Octave 3.8, jcobi/2

Percentage Accurate: 63.8% → 97.2%

Time: 21.4s

Alternatives: 13

Speedup: 4.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 63.8% 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: 97.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -1.0)
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(Float64(4.0 * Float64(i / alpha)) + Float64(2.0 * Float64(1.0 / alpha)))) / 2.0);
	else
		tmp = Float64(fma(Float64(alpha + beta), Float64(Float64(Float64(beta - alpha) * Float64(1.0 / Float64(alpha + Float64(beta + fma(2.0, i, 2.0))))) / 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[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(alpha + beta), $MachinePrecision] * N[(N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -1

   1. Initial program 1.7%

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

   3. Simplified 9.1%

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

   5. Taylor expanded in alpha around inf 96.5%

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

   6. Taylor expanded in beta around 0 96.5%

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

   if -1 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 79.0%

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

   3. Simplified 99.5%

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

      1. div-inv 99.5%

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

   6. Applied egg-rr 99.5%

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

4. Final simplification 98.9%

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

Alternative 2: 97.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -1.0)
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(Float64(4.0 * Float64(i / alpha)) + Float64(2.0 * Float64(1.0 / alpha)))) / 2.0);
	else
		tmp = Float64(fma(Float64(alpha + beta), Float64(Float64(Float64(beta - alpha) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))) / 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[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(alpha + beta), $MachinePrecision] * N[(N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -1

   1. Initial program 1.7%

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

   3. Simplified 9.1%

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

   5. Taylor expanded in alpha around inf 96.5%

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

   6. Taylor expanded in beta around 0 96.5%

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

   if -1 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 79.0%

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

   3. Simplified 99.5%

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

4. Final simplification 98.8%

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

Alternative 3: 97.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -1

   1. Initial program 1.7%

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

   3. Simplified 9.1%

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

   5. Taylor expanded in alpha around inf 96.5%

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

   6. Taylor expanded in beta around 0 96.5%

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

   if -1 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 79.0%

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

   3. Simplified 99.5%

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

4. Final simplification 98.8%

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

Alternative 4: 96.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -1

   1. Initial program 1.7%

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

   3. Simplified 9.1%

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

   5. Taylor expanded in alpha around inf 96.5%

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

   6. Taylor expanded in beta around 0 96.5%

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

   if -1 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 79.0%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in alpha around 0 98.6%

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

4. Final simplification 98.1%

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

Alternative 5: 95.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_1}}{2 + t\_1}\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \mathbf{elif}\;t\_2 \leq 10^{-41}:\\ \;\;\;\;\frac{t\_2 + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\frac{\beta}{t\_0} - \frac{\alpha}{t\_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ 2.0 t_1))))
   (if (<= t_2 -1.0)
     (/
      (+ (* 2.0 (/ beta alpha)) (+ (* 4.0 (/ i alpha)) (* 2.0 (/ 1.0 alpha))))
      2.0)
     (if (<= t_2 1e-41)
       (/ (+ t_2 1.0) 2.0)
       (/ (+ 1.0 (- (/ beta t_0) (/ alpha t_0))) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + 2.0;
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = (((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1);
	double tmp;
	if (t_2 <= -1.0) {
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0;
	} else if (t_2 <= 1e-41) {
		tmp = (t_2 + 1.0) / 2.0;
	} else {
		tmp = (1.0 + ((beta / t_0) - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (alpha + beta) + 2.0d0
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = (((alpha + beta) * (beta - alpha)) / t_1) / (2.0d0 + t_1)
    if (t_2 <= (-1.0d0)) then
        tmp = ((2.0d0 * (beta / alpha)) + ((4.0d0 * (i / alpha)) + (2.0d0 * (1.0d0 / alpha)))) / 2.0d0
    else if (t_2 <= 1d-41) then
        tmp = (t_2 + 1.0d0) / 2.0d0
    else
        tmp = (1.0d0 + ((beta / t_0) - (alpha / t_0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + 2.0;
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = (((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1);
	double tmp;
	if (t_2 <= -1.0) {
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0;
	} else if (t_2 <= 1e-41) {
		tmp = (t_2 + 1.0) / 2.0;
	} else {
		tmp = (1.0 + ((beta / t_0) - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + 2.0
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = (((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)
	tmp = 0
	if t_2 <= -1.0:
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0
	elif t_2 <= 1e-41:
		tmp = (t_2 + 1.0) / 2.0
	else:
		tmp = (1.0 + ((beta / t_0) - (alpha / t_0))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(2.0 + t_1))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(Float64(4.0 * Float64(i / alpha)) + Float64(2.0 * Float64(1.0 / alpha)))) / 2.0);
	elseif (t_2 <= 1e-41)
		tmp = Float64(Float64(t_2 + 1.0) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta / t_0) - Float64(alpha / t_0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + 2.0;
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = (((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1);
	tmp = 0.0;
	if (t_2 <= -1.0)
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0;
	elseif (t_2 <= 1e-41)
		tmp = (t_2 + 1.0) / 2.0;
	else
		tmp = (1.0 + ((beta / t_0) - (alpha / t_0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$2, 1e-41], N[(N[(t$95$2 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta / t$95$0), $MachinePrecision] - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -1

