Octave 3.8, jcobi/2

Percentage Accurate: 62.9% → 97.8%

Time: 20.4s

Alternatives: 12

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: 62.9% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.999998:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, \frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.999998)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (log
       (exp
        (fma
         (/ (+ alpha beta) (+ alpha (+ beta (fma 2.0 i 2.0))))
         (/ (- beta alpha) (+ alpha (fma 2.0 i beta)))
         1.0)))
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.999998) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = log(exp(fma(((alpha + beta) / (alpha + (beta + fma(2.0, i, 2.0)))), ((beta - alpha) / (alpha + fma(2.0, i, beta))), 1.0))) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(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)) <= -0.999998)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(log(exp(fma(Float64(Float64(alpha + beta) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))), Float64(Float64(beta - alpha) / Float64(alpha + fma(2.0, i, beta))), 1.0))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(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], -0.999998], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Log[N[Exp[N[(N[(N[(alpha + beta), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $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)) < -0.999998000000000054

   1. Initial program 2.9%

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

      1. associate-/l* 22.5%

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

      2. associate-/l/ 22.7%

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

      3. associate-+l+ 22.7%

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

      4. associate-+l+ 22.7%

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

      5. fma-def 22.7%

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

      6. +-commutative 22.7%

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

      7. fma-def 22.7%

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

   3. Simplified 22.7%

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

   4. Taylor expanded in alpha around inf 21.3%

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

   5. Taylor expanded in alpha around -inf 85.5%

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

   if -0.999998000000000054 < (/.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 80.0%

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

      1. associate-/l* 99.9%

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

      2. associate-/l/ 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+l+ 99.9%

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

      5. fma-def 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-def 99.9%

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

   3. Simplified 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999998:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, \frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}\right)}{2}\\ \end{array} \]

Alternative 2: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.999998:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{\alpha + \beta}{\frac{\alpha + \left(\beta + 2 \cdot i\right)}{\beta - \alpha}}}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}}{2}\\ \end{array} \end{array} \]

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)) -0.999998)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (+
       1.0
       (/
        (/ (+ alpha beta) (/ (+ alpha (+ beta (* 2.0 i))) (- beta alpha)))
        (+ (+ alpha beta) (+ 2.0 (* 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)) <= -0.999998) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((alpha + beta) / ((alpha + (beta + (2.0 * i))) / (beta - alpha))) / ((alpha + beta) + (2.0 + (2.0 * i))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)) <= (-0.999998d0)) then
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + (((alpha + beta) / ((alpha + (beta + (2.0d0 * i))) / (beta - alpha))) / ((alpha + beta) + (2.0d0 + (2.0d0 * i))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.999998) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((alpha + beta) / ((alpha + (beta + (2.0 * i))) / (beta - alpha))) / ((alpha + beta) + (2.0 + (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)) <= -0.999998:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	else:
		tmp = (1.0 + (((alpha + beta) / ((alpha + (beta + (2.0 * i))) / (beta - alpha))) / ((alpha + beta) + (2.0 + (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)) <= -0.999998)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(alpha + beta) / Float64(Float64(alpha + Float64(beta + Float64(2.0 * i))) / Float64(beta - alpha))) / Float64(Float64(alpha + beta) + Float64(2.0 + Float64(2.0 * i))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.999998)
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	else
		tmp = (1.0 + (((alpha + beta) / ((alpha + (beta + (2.0 * i))) / (beta - alpha))) / ((alpha + beta) + (2.0 + (2.0 * i))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(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], -0.999998], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(alpha + beta), $MachinePrecision] / N[(N[(alpha + N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.999998000000000054

   1. Initial program 2.9%

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

      1. associate-/l* 22.5%

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

      2. associate-/l/ 22.7%

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

      3. associate-+l+ 22.7%

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

      4. associate-+l+ 22.7%

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

      5. fma-def 22.7%

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

      6. +-commutative 22.7%

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

      7. fma-def 22.7%

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

   3. Simplified 22.7%

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

   4. Taylor expanded in alpha around inf 21.3%

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

   5. Taylor expanded in alpha around -inf 85.5%

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

   if -0.999998000000000054 < (/.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 80.0%

