Octave 3.8, jcobi/2

Percentage Accurate: 64.1% → 97.8%

Time: 17.3s

Alternatives: 9

Speedup: 9.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 64.1% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{\beta \cdot 2 + \left(2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, \frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))
        -0.9999999)
     (/ (/ (+ (* beta 2.0) (+ 2.0 (* i 4.0))) alpha) 2.0)
     (/
      (fma
       (/ (+ alpha beta) (+ alpha (+ beta (fma 2.0 i 2.0))))
       (/ (- beta alpha) (+ alpha (fma 2.0 i beta)))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.9999999) {
		tmp = (((beta * 2.0) + (2.0 + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = fma(((alpha + beta) / (alpha + (beta + fma(2.0, i, 2.0)))), ((beta - alpha) / (alpha + fma(2.0, i, beta))), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(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)) <= -0.9999999)
		tmp = Float64(Float64(Float64(Float64(beta * 2.0) + Float64(2.0 + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(fma(Float64(Float64(alpha + beta) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))), Float64(Float64(beta - alpha) / Float64(alpha + fma(2.0, i, beta))), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(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], -0.9999999], N[(N[(N[(N[(beta * 2.0), $MachinePrecision] + N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(alpha + beta), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.999999900000000053

   1. Initial program 2.8%

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

   3. Simplified 14.1%

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

   5. Applied egg-rr 2.5%

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

   6. Taylor expanded in alpha around inf 91.6%

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

   7. Taylor expanded in i around 0 91.6%

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

      1. *-commutative 91.6%

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

   9. Simplified 91.6%

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

   if -0.999999900000000053 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 82.5%

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

   3. Simplified 99.7%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{\beta \cdot 2 + \left(2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, \frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \]

Alternative 2: 97.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{\beta \cdot 2 + \left(2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{2}\\ \end{array} \end{array} \]

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))
        -0.9999999)
     (/ (/ (+ (* beta 2.0) (+ 2.0 (* i 4.0))) alpha) 2.0)
     (/
      (+
       1.0
       (*
        (/ (+ alpha beta) (+ alpha (+ beta (fma 2.0 i 2.0))))
        (/ (- beta alpha) (fma 2.0 i (+ alpha beta)))))
      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)) <= -0.9999999) {
		tmp = (((beta * 2.0) + (2.0 + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((alpha + beta) / (alpha + (beta + fma(2.0, i, 2.0)))) * ((beta - alpha) / fma(2.0, i, (alpha + beta))))) / 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)) <= -0.9999999)
		tmp = Float64(Float64(Float64(Float64(beta * 2.0) + Float64(2.0 + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(alpha + beta) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))) * Float64(Float64(beta - alpha) / fma(2.0, i, Float64(alpha + beta))))) / 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], -0.9999999], N[(N[(N[(N[(beta * 2.0), $MachinePrecision] + N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + 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[(2.0 * i + N[(alpha + beta), $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)) < -0.999999900000000053

   1. Initial program 2.8%

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

   3. Simplified 14.1%

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

   5. Applied egg-rr 2.5%

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

   6. Taylor expanded in alpha around inf 91.6%

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

   7. Taylor expanded in i around 0 91.6%

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

      1. *-commutative 91.6%

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

   9. Simplified 91.6%

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

   if -0.999999900000000053 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 82.5%

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

   3. Simplified 99.7%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{\beta \cdot 2 + \left(2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{2}\\ \end{array} \]

Alternative 3: 96.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.5:\\ \;\;\;\;\frac{\frac{\beta \cdot 2 + \left(2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{t_1}}{2}\\ \end{array} \end{array} \]

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) -0.5)
     (/ (/ (+ (* beta 2.0) (+ 2.0 (* i 4.0))) alpha) 2.0)
     (/ (+ 1.0 (/ beta 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) <= -0.5) {
		tmp = (((beta * 2.0) + (2.0 + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / 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) <= (-0.5d0)) then
        tmp = (((beta * 2.0d0) + (2.0d0 + (i * 4.0d0))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + (beta / 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) <= -0.5) {
		tmp = (((beta * 2.0) + (2.0 + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / 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) <= -0.5:
		tmp = (((beta * 2.0) + (2.0 + (i * 4.0))) / alpha) / 2.0
	else:
		tmp = (1.0 + (beta / 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) <= -0.5)
		tmp = Float64(Float64(Float64(Float64(beta * 2.0) + Float64(2.0 + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(beta / 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) <= -0.5)
		tmp = (((beta * 2.0) + (2.0 + (i * 4.0))) / alpha) / 2.0;
	else
		tmp = (1.0 + (beta / 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], -0.5], N[(N[(N[(N[(beta * 2.0), $MachinePrecision] + N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(beta / 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)) < -0.5