   1. Initial program 1.7%

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

   3. Simplified 9.1%

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

   5. Taylor expanded in alpha around inf 96.5%

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

   6. Taylor expanded in beta around 0 96.5%

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

   if -1 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 1.00000000000000001e-41

   1. Initial program 99.3%

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

   if 1.00000000000000001e-41 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 45.0%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in i around 0 93.9%

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

      1. +-commutative 93.9%

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

      2. sub-div 93.9%

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

   7. Applied egg-rr 93.9%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -1:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 10^{-41}:\\ \;\;\;\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.3 \cdot 10^{+30}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 3.7 \cdot 10^{+70}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\beta}{\alpha} + \frac{1}{\alpha}\right)}{2}\\ \mathbf{elif}\;\alpha \leq 5.7 \cdot 10^{+91}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.3e+30)
   (/ (+ 1.0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))) 2.0)
   (if (<= alpha 3.7e+70)
     (/ (* 2.0 (+ (/ beta alpha) (/ 1.0 alpha))) 2.0)
     (if (<= alpha 5.7e+91)
       (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
       (/
        (+
         (* 2.0 (/ beta alpha))
         (+ (* 4.0 (/ i alpha)) (* 2.0 (/ 1.0 alpha))))
        2.0)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.3e+30) {
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	} else if (alpha <= 3.7e+70) {
		tmp = (2.0 * ((beta / alpha) + (1.0 / alpha))) / 2.0;
	} else if (alpha <= 5.7e+91) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / 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 <= 2.3d+30) then
        tmp = (1.0d0 + ((beta - alpha) / ((alpha + beta) + 2.0d0))) / 2.0d0
    else if (alpha <= 3.7d+70) then
        tmp = (2.0d0 * ((beta / alpha) + (1.0d0 / alpha))) / 2.0d0
    else if (alpha <= 5.7d+91) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 * (beta / alpha)) + ((4.0d0 * (i / alpha)) + (2.0d0 * (1.0d0 / alpha)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.3e+30) {
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	} else if (alpha <= 3.7e+70) {
		tmp = (2.0 * ((beta / alpha) + (1.0 / alpha))) / 2.0;
	} else if (alpha <= 5.7e+91) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.3e+30:
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0
	elif alpha <= 3.7e+70:
		tmp = (2.0 * ((beta / alpha) + (1.0 / alpha))) / 2.0
	elif alpha <= 5.7e+91:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.3e+30)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0))) / 2.0);
	elseif (alpha <= 3.7e+70)
		tmp = Float64(Float64(2.0 * Float64(Float64(beta / alpha) + Float64(1.0 / alpha))) / 2.0);
	elseif (alpha <= 5.7e+91)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(Float64(4.0 * Float64(i / alpha)) + Float64(2.0 * Float64(1.0 / alpha)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.3e+30)
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	elseif (alpha <= 3.7e+70)
		tmp = (2.0 * ((beta / alpha) + (1.0 / alpha))) / 2.0;
	elseif (alpha <= 5.7e+91)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.3e+30], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 3.7e+70], N[(N[(2.0 * N[(N[(beta / alpha), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 5.7e+91], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if alpha < 2.3e30