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

      1. associate-/l* 99.9%

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

      2. associate-+l+ 99.9%

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

      3. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999998:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{\alpha + \beta}{\frac{\alpha + \left(\beta + 2 \cdot i\right)}{\beta - \alpha}}}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}}{2}\\ \end{array} \]

Alternative 3: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.999998:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{t_1}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))) (t_1 (+ 2.0 t_0)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1) -0.999998)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/ (+ 1.0 (/ (- beta alpha) t_1)) 2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.999998) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / t_1)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = 2.0d0 + t_0
    if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= (-0.999998d0)) then
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + ((beta - alpha) / t_1)) / 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 t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.999998) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / t_1)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = 2.0 + t_0
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.999998:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	else:
		tmp = (1.0 + ((beta - alpha) / t_1)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(2.0 + t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / t_1) <= -0.999998)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / t_1)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = 2.0 + t_0;
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.999998)
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	else
		tmp = (1.0 + ((beta - alpha) / t_1)) / 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]}, Block[{t$95$1 = N[(2.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], -0.999998], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / t$95$1), $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)) < -0.999998000000000054

   1. Initial program 2.9%

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

      1. associate-/l* 22.5%

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

      2. associate-/l/ 22.7%

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

      3. associate-+l+ 22.7%

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

      4. associate-+l+ 22.7%

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

      5. fma-def 22.7%

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

      6. +-commutative 22.7%

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

      7. fma-def 22.7%

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

   3. Simplified 22.7%

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

   4. Taylor expanded in alpha around inf 21.3%

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

   5. Taylor expanded in alpha around -inf 85.5%

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

   if -0.999998000000000054 < (/.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 80.0%

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

   2. Taylor expanded in i around 0 98.4%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999998:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array} \]

Alternative 4: 81.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3 \cdot 10^{+71}:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta - \alpha}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 3e+71)
   (/ (+ 1.0 (/ 1.0 (/ (+ (+ alpha beta) 2.0) (- beta alpha)))) 2.0)
   (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3e+71) {
		tmp = (1.0 + (1.0 / (((alpha + beta) + 2.0) / (beta - alpha)))) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 3d+71) then
        tmp = (1.0d0 + (1.0d0 / (((alpha + beta) + 2.0d0) / (beta - alpha)))) / 2.0d0
    else
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3e+71) {
		tmp = (1.0 + (1.0 / (((alpha + beta) + 2.0) / (beta - alpha)))) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 3e+71:
		tmp = (1.0 + (1.0 / (((alpha + beta) + 2.0) / (beta - alpha)))) / 2.0
	else:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 3e+71)
		tmp = Float64(Float64(1.0 + Float64(1.0 / Float64(Float64(Float64(alpha + beta) + 2.0) / Float64(beta - alpha)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 3e+71)
		tmp = (1.0 + (1.0 / (((alpha + beta) + 2.0) / (beta - alpha)))) / 2.0;
	else
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 3e+71], N[(N[(1.0 + N[(1.0 / N[(N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3.00000000000000013e71

