((beta + 2.0) + (i * 4.0)))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if (alpha <= 2.15e+75) or (not (alpha <= 1.05e+112) and (alpha <= 3e+144)):
		tmp = (1.0 + (beta / ((beta + 2.0) + (i * 4.0)))) / 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 <= 2.15e+75) || (!(alpha <= 1.05e+112) && (alpha <= 3e+144)))
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(Float64(beta + 2.0) + Float64(i * 4.0)))) / 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 <= 2.15e+75) || (~((alpha <= 1.05e+112)) && (alpha <= 3e+144)))
		tmp = (1.0 + (beta / ((beta + 2.0) + (i * 4.0)))) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[Or[LessEqual[alpha, 2.15e+75], And[N[Not[LessEqual[alpha, 1.05e+112]], $MachinePrecision], LessEqual[alpha, 3e+144]]], N[(N[(1.0 + N[(beta / N[(N[(beta + 2.0), $MachinePrecision] + N[(i * 4.0), $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 < 2.1500000000000001e75 or 1.0499999999999999e112 < alpha < 2.9999999999999999e144

   1. Initial program 74.6%

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

      1. associate-/l* 95.5%

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

      2. associate-+l+ 95.5%

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

      3. associate-+l+ 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in beta around inf 94.6%

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

   5. Taylor expanded in alpha around 0 94.6%

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

   6. Taylor expanded in beta around inf 94.2%

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

      1. associate-+r+ 94.2%

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

      2. +-commutative 94.2%

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

      3. *-commutative 94.2%

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

   8. Simplified 94.2%

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

   if 2.1500000000000001e75 < alpha < 1.0499999999999999e112 or 2.9999999999999999e144 < alpha

   1. Initial program 3.9%

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

      1. associate-/l* 21.7%

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

      2. associate-+l+ 21.7%

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

      3. associate-+l+ 21.7%

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

   3. Simplified 21.7%

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

   4. Taylor expanded in beta around 0 16.0%

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

      1. associate-*r/ 16.0%

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

      2. mul-1-neg 16.0%

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

      3. +-commutative 16.0%

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

   6. Simplified 16.0%

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

   7. Taylor expanded in alpha around inf 66.3%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.15 \cdot 10^{+75} \lor \neg \left(\alpha \leq 1.05 \cdot 10^{+112}\right) \land \alpha \leq 3 \cdot 10^{+144}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\beta + 2\right) + i \cdot 4}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 6: 80.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.1 \cdot 10^{+75} \lor \neg \left(\alpha \leq 3 \cdot 10^{+119}\right) \land \alpha \leq 2.95 \cdot 10^{+144}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (or (<= alpha 2.1e+75) (and (not (<= alpha 3e+119)) (<= alpha 2.95e+144)))
   (/ (+ 1.0 (/ beta (+ (+ alpha 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 <= 2.1e+75) || (!(alpha <= 3e+119) && (alpha <= 2.95e+144))) {
		tmp = (1.0 + (beta / ((alpha + 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 <= 2.1d+75) .or. (.not. (alpha <= 3d+119)) .and. (alpha <= 2.95d+144)) then
        tmp = (1.0d0 + (beta / ((alpha + 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 <= 2.1e+75) || (!(alpha <= 3e+119) && (alpha <= 2.95e+144))) {
		tmp = (1.0 + (beta / ((alpha + 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 <= 2.1e+75) or (not (alpha <= 3e+119) and (alpha <= 2.95e+144)):
		tmp = (1.0 + (beta / ((alpha + 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 <= 2.1e+75) || (!(alpha <= 3e+119) && (alpha <= 2.95e+144)))
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(Float64(alpha + 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 <= 2.1e+75) || (~((alpha <= 3e+119)) && (alpha <= 2.95e+144)))
		tmp = (1.0 + (beta / ((alpha + 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[Or[LessEqual[alpha, 2.1e+75], And[N[Not[LessEqual[alpha, 3e+119]], $MachinePrecision], LessEqual[alpha, 2.95e+144]]], N[(N[(1.0 + N[(beta / N[(N[(alpha + beta), $MachinePrecision] + 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 < 2.09999999999999999e75 or 3.00000000000000001e119 < alpha < 2.94999999999999994e144

   1. Initial program 74.6%

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

   2. Taylor expanded in beta around inf 94.0%

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

   3. Taylor expanded in i around 0 85.2%

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

      1. +-commutative 85.2%

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

   5. Simplified 85.2%

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

   if 2.09999999999999999e75 < alpha < 3.00000000000000001e119 or 2.94999999999999994e144 < alpha