   1. Initial program 82.6%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in i around 0 93.1%

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

   if 2.3e30 < alpha < 3.69999999999999989e70

   1. Initial program 28.1%

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

   3. Simplified 27.7%

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

   5. Taylor expanded in alpha around inf 76.4%

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

   6. Taylor expanded in i around 0 76.7%

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

      1. distribute-rgt1-in 76.7%

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

      2. metadata-eval 76.7%

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

      3. mul0-lft 76.7%

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

      4. mul-1-neg 76.7%

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

      5. *-commutative 76.7%

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

   8. Simplified 76.7%

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

   9. Taylor expanded in beta around 0 76.9%

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

       1. distribute-lft-out 76.9%

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

   11. Applied egg-rr 76.9%

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

   if 3.69999999999999989e70 < alpha < 5.69999999999999964e91

   1. Initial program 58.6%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in i around 0 30.9%

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

   6. Taylor expanded in alpha around 0 86.6%

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

   if 5.69999999999999964e91 < alpha

   1. Initial program 6.4%

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

   3. Simplified 27.0%

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

   5. Taylor expanded in alpha around inf 78.9%

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

   6. Taylor expanded in beta around 0 78.9%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.3 \cdot 10^{+30}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 3.7 \cdot 10^{+70}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\beta}{\alpha} + \frac{1}{\alpha}\right)}{2}\\ \mathbf{elif}\;\alpha \leq 5.7 \cdot 10^{+91}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 9 \cdot 10^{+165}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 7.5 \cdot 10^{+192} \lor \neg \left(\alpha \leq 1.22 \cdot 10^{+234}\right) \land \alpha \leq 3.8 \cdot 10^{+242}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 9e+165)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (if (or (<= alpha 7.5e+192)
           (and (not (<= alpha 1.22e+234)) (<= alpha 3.8e+242)))
     (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 9e+165) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 7.5e+192) || (!(alpha <= 1.22e+234) && (alpha <= 3.8e+242))) {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 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 <= 9d+165) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else if ((alpha <= 7.5d+192) .or. (.not. (alpha <= 1.22d+234)) .and. (alpha <= 3.8d+242)) then
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 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 <= 9e+165) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 7.5e+192) || (!(alpha <= 1.22e+234) && (alpha <= 3.8e+242))) {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 9e+165:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	elif (alpha <= 7.5e+192) or (not (alpha <= 1.22e+234) and (alpha <= 3.8e+242)):
		tmp = ((2.0 + (i * 4.0)) / alpha) / 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 <= 9e+165)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	elseif ((alpha <= 7.5e+192) || (!(alpha <= 1.22e+234) && (alpha <= 3.8e+242)))
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 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 <= 9e+165)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	elseif ((alpha <= 7.5e+192) || (~((alpha <= 1.22e+234)) && (alpha <= 3.8e+242)))
		tmp = ((2.0 + (i * 4.0)) / alpha) / 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, 9e+165], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 7.5e+192], And[N[Not[LessEqual[alpha, 1.22e+234]], $MachinePrecision], LessEqual[alpha, 3.8e+242]]], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $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 3 regimes

2. if alpha < 8.9999999999999993e165

   1. Initial program 74.7%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in i around 0 83.7%

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

   6. Taylor expanded in alpha around 0 86.5%

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

   if 8.9999999999999993e165 < alpha < 7.5e192 or 1.22000000000000006e234 < alpha < 3.80000000000000008e242

