Octave 3.8, jcobi/2

Percentage Accurate: 62.7% → 97.8%

Time: 15.2s

Alternatives: 9

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: 62.7% 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: 97.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.99999998:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + \beta}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))
        -0.99999998)
     (/ (/ (+ (* i 4.0) (+ 2.0 (* beta 2.0))) alpha) 2.0)
     (/
      (+
       (*
        (/ (+ alpha beta) (+ beta (+ alpha (fma 2.0 i 2.0))))
        (/ (- beta alpha) (fma 2.0 i (+ alpha 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)) <= -0.99999998) {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	} else {
		tmp = ((((alpha + beta) / (beta + (alpha + fma(2.0, i, 2.0)))) * ((beta - alpha) / fma(2.0, i, (alpha + 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)) <= -0.99999998)
		tmp = Float64(Float64(Float64(Float64(i * 4.0) + Float64(2.0 + Float64(beta * 2.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha + beta) / Float64(beta + Float64(alpha + fma(2.0, i, 2.0)))) * Float64(Float64(beta - alpha) / fma(2.0, i, Float64(alpha + 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], -0.99999998], N[(N[(N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] / N[(beta + N[(alpha + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 2.7%

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

   3. Simplified 16.5%

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

   4. Taylor expanded in beta around 0 16.5%

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

      1. associate-+r+ 16.5%

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

   6. Simplified 16.5%

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

   7. Taylor expanded in alpha around inf 88.8%

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

   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 81.3%

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

   3. Simplified 99.7%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999998:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + \beta}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}\\ \end{array} \]

Alternative 2: 96.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.5) {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta / (beta + (2.0 + (2.0 * i)))) * (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) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)) <= (-0.5d0)) then
        tmp = (((i * 4.0d0) + (2.0d0 + (beta * 2.0d0))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + ((beta / (beta + (2.0d0 + (2.0d0 * i)))) * (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 t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.5) {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta / (beta + (2.0 + (2.0 * i)))) * (beta / (beta + (2.0 * i))))) / 2.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta / N[(beta + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta / N[(beta + 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)) < -0.5

   1. Initial program 4.7%

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

   3. Simplified 18.1%

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

   4. Taylor expanded in beta around 0 18.1%

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

      1. associate-+r+ 18.1%

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

   6. Simplified 18.1%

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

   7. Taylor expanded in alpha around inf 87.7%

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

   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 81.4%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in alpha around 0 99.1%

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

   5. Taylor expanded in alpha around 0 99.1%

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

4. Final simplification 96.1%

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

Alternative 3: 74.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5.5 \cdot 10^{+52}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2.8 \cdot 10^{+130} \lor \neg \left(\alpha \leq 10^{+198}\right) \land \left(\alpha \leq 1.5 \cdot 10^{+255} \lor \neg \left(\alpha \leq 2.85 \cdot 10^{+281}\right)\right):\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 5.5e+52)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (if (or (<= alpha 2.8e+130)
           (and (not (<= alpha 1e+198))
                (or (<= alpha 1.5e+255) (not (<= alpha 2.85e+281)))))
     (/ (/ (+ 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 <= 5.5e+52) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 2.8e+130) || (!(alpha <= 1e+198) && ((alpha <= 1.5e+255) || !(alpha <= 2.85e+281)))) {
		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 <= 5.5d+52) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else if ((alpha <= 2.8d+130) .or. (.not. (alpha <= 1d+198)) .and. (alpha <= 1.5d+255) .or. (.not. (alpha <= 2.85d+281))) 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 <= 5.5e+52) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 2.8e+130) || (!(alpha <= 1e+198) && ((alpha <= 1.5e+255) || !(alpha <= 2.85e+281)))) {
		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 <= 5.5e+52:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	elif (alpha <= 2.8e+130) or (not (alpha <= 1e+198) and ((alpha <= 1.5e+255) or not (alpha <= 2.85e+281))):
		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 <= 5.5e+52)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	elseif ((alpha <= 2.8e+130) || (!(alpha <= 1e+198) && ((alpha <= 1.5e+255) || !(alpha <= 2.85e+281))))
		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 <= 5.5e+52)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	elseif ((alpha <= 2.8e+130) || (~((alpha <= 1e+198)) && ((alpha <= 1.5e+255) || ~((alpha <= 2.85e+281)))))
		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, 5.5e+52], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 2.8e+130], And[N[Not[LessEqual[alpha, 1e+198]], $MachinePrecision], Or[LessEqual[alpha, 1.5e+255], N[Not[LessEqual[alpha, 2.85e+281]], $MachinePrecision]]]], 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 < 5.49999999999999996e52

   1. Initial program 81.0%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in alpha around 0 96.6%

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

   5. Taylor expanded in alpha around 0 96.6%

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

   6. Taylor expanded in i around 0 89.6%

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

   if 5.49999999999999996e52 < alpha < 2.7999999999999999e130 or 1.00000000000000002e198 < alpha < 1.50000000000000009e255 or 2.84999999999999993e281 < alpha

   1. Initial program 6.7%

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

   3. Simplified 22.4%

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

   4. Taylor expanded in beta around 0 22.4%

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

      1. associate-+r+ 22.4%

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

   6. Simplified 22.4%

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

   7. Taylor expanded in alpha around inf 82.9%

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

   8. Taylor expanded in i around 0 73.8%

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

      1. *-commutative 73.8%

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

   10. Simplified 73.8%

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

   if 2.7999999999999999e130 < alpha < 1.00000000000000002e198 or 1.50000000000000009e255 < alpha < 2.84999999999999993e281