   1. Initial program 3.9%

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

      1. associate-/l* 21.7%

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

      2. associate-+l+ 21.7%

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

      3. associate-+l+ 21.7%

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

   3. Simplified 21.7%

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

   4. Taylor expanded in beta around 0 16.0%

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

      1. associate-*r/ 16.0%

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

      2. mul-1-neg 16.0%

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

      3. +-commutative 16.0%

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

   6. Simplified 16.0%

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

   7. Taylor expanded in alpha around inf 66.3%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.1 \cdot 10^{+75} \lor \neg \left(\alpha \leq 3 \cdot 10^{+119}\right) \land \alpha \leq 2.95 \cdot 10^{+144}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 80.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.8 \cdot 10^{+75} \lor \neg \left(\alpha \leq 1.25 \cdot 10^{+113}\right) \land \alpha \leq 2.7 \cdot 10^{+144}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (or (<= alpha 1.8e+75)
         (and (not (<= alpha 1.25e+113)) (<= alpha 2.7e+144)))
   (/ (+ 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 <= 1.8e+75) || (!(alpha <= 1.25e+113) && (alpha <= 2.7e+144))) {
		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 <= 1.8d+75) .or. (.not. (alpha <= 1.25d+113)) .and. (alpha <= 2.7d+144)) 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 <= 1.8e+75) || (!(alpha <= 1.25e+113) && (alpha <= 2.7e+144))) {
		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 <= 1.8e+75) or (not (alpha <= 1.25e+113) and (alpha <= 2.7e+144)):
		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 <= 1.8e+75) || (!(alpha <= 1.25e+113) && (alpha <= 2.7e+144)))
		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 <= 1.8e+75) || (~((alpha <= 1.25e+113)) && (alpha <= 2.7e+144)))
		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[Or[LessEqual[alpha, 1.8e+75], And[N[Not[LessEqual[alpha, 1.25e+113]], $MachinePrecision], LessEqual[alpha, 2.7e+144]]], 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 < 1.8e75 or 1.25e113 < alpha < 2.70000000000000015e144

   1. Initial program 74.6%

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

      1. associate-/l* 95.5%

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

      2. associate-+l+ 95.5%

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

      3. associate-+l+ 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in beta around inf 94.6%

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

   5. Taylor expanded in alpha around 0 94.6%

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

   6. Taylor expanded in i around 0 84.6%

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

      1. +-commutative 84.6%

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

   8. Simplified 84.6%

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

   if 1.8e75 < alpha < 1.25e113 or 2.70000000000000015e144 < alpha

   1. Initial program 3.9%

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

      1. associate-/l* 21.7%

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

      2. associate-+l+ 21.7%

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

      3. associate-+l+ 21.7%

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

   3. Simplified 21.7%

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

   4. Taylor expanded in beta around 0 16.0%

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

      1. associate-*r/ 16.0%

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

      2. mul-1-neg 16.0%

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

      3. +-commutative 16.0%

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

   6. Simplified 16.0%

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

   7. Taylor expanded in alpha around inf 66.3%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.8 \cdot 10^{+75} \lor \neg \left(\alpha \leq 1.25 \cdot 10^{+113}\right) \land \alpha \leq 2.7 \cdot 10^{+144}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 79.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot i \leq 10^{+45}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2 \cdot i}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= (* 2.0 i) 1e+45)
   (/ (+ 1.0 (/ beta (+ (+ alpha beta) 2.0))) 2.0)
   (/ (+ 1.0 (/ beta (+ beta (* 2.0 i)))) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if ((2.0 * i) <= 1e+45) {
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	} else {
		tmp = (1.0 + (beta / (beta + (2.0 * i)))) / 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 ((2.0d0 * i) <= 1d+45) then
        tmp = (1.0d0 + (beta / ((alpha + beta) + 2.0d0))) / 2.0d0
    else
        tmp = (1.0d0 + (beta / (beta + (2.0d0 * i)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if ((2.0 * i) <= 1e+45) {
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	} else {
		tmp = (1.0 + (beta / (beta + (2.0 * i)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if (2.0 * i) <= 1e+45:
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0
	else:
		tmp = (1.0 + (beta / (beta + (2.0 * i)))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (Float64(2.0 * i) <= 1e+45)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(Float64(alpha + beta) + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + Float64(2.0 * i)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if ((2.0 * i) <= 1e+45)
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	else
		tmp = (1.0 + (beta / (beta + (2.0 * i)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[N[(2.0 * i), $MachinePrecision], 1e+45], N[(N[(1.0 + N[(beta / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(beta / N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 2 i) < 9.9999999999999993e44

   1. Initial program 60.0%

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

   2. Taylor expanded in beta around inf 76.5%

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

   3. Taylor expanded in i around 0 76.5%

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

      1. +-commutative 76.5%

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

   5. Simplified 76.5%

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

   if 9.9999999999999993e44 < (*.f64 2 i)

   1. Initial program 62.8%

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

      1. associate-/l* 86.5%

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

      2. associate-+l+ 86.5%

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

      3. associate-+l+ 86.5%

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

   3. Simplified 86.5%

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

   4. Taylor expanded in alpha around 0 85.4%

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

   5. Taylor expanded in alpha around inf 84.4%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot i \leq 10^{+45}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2 \cdot i}}{2}\\ \end{array} \]

Alternative 9: 75.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 2.1 \cdot 10^{+212}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 2.1e+212) (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 2.1e+212) {
		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 <= 2.1d+212) 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 <= 2.1e+212) {
		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 <= 2.1e+212:
		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 <= 2.1e+212)
		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 <= 2.1e+212)
		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, 2.1e+212], 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 < 2.1e212

   1. Initial program 58.9%

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

      1. associate-/l* 78.3%

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

      2. associate-+l+ 78.3%

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

      3. associate-+l+ 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in beta around inf 77.5%

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

   5. Taylor expanded in alpha around 0 77.5%

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

   6. Taylor expanded in i around 0 74.2%

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

      1. +-commutative 74.2%

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

   8. Simplified 74.2%

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

   if 2.1e212 < i

   1. Initial program 71.2%

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

   2. Taylor expanded in i around inf 89.9%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2.1 \cdot 10^{+212}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 10: 70.8% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+165}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 6e+165) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6e+165) {
		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 <= 6d+165) 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 <= 6e+165) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 6e+165:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6e+165)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 6e+165)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 6e+165], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 5.99999999999999981e165

   1. Initial program 74.1%

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

   2. Taylor expanded in i around inf 71.8%

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

   if 5.99999999999999981e165 < beta

   1. Initial program 3.1%

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

   2. Taylor expanded in beta around inf 86.6%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+165}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11: 61.6% 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 61.4%

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

2. Taylor expanded in i around inf 63.9%

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

3. Final simplification 63.9%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023319 
(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))