   1. Initial program 1.1%

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

   3. Simplified 19.1%

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

   5. Taylor expanded in alpha around inf 85.7%

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

   6. Taylor expanded in beta around 0 85.7%

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

      1. *-commutative 85.7%

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

   8. Simplified 85.7%

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

   if 7.5e192 < alpha < 1.22000000000000006e234 or 3.80000000000000008e242 < alpha

   1. Initial program 1.2%

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

   3. Simplified 12.9%

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

   5. Taylor expanded in alpha around inf 94.0%

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

   6. Taylor expanded in i around 0 63.4%

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

      1. distribute-rgt1-in 63.4%

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

      2. metadata-eval 63.4%

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

      3. mul0-lft 63.4%

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

      4. mul-1-neg 63.4%

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

      5. *-commutative 63.4%

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

   8. Simplified 63.4%

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

   9. Taylor expanded in alpha around 0 63.4%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 9 \cdot 10^{+165}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 7.5 \cdot 10^{+192} \lor \neg \left(\alpha \leq 1.22 \cdot 10^{+234}\right) \land \alpha \leq 3.8 \cdot 10^{+242}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.22 \cdot 10^{+30}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 7.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\beta}{\alpha} + \frac{1}{\alpha}\right)}{2}\\ \mathbf{elif}\;\alpha \leq 2 \cdot 10^{+166}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.22e+30)
   (/ (+ 1.0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))) 2.0)
   (if (<= alpha 7.6e+69)
     (/ (* 2.0 (+ (/ beta alpha) (/ 1.0 alpha))) 2.0)
     (if (<= alpha 2e+166)
       (/ (+ 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 <= 1.22e+30) {
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	} else if (alpha <= 7.6e+69) {
		tmp = (2.0 * ((beta / alpha) + (1.0 / alpha))) / 2.0;
	} else if (alpha <= 2e+166) {
		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 <= 1.22d+30) then
        tmp = (1.0d0 + ((beta - alpha) / ((alpha + beta) + 2.0d0))) / 2.0d0
    else if (alpha <= 7.6d+69) then
        tmp = (2.0d0 * ((beta / alpha) + (1.0d0 / alpha))) / 2.0d0
    else if (alpha <= 2d+166) 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 <= 1.22e+30) {
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	} else if (alpha <= 7.6e+69) {
		tmp = (2.0 * ((beta / alpha) + (1.0 / alpha))) / 2.0;
	} else if (alpha <= 2e+166) {
		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 <= 1.22e+30:
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0
	elif alpha <= 7.6e+69:
		tmp = (2.0 * ((beta / alpha) + (1.0 / alpha))) / 2.0
	elif alpha <= 2e+166:
		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 <= 1.22e+30)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0))) / 2.0);
	elseif (alpha <= 7.6e+69)
		tmp = Float64(Float64(2.0 * Float64(Float64(beta / alpha) + Float64(1.0 / alpha))) / 2.0);
	elseif (alpha <= 2e+166)
		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 <= 1.22e+30)
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	elseif (alpha <= 7.6e+69)
		tmp = (2.0 * ((beta / alpha) + (1.0 / alpha))) / 2.0;
	elseif (alpha <= 2e+166)
		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, 1.22e+30], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 7.6e+69], N[(N[(2.0 * N[(N[(beta / alpha), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 2e+166], 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 4 regimes