   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

      1. associate-/l* 97.4%

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

      2. associate-/l/ 97.4%

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

      3. associate-+l+ 97.4%

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

      4. associate-+l+ 97.4%

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

      5. fma-def 97.4%

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

      6. +-commutative 97.4%

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

      7. fma-def 97.4%

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

   3. Simplified 97.4%

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

   5. Applied egg-rr 97.4%

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

      1. unpow-1 97.4%

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

      2. associate-/r* 97.4%

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

      3. associate-+r+ 97.4%

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

      4. +-commutative 97.4%

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

      5. +-commutative 97.4%

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

      6. fma-udef 97.4%

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

      7. +-commutative 97.4%

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

      8. +-commutative 97.4%

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

      9. +-commutative 97.4%

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

      10. fma-udef 97.4%

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

   7. Simplified 97.4%

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

   8. Taylor expanded in i around 0 84.7%

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

   if 3.00000000000000013e71 < alpha

   1. Initial program 12.5%

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

      1. associate-/l* 36.8%

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

      2. associate-/l/ 36.9%

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

      3. associate-+l+ 36.9%

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

      4. associate-+l+ 36.9%

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

      5. fma-def 36.9%

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

      6. +-commutative 36.9%

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

      7. fma-def 36.9%

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

   3. Simplified 36.9%

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

   4. Taylor expanded in alpha around inf 28.9%

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

   5. Taylor expanded in alpha around -inf 71.3%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3 \cdot 10^{+71}:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta - \alpha}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 5: 88.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.4 \cdot 10^{+137}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{2 + \left(\beta + i \cdot 4\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.4e+137)
   (/ (+ 1.0 (/ (+ alpha beta) (+ 2.0 (+ beta (* i 4.0))))) 2.0)
   (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.4e+137) {
		tmp = (1.0 + ((alpha + beta) / (2.0 + (beta + (i * 4.0))))) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.4d+137) then
        tmp = (1.0d0 + ((alpha + beta) / (2.0d0 + (beta + (i * 4.0d0))))) / 2.0d0
    else
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.4e+137) {
		tmp = (1.0 + ((alpha + beta) / (2.0 + (beta + (i * 4.0))))) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.4e+137:
		tmp = (1.0 + ((alpha + beta) / (2.0 + (beta + (i * 4.0))))) / 2.0
	else:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.4e+137)
		tmp = Float64(Float64(1.0 + Float64(Float64(alpha + beta) / Float64(2.0 + Float64(beta + Float64(i * 4.0))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.4e+137)
		tmp = (1.0 + ((alpha + beta) / (2.0 + (beta + (i * 4.0))))) / 2.0;
	else
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.4e+137], N[(N[(1.0 + N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 + N[(beta + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.4e137

   1. Initial program 76.8%

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

      1. associate-/l* 94.9%

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

      2. associate-/l/ 94.9%

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

      3. associate-+l+ 94.9%

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

      4. associate-+l+ 94.9%

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

      5. fma-def 94.9%

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

      6. +-commutative 94.9%

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

      7. fma-def 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in alpha around 0 80.7%

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

   5. Taylor expanded in beta around inf 93.3%

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

      1. *-commutative 93.3%

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

   7. Simplified 93.3%

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

   if 1.4e137 < alpha

   1. Initial program 4.2%

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

      1. associate-/l* 31.1%

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

      2. associate-/l/ 31.2%

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

      3. associate-+l+ 31.2%

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

      4. associate-+l+ 31.2%

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

      5. fma-def 31.2%

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

      6. +-commutative 31.2%

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

      7. fma-def 31.2%

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

   3. Simplified 31.2%

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

   4. Taylor expanded in alpha around inf 26.2%

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

   5. Taylor expanded in alpha around -inf 77.4%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.4 \cdot 10^{+137}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{2 + \left(\beta + i \cdot 4\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 6: 76.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3.3 \cdot 10^{+71}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \left(\beta + i\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 3.3e+71)
   (/ (+ 1.0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))) 2.0)
   (/ (/ (+ 2.0 (* 2.0 (+ beta i))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.3e+71) {
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (2.0 * (beta + 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 <= 3.3d+71) then
        tmp = (1.0d0 + ((beta - alpha) / ((alpha + beta) + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (2.0d0 * (beta + 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 <= 3.3e+71) {
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (2.0 * (beta + i))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 3.3e+71:
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (2.0 * (beta + i))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 3.3e+71)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(2.0 * Float64(beta + i))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 3.3e+71)
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (2.0 * (beta + i))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 3.3e+71], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(2.0 * N[(beta + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3.2999999999999998e71

   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

      1. associate-/l* 97.4%

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

      2. associate-/l/ 97.4%

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

      3. associate-+l+ 97.4%

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

      4. associate-+l+ 97.4%

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

      5. fma-def 97.4%

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

      6. +-commutative 97.4%

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

      7. fma-def 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in i around 0 84.7%