   1. Initial program 5.2%

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

   3. Simplified 16.1%

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

   5. Applied egg-rr 5.0%

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

   6. Taylor expanded in alpha around inf 90.1%

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

   7. Taylor expanded in i around 0 90.1%

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

      1. *-commutative 90.1%

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

   9. Simplified 90.1%

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

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

   1. Initial program 82.5%

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

   2. Taylor expanded in beta around inf 99.7%

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

4. Final simplification 97.3%

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

Alternative 4: 72.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{if}\;2 \cdot i \leq 10^{-151}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;2 \cdot i \leq 5 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \mathbf{elif}\;2 \cdot i \leq 9.2 \cdot 10^{+259}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (/ beta (+ beta (+ alpha 2.0)))) 2.0)))
   (if (<= (* 2.0 i) 1e-151)
     t_0
     (if (<= (* 2.0 i) 5e-125)
       (/ (/ (+ beta 2.0) alpha) 2.0)
       (if (<= (* 2.0 i) 9.2e+259) t_0 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + (beta / (beta + (alpha + 2.0)))) / 2.0;
	double tmp;
	if ((2.0 * i) <= 1e-151) {
		tmp = t_0;
	} else if ((2.0 * i) <= 5e-125) {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	} else if ((2.0 * i) <= 9.2e+259) {
		tmp = t_0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (beta / (beta + (alpha + 2.0d0)))) / 2.0d0
    if ((2.0d0 * i) <= 1d-151) then
        tmp = t_0
    else if ((2.0d0 * i) <= 5d-125) then
        tmp = ((beta + 2.0d0) / alpha) / 2.0d0
    else if ((2.0d0 * i) <= 9.2d+259) then
        tmp = t_0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + (beta / (beta + (alpha + 2.0)))) / 2.0;
	double tmp;
	if ((2.0 * i) <= 1e-151) {
		tmp = t_0;
	} else if ((2.0 * i) <= 5e-125) {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	} else if ((2.0 * i) <= 9.2e+259) {
		tmp = t_0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (1.0 + (beta / (beta + (alpha + 2.0)))) / 2.0
	tmp = 0
	if (2.0 * i) <= 1e-151:
		tmp = t_0
	elif (2.0 * i) <= 5e-125:
		tmp = ((beta + 2.0) / alpha) / 2.0
	elif (2.0 * i) <= 9.2e+259:
		tmp = t_0
	else:
		tmp = 0.5
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(1.0 + Float64(beta / Float64(beta + Float64(alpha + 2.0)))) / 2.0)
	tmp = 0.0
	if (Float64(2.0 * i) <= 1e-151)
		tmp = t_0;
	elseif (Float64(2.0 * i) <= 5e-125)
		tmp = Float64(Float64(Float64(beta + 2.0) / alpha) / 2.0);
	elseif (Float64(2.0 * i) <= 9.2e+259)
		tmp = t_0;
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (1.0 + (beta / (beta + (alpha + 2.0)))) / 2.0;
	tmp = 0.0;
	if ((2.0 * i) <= 1e-151)
		tmp = t_0;
	elseif ((2.0 * i) <= 5e-125)
		tmp = ((beta + 2.0) / alpha) / 2.0;
	elseif ((2.0 * i) <= 9.2e+259)
		tmp = t_0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[N[(2.0 * i), $MachinePrecision], 1e-151], t$95$0, If[LessEqual[N[(2.0 * i), $MachinePrecision], 5e-125], N[(N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[N[(2.0 * i), $MachinePrecision], 9.2e+259], t$95$0, 0.5]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 2 i) < 9.9999999999999994e-152 or 4.99999999999999967e-125 < (*.f64 2 i) < 9.2000000000000004e259