2. if alpha < 1.22e30

   1. Initial program 82.6%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in i around 0 93.1%

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

   if 1.22e30 < alpha < 7.60000000000000055e69

   1. Initial program 28.1%

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

   3. Simplified 27.7%

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

   5. Taylor expanded in alpha around inf 76.4%

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

   6. Taylor expanded in i around 0 76.7%

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

      1. distribute-rgt1-in 76.7%

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

      2. metadata-eval 76.7%

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

      3. mul0-lft 76.7%

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

      4. mul-1-neg 76.7%

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

      5. *-commutative 76.7%

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

   8. Simplified 76.7%

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

   9. Taylor expanded in beta around 0 76.9%

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

       1. distribute-lft-out 76.9%

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

   11. Applied egg-rr 76.9%

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

   if 7.60000000000000055e69 < alpha < 1.99999999999999988e166

   1. Initial program 31.7%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in i around 0 36.4%

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

   6. Taylor expanded in alpha around 0 58.4%

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

   if 1.99999999999999988e166 < alpha

   1. Initial program 1.1%

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

   3. Simplified 14.6%

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

   5. Taylor expanded in alpha around inf 91.7%

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

   6. Taylor expanded in beta around 0 67.4%

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

      1. *-commutative 67.4%

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

   8. Simplified 67.4%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.22 \cdot 10^{+30}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 7.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\beta}{\alpha} + \frac{1}{\alpha}\right)}{2}\\ \mathbf{elif}\;\alpha \leq 2 \cdot 10^{+166}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 9.2 \cdot 10^{+166}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2 \cdot 10^{+251} \lor \neg \left(\alpha \leq 4.1 \cdot 10^{+298}\right):\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 9.2e+166)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (if (or (<= alpha 2e+251) (not (<= alpha 4.1e+298)))
     (/ (* 4.0 (/ i alpha)) 2.0)
     (/ (/ (* beta 2.0) alpha) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 9.2e+166) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 2e+251) || !(alpha <= 4.1e+298)) {
		tmp = (4.0 * (i / alpha)) / 2.0;
	} else {
		tmp = ((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 <= 9.2d+166) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else if ((alpha <= 2d+251) .or. (.not. (alpha <= 4.1d+298))) then
        tmp = (4.0d0 * (i / alpha)) / 2.0d0
    else
        tmp = ((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 <= 9.2e+166) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 2e+251) || !(alpha <= 4.1e+298)) {
		tmp = (4.0 * (i / alpha)) / 2.0;
	} else {
		tmp = ((beta * 2.0) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 9.2e+166:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	elif (alpha <= 2e+251) or not (alpha <= 4.1e+298):
		tmp = (4.0 * (i / alpha)) / 2.0
	else:
		tmp = ((beta * 2.0) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 9.2e+166)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	elseif ((alpha <= 2e+251) || !(alpha <= 4.1e+298))
		tmp = Float64(Float64(4.0 * Float64(i / alpha)) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta * 2.0) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 9.2e+166)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	elseif ((alpha <= 2e+251) || ~((alpha <= 4.1e+298)))
		tmp = (4.0 * (i / alpha)) / 2.0;
	else
		tmp = ((beta * 2.0) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 9.2e+166], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 2e+251], N[Not[LessEqual[alpha, 4.1e+298]], $MachinePrecision]], N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta * 2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 9.2000000000000003e166

   1. Initial program 74.7%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in i around 0 83.7%

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

   6. Taylor expanded in alpha around 0 86.5%

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

   if 9.2000000000000003e166 < alpha < 2.0000000000000001e251 or 4.10000000000000016e298 < alpha

   1. Initial program 1.1%

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

   3. Simplified 14.8%

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

   5. Taylor expanded in alpha around inf 91.0%

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

   6. Taylor expanded in i around inf 51.7%

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

   if 2.0000000000000001e251 < alpha < 4.10000000000000016e298

   1. Initial program 1.3%

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

   3. Simplified 14.4%

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

   5. Taylor expanded in alpha around inf 93.3%

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

   6. Taylor expanded in beta around inf 46.9%

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

      1. associate-*r/ 46.9%

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

      2. *-commutative 46.9%

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

   8. Simplified 46.9%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 9.2 \cdot 10^{+166}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2 \cdot 10^{+251} \lor \neg \left(\alpha \leq 4.1 \cdot 10^{+298}\right):\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3.2 \cdot 10^{+180}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 4 \cdot 10^{+298}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 3.2e+180)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (if (<= alpha 4e+298)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/ (* 4.0 (/ i alpha)) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.2e+180) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if (alpha <= 4e+298) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (4.0 * (i / 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 <= 3.2d+180) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else if (alpha <= 4d+298) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (4.0d0 * (i / alpha)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.2e+180) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if (alpha <= 4e+298) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (4.0 * (i / alpha)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 3.2e+180:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	elif alpha <= 4e+298:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = (4.0 * (i / alpha)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 3.2e+180)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	elseif (alpha <= 4e+298)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(4.0 * Float64(i / alpha)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 3.2e+180)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	elseif (alpha <= 4e+298)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = (4.0 * (i / alpha)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 3.2e+180], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 4e+298], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 3.19999999999999994e180