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

      1. +-commutative 84.7%

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

   6. Simplified 84.7%

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

   if 3.2999999999999998e71 < alpha

   1. Initial program 12.5%

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

   2. Taylor expanded in i around 0 33.3%

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

   3. Taylor expanded in alpha around inf 55.2%

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

      1. distribute-lft-out 55.2%

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

   5. Simplified 55.2%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.3 \cdot 10^{+71}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \left(\beta + i\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 81.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3.1 \cdot 10^{+71}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 3.1e+71)
   (/ (+ 1.0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))) 2.0)
   (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.1e+71) {
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 3.1d+71) then
        tmp = (1.0d0 + ((beta - alpha) / ((alpha + beta) + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.1e+71) {
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 3.1e+71:
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0
	else:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 3.1e+71)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 3.1e+71)
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	else
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 3.1e+71], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3.10000000000000018e71

   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

      1. associate-/l* 97.4%

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

      2. associate-/l/ 97.4%

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

      3. associate-+l+ 97.4%

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

      4. associate-+l+ 97.4%

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

      5. fma-def 97.4%

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

      6. +-commutative 97.4%

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

      7. fma-def 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in i around 0 84.7%

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

      1. +-commutative 84.7%

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

   6. Simplified 84.7%

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

   if 3.10000000000000018e71 < alpha

   1. Initial program 12.5%

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

      1. associate-/l* 36.8%

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

      2. associate-/l/ 36.9%

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

      3. associate-+l+ 36.9%

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

      4. associate-+l+ 36.9%

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

      5. fma-def 36.9%

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

      6. +-commutative 36.9%

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

      7. fma-def 36.9%

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

   3. Simplified 36.9%

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

   4. Taylor expanded in alpha around inf 28.9%

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

   5. Taylor expanded in alpha around -inf 71.3%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.1 \cdot 10^{+71}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 77.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.3 \cdot 10^{+74}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 4.3e+74)
   (/ (+ 1.0 (/ (+ alpha 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 <= 4.3e+74) {
		tmp = (1.0 + ((alpha + 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 <= 4.3d+74) then
        tmp = (1.0d0 + ((alpha + 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 <= 4.3e+74) {
		tmp = (1.0 + ((alpha + 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 <= 4.3e+74:
		tmp = (1.0 + ((alpha + 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 <= 4.3e+74)
		tmp = Float64(Float64(1.0 + Float64(Float64(alpha + 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 <= 4.3e+74)
		tmp = (1.0 + ((alpha + 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, 4.3e+74], N[(N[(1.0 + N[(N[(alpha + beta), $MachinePrecision] / 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 < 4.30000000000000001e74

   1. Initial program 79.1%

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

      1. associate-/l* 97.4%

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

      2. associate-/l/ 97.4%

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

      3. associate-+l+ 97.4%

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

      4. associate-+l+ 97.4%

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

      5. fma-def 97.4%

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

      6. +-commutative 97.4%

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

      7. fma-def 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in alpha around 0 83.3%

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

   5. Taylor expanded in i around 0 83.9%

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

   if 4.30000000000000001e74 < alpha

   1. Initial program 11.0%

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

      1. associate-/l* 35.8%

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

      2. associate-/l/ 35.9%

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

      3. associate-+l+ 35.9%

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

      4. associate-+l+ 35.9%

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

      5. fma-def 35.9%

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

      6. +-commutative 35.9%

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

      7. fma-def 35.9%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in alpha around inf 72.4%

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

   5. Taylor expanded in beta around 0 55.2%

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

      1. *-commutative 55.2%

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

   7. Simplified 55.2%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.3 \cdot 10^{+74}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 9: 75.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 9.2 \cdot 10^{+76}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \left(\beta + i\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 9.2e+76)
   (/ (+ 1.0 (/ (+ alpha beta) (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* 2.0 (+ beta i))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 9.2e+76) {
		tmp = (1.0 + ((alpha + beta) / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (2.0 * (beta + 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 <= 9.2d+76) then
        tmp = (1.0d0 + ((alpha + beta) / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (2.0d0 * (beta + 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 <= 9.2e+76) {
		tmp = (1.0 + ((alpha + beta) / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (2.0 * (beta + i))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 9.2e+76:
		tmp = (1.0 + ((alpha + beta) / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (2.0 * (beta + i))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 9.2e+76)
		tmp = Float64(Float64(1.0 + Float64(Float64(alpha + beta) / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(2.0 * Float64(beta + i))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 9.2e+76)
		tmp = (1.0 + ((alpha + beta) / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (2.0 * (beta + i))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 9.2e+76], N[(N[(1.0 + N[(N[(alpha + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(2.0 * N[(beta + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 9.20000000000000005e76