   1. Initial program 1.8%

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

   3. Simplified 24.1%

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

   4. Taylor expanded in beta around 0 24.1%

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

      1. associate-+r+ 24.1%

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

   6. Simplified 24.1%

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

   7. Taylor expanded in alpha around inf 81.5%

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

   8. Taylor expanded in i around inf 68.2%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5.5 \cdot 10^{+52}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2.8 \cdot 10^{+130} \lor \neg \left(\alpha \leq 10^{+198}\right) \land \left(\alpha \leq 1.5 \cdot 10^{+255} \lor \neg \left(\alpha \leq 2.85 \cdot 10^{+281}\right)\right):\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \end{array} \]

Alternative 4: 88.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 4.2e+38)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (/ (/ (+ (* i 4.0) (+ 2.0 (* beta 2.0))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.2e+38) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = (((i * 4.0) + (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 <= 4.2d+38) then
        tmp = (1.0d0 + (beta / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = (((i * 4.0d0) + (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 <= 4.2e+38) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 4.2e+38:
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 4.2e+38)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(i * 4.0) + 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 <= 4.2e+38)
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 4.2e+38], N[(N[(1.0 + N[(beta / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.2e38

   1. Initial program 82.9%

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

   2. Taylor expanded in beta around inf 96.6%

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

   if 4.2e38 < alpha

   1. Initial program 6.2%

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

   3. Simplified 25.4%

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

   4. Taylor expanded in beta around 0 24.3%

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

      1. associate-+r+ 24.3%

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

   6. Simplified 24.3%

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

   7. Taylor expanded in alpha around inf 79.9%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 5: 83.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.8 \cdot 10^{+38}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 4.8e+38)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ (* i 4.0) (+ 2.0 (* beta 2.0))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.8e+38) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (((i * 4.0) + (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 <= 4.8d+38) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (((i * 4.0d0) + (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 <= 4.8e+38) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 4.8e+38:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 4.8e+38)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(i * 4.0) + 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 <= 4.8e+38)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 4.8e+38], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.80000000000000035e38

   1. Initial program 82.9%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in alpha around 0 98.0%

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

   5. Taylor expanded in alpha around 0 98.0%

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

   6. Taylor expanded in i around 0 91.0%

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

   if 4.80000000000000035e38 < alpha

   1. Initial program 6.2%

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

   3. Simplified 25.4%

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

   4. Taylor expanded in beta around 0 24.3%

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

      1. associate-+r+ 24.3%

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

   6. Simplified 24.3%

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

   7. Taylor expanded in alpha around inf 79.9%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.8 \cdot 10^{+38}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 6: 73.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5.4 \cdot 10^{+116}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 5.4e+116)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (* 4.0 (/ i alpha)) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5.4e+116) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 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 <= 5.4d+116) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 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 <= 5.4e+116) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (4.0 * (i / alpha)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 5.4e+116:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (4.0 * (i / alpha)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 5.4e+116)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 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 <= 5.4e+116)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (4.0 * (i / alpha)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 5.4e+116], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 5.3999999999999999e116

   1. Initial program 77.0%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in alpha around 0 91.7%

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

   5. Taylor expanded in alpha around 0 91.7%

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

   6. Taylor expanded in i around 0 85.2%

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

   if 5.3999999999999999e116 < alpha

   1. Initial program 1.5%

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

   3. Simplified 24.1%

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

   4. Taylor expanded in beta around 0 24.1%

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

      1. associate-+r+ 24.1%

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

   6. Simplified 24.1%

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

   7. Taylor expanded in alpha around inf 81.6%

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

   8. Taylor expanded in i around inf 39.4%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5.4 \cdot 10^{+116}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 79.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.8 \cdot 10^{+38}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 4.8e+38)
   (/ (+ 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 <= 4.8e+38) {
		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 <= 4.8d+38) 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 <= 4.8e+38) {
		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 <= 4.8e+38:
		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 <= 4.8e+38)
		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 <= 4.8e+38)
		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, 4.8e+38], 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 < 4.80000000000000035e38

   1. Initial program 82.9%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in alpha around 0 98.0%

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

   5. Taylor expanded in alpha around 0 98.0%

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

   6. Taylor expanded in i around 0 91.0%

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

   if 4.80000000000000035e38 < alpha

   1. Initial program 6.2%

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

   3. Simplified 25.4%

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

   4. Taylor expanded in beta around 0 24.3%

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

      1. associate-+r+ 24.3%

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

   6. Simplified 24.3%

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

   7. Taylor expanded in alpha around inf 79.9%

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

   8. Taylor expanded in beta around 0 62.6%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.8 \cdot 10^{+38}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 71.8% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.9 \cdot 10^{+25}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 1.9e+25) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.9e+25) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.9d+25) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.9e+25) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.9e+25:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.9e+25)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.9e+25)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.9e+25], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 1.9e25

   1. Initial program 69.9%

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

   3. Simplified 74.8%

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

   4. Taylor expanded in i around inf 71.4%

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

   if 1.9e25 < beta

   1. Initial program 40.6%

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

   3. Simplified 86.3%

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

   4. Taylor expanded in beta around inf 69.9%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.9 \cdot 10^{+25}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 61.5% 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 61.1%

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

3. Simplified 78.3%

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

4. Taylor expanded in i around inf 58.8%

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

5. Final simplification 58.8%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023272 
(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))