   1. Initial program 72.6%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in i around 0 81.5%

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

   6. Taylor expanded in alpha around 0 84.7%

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

   if 3.19999999999999994e180 < alpha < 3.9999999999999998e298

   1. Initial program 1.1%

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

   3. Simplified 12.6%

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

   5. Taylor expanded in alpha around inf 94.0%

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

   6. Taylor expanded in i around 0 63.3%

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

      1. distribute-rgt1-in 63.3%

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

      2. metadata-eval 63.3%

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

      3. mul0-lft 63.3%

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

      4. mul-1-neg 63.3%

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

      5. *-commutative 63.3%

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

   8. Simplified 63.3%

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

   9. Taylor expanded in alpha around 0 63.3%

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

   if 3.9999999999999998e298 < alpha

   1. Initial program 1.1%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in alpha around inf 100.0%

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

   6. Taylor expanded in i around inf 60.4%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.2 \cdot 10^{+180}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 4 \cdot 10^{+298}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 66000000000000:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 1.25 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{\beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 66000000000000.0)
   0.5
   (if (<= beta 1.25e+30) (/ (/ (* beta 2.0) alpha) 2.0) 1.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 66000000000000.0) {
		tmp = 0.5;
	} else if (beta <= 1.25e+30) {
		tmp = ((beta * 2.0) / alpha) / 2.0;
	} 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 <= 66000000000000.0d0) then
        tmp = 0.5d0
    else if (beta <= 1.25d+30) then
        tmp = ((beta * 2.0d0) / alpha) / 2.0d0
    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 <= 66000000000000.0) {
		tmp = 0.5;
	} else if (beta <= 1.25e+30) {
		tmp = ((beta * 2.0) / alpha) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 66000000000000.0:
		tmp = 0.5
	elif beta <= 1.25e+30:
		tmp = ((beta * 2.0) / alpha) / 2.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 66000000000000.0)
		tmp = 0.5;
	elseif (beta <= 1.25e+30)
		tmp = Float64(Float64(Float64(beta * 2.0) / alpha) / 2.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 66000000000000.0)
		tmp = 0.5;
	elseif (beta <= 1.25e+30)
		tmp = ((beta * 2.0) / alpha) / 2.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 66000000000000.0], 0.5, If[LessEqual[beta, 1.25e+30], N[(N[(N[(beta * 2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if beta < 6.6e13

   1. Initial program 76.1%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in i around inf 75.7%

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

   if 6.6e13 < beta < 1.25e30

   1. Initial program 18.5%

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

   3. Simplified 20.7%

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

   5. Taylor expanded in alpha around inf 83.5%

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

   6. Taylor expanded in beta around inf 81.2%

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

      1. associate-*r/ 81.2%

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

      2. *-commutative 81.2%

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

   8. Simplified 81.2%

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

   if 1.25e30 < beta

   1. Initial program 34.7%

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

   3. Simplified 85.4%

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

   5. Taylor expanded in beta around inf 73.1%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 66000000000000:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 1.25 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{\beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 22000000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 22000000000.0) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 22000000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 22000000000.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 22000000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 22000000000.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 22000000000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 22000000000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 22000000000.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2e10

   1. Initial program 75.9%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in i around inf 75.6%

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

   if 2.2e10 < beta

   1. Initial program 34.3%

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

   3. Simplified 81.2%

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

   5. Taylor expanded in beta around inf 69.0%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 22000000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.8% 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 61.5%

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

3. Simplified 79.0%

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

5. Taylor expanded in i around inf 58.4%

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

6. Final simplification 58.4%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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))