   1. Initial program 79.1%

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

      1. associate-/l* 97.4%

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

      2. associate-/l/ 97.4%

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

      3. associate-+l+ 97.4%

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

      4. associate-+l+ 97.4%

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

      5. fma-def 97.4%

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

      6. +-commutative 97.4%

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

      7. fma-def 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in alpha around 0 83.3%

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

   5. Taylor expanded in i around 0 83.9%

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

   if 9.20000000000000005e76 < alpha

   1. Initial program 11.0%

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

   2. Taylor expanded in i around 0 32.2%

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

   3. Taylor expanded in alpha around inf 56.1%

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

      1. distribute-lft-out 56.1%

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

   5. Simplified 56.1%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 9.2 \cdot 10^{+76}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \left(\beta + i\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 10: 72.1% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+66}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 5e+66) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5e+66) {
		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 <= 5d+66) 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 <= 5e+66) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 5e+66:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5e+66)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5e+66)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 5e+66], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 4.99999999999999991e66

   1. Initial program 75.5%

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

      1. associate-/l* 78.9%

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

      2. associate-/l/ 78.9%

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

      3. associate-+l+ 78.9%

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

      4. associate-+l+ 78.9%

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

      5. fma-def 78.9%

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

      6. +-commutative 78.9%

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

      7. fma-def 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in i around inf 73.0%

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

   if 4.99999999999999991e66 < beta

   1. Initial program 33.8%

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

      1. associate-/l* 92.5%

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

      2. associate-/l/ 92.6%

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

      3. associate-+l+ 92.6%

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

      4. associate-+l+ 92.6%

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

      5. fma-def 92.6%

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

      6. +-commutative 92.6%

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

      7. fma-def 92.6%

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

   3. Simplified 92.6%

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

   4. Taylor expanded in beta around inf 76.8%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+66}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11: 16.3% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 0.3333333333333333)

C

double code(double alpha, double beta, double i) {
	return 0.3333333333333333;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.3333333333333333d0
end function

Java

public static double code(double alpha, double beta, double i) {
	return 0.3333333333333333;
}

Python

def code(alpha, beta, i):
	return 0.3333333333333333

Julia

function code(alpha, beta, i)
	return 0.3333333333333333
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 0.3333333333333333;
end

Wolfram

code[alpha_, beta_, i_] := 0.3333333333333333

Derivation

1. Initial program 63.1%

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

   1. associate-/l* 83.0%

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

   2. associate-/l/ 83.0%

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

   3. associate-+l+ 83.0%

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

   4. associate-+l+ 83.0%

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

   5. fma-def 83.0%

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

   6. +-commutative 83.0%

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

   7. fma-def 83.0%

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

3. Simplified 83.0%

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

4. Taylor expanded in alpha around inf 60.6%

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

5. Taylor expanded in beta around inf 16.5%

   \[\leadsto \frac{\color{blue}{0.6666666666666666}}{2} \]

6. Final simplification 16.5%

   \[\leadsto 0.3333333333333333 \]

Alternative 12: 61.4% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 0.5)

C

double code(double alpha, double beta, double i) {
	return 0.5;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.5d0
end function

Java

public static double code(double alpha, double beta, double i) {
	return 0.5;
}

Python

def code(alpha, beta, i):
	return 0.5

Julia

function code(alpha, beta, i)
	return 0.5
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 0.5;
end

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 63.1%

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

   1. associate-/l* 83.0%

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

   2. associate-/l/ 83.0%

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

   3. associate-+l+ 83.0%

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

   4. associate-+l+ 83.0%

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

   5. fma-def 83.0%

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

   6. +-commutative 83.0%

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

   7. fma-def 83.0%

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

3. Simplified 83.0%

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

4. Taylor expanded in i around inf 59.9%

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

5. Final simplification 59.9%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023308 
(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))