   1. Initial program 65.2%

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

   2. Taylor expanded in beta around inf 79.2%

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

   3. Taylor expanded in i around 0 76.2%

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

      1. associate-+r+ 76.2%

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

   5. Simplified 76.2%

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

   if 9.9999999999999994e-152 < (*.f64 2 i) < 4.99999999999999967e-125

   1. Initial program 21.6%

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

   2. Taylor expanded in alpha around inf 24.1%

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

      1. neg-mul-1 24.1%

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

   4. Simplified 24.1%

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

   5. Taylor expanded in alpha around inf 72.6%

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

   6. Taylor expanded in i around 0 72.6%

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

      1. +-commutative 72.6%

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

   8. Simplified 72.6%

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

   if 9.2000000000000004e259 < (*.f64 2 i)

   1. Initial program 69.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in i around inf 100.0%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot i \leq 10^{-151}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{elif}\;2 \cdot i \leq 5 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \mathbf{elif}\;2 \cdot i \leq 9.2 \cdot 10^{+259}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 5: 73.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;2 \cdot i \leq 10^{-151}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;2 \cdot i \leq 5 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \mathbf{elif}\;2 \cdot i \leq 2 \cdot 10^{+182}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)))
   (if (<= (* 2.0 i) 1e-151)
     t_0
     (if (<= (* 2.0 i) 5e-125)
       (/ (/ (+ beta 2.0) alpha) 2.0)
       (if (<= (* 2.0 i) 2e+182) t_0 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if ((2.0 * i) <= 1e-151) {
		tmp = t_0;
	} else if ((2.0 * i) <= 5e-125) {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	} else if ((2.0 * i) <= 2e+182) {
		tmp = t_0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    if ((2.0d0 * i) <= 1d-151) then
        tmp = t_0
    else if ((2.0d0 * i) <= 5d-125) then
        tmp = ((beta + 2.0d0) / alpha) / 2.0d0
    else if ((2.0d0 * i) <= 2d+182) then
        tmp = t_0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if ((2.0 * i) <= 1e-151) {
		tmp = t_0;
	} else if ((2.0 * i) <= 5e-125) {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	} else if ((2.0 * i) <= 2e+182) {
		tmp = t_0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0
	tmp = 0
	if (2.0 * i) <= 1e-151:
		tmp = t_0
	elif (2.0 * i) <= 5e-125:
		tmp = ((beta + 2.0) / alpha) / 2.0
	elif (2.0 * i) <= 2e+182:
		tmp = t_0
	else:
		tmp = 0.5
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0)
	tmp = 0.0
	if (Float64(2.0 * i) <= 1e-151)
		tmp = t_0;
	elseif (Float64(2.0 * i) <= 5e-125)
		tmp = Float64(Float64(Float64(beta + 2.0) / alpha) / 2.0);
	elseif (Float64(2.0 * i) <= 2e+182)
		tmp = t_0;
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	tmp = 0.0;
	if ((2.0 * i) <= 1e-151)
		tmp = t_0;
	elseif ((2.0 * i) <= 5e-125)
		tmp = ((beta + 2.0) / alpha) / 2.0;
	elseif ((2.0 * i) <= 2e+182)
		tmp = t_0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[N[(2.0 * i), $MachinePrecision], 1e-151], t$95$0, If[LessEqual[N[(2.0 * i), $MachinePrecision], 5e-125], N[(N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[N[(2.0 * i), $MachinePrecision], 2e+182], t$95$0, 0.5]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 2 i) < 9.9999999999999994e-152 or 4.99999999999999967e-125 < (*.f64 2 i) < 2.0000000000000001e182

   1. Initial program 64.8%

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

   2. Taylor expanded in beta around inf 77.5%

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

   3. Taylor expanded in alpha around 0 77.5%

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

   4. Taylor expanded in i around 0 74.8%

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

      1. +-commutative 74.8%

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

   6. Simplified 74.8%

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

   if 9.9999999999999994e-152 < (*.f64 2 i) < 4.99999999999999967e-125

   1. Initial program 21.6%

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

   2. Taylor expanded in alpha around inf 24.1%

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

      1. neg-mul-1 24.1%

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

   4. Simplified 24.1%

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

   5. Taylor expanded in alpha around inf 72.6%

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

   6. Taylor expanded in i around 0 72.6%

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

      1. +-commutative 72.6%

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

   8. Simplified 72.6%

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

   if 2.0000000000000001e182 < (*.f64 2 i)

   1. Initial program 67.9%

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

   3. Simplified 93.0%

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

   4. Taylor expanded in i around inf 87.4%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot i \leq 10^{-151}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;2 \cdot i \leq 5 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \mathbf{elif}\;2 \cdot i \leq 2 \cdot 10^{+182}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 6: 82.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4 \cdot 10^{+88}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\beta + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 5.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot i}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 1.8 \cdot 10^{+140}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta \cdot 2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 4e+88)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ beta (* 2.0 i))))) 2.0)
   (if (<= alpha 5.5e+112)
     (/ (/ (+ 2.0 (* 2.0 i)) alpha) 2.0)
     (if (<= alpha 1.8e+140)
       (/ (+ 1.0 (/ beta (+ 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 <= 4e+88) {
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 2.0;
	} else if (alpha <= 5.5e+112) {
		tmp = ((2.0 + (2.0 * i)) / alpha) / 2.0;
	} else if (alpha <= 1.8e+140) {
		tmp = (1.0 + (beta / (beta + (alpha + 2.0)))) / 2.0;
	} else {
		tmp = (((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 <= 4d+88) then
        tmp = (1.0d0 + (beta / (2.0d0 + (beta + (2.0d0 * i))))) / 2.0d0
    else if (alpha <= 5.5d+112) then
        tmp = ((2.0d0 + (2.0d0 * i)) / alpha) / 2.0d0
    else if (alpha <= 1.8d+140) then
        tmp = (1.0d0 + (beta / (beta + (alpha + 2.0d0)))) / 2.0d0
    else
        tmp = (((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 <= 4e+88) {
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 2.0;
	} else if (alpha <= 5.5e+112) {
		tmp = ((2.0 + (2.0 * i)) / alpha) / 2.0;
	} else if (alpha <= 1.8e+140) {
		tmp = (1.0 + (beta / (beta + (alpha + 2.0)))) / 2.0;
	} else {
		tmp = (((beta * 2.0) + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 4e+88:
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 2.0
	elif alpha <= 5.5e+112:
		tmp = ((2.0 + (2.0 * i)) / alpha) / 2.0
	elif alpha <= 1.8e+140:
		tmp = (1.0 + (beta / (beta + (alpha + 2.0)))) / 2.0
	else:
		tmp = (((beta * 2.0) + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 4e+88)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + Float64(beta + Float64(2.0 * i))))) / 2.0);
	elseif (alpha <= 5.5e+112)
		tmp = Float64(Float64(Float64(2.0 + Float64(2.0 * i)) / alpha) / 2.0);
	elseif (alpha <= 1.8e+140)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + Float64(alpha + 2.0)))) / 2.0);
	else
		tmp = Float64(Float64(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 <= 4e+88)
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 2.0;
	elseif (alpha <= 5.5e+112)
		tmp = ((2.0 + (2.0 * i)) / alpha) / 2.0;
	elseif (alpha <= 1.8e+140)
		tmp = (1.0 + (beta / (beta + (alpha + 2.0)))) / 2.0;
	else
		tmp = (((beta * 2.0) + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 4e+88], N[(N[(1.0 + N[(beta / N[(2.0 + N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 5.5e+112], N[(N[(N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 1.8e+140], N[(N[(1.0 + N[(beta / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if alpha < 3.99999999999999984e88

   1. Initial program 81.7%

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

   2. Taylor expanded in beta around inf 94.3%

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

   3. Taylor expanded in alpha around 0 94.3%

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

   if 3.99999999999999984e88 < alpha < 5.50000000000000026e112

   1. Initial program 3.6%

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

   2. Taylor expanded in alpha around inf 4.0%

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

      1. neg-mul-1 4.0%

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

   4. Simplified 4.0%

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

   5. Taylor expanded in alpha around inf 86.5%

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

   6. Taylor expanded in beta around 0 86.5%

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

   if 5.50000000000000026e112 < alpha < 1.8e140

   1. Initial program 45.0%

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

   2. Taylor expanded in beta around inf 72.9%

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

   3. Taylor expanded in i around 0 72.9%

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

      1. associate-+r+ 72.9%

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

   5. Simplified 72.9%

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

   if 1.8e140 < alpha

   1. Initial program 1.4%

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

   3. Simplified 27.7%

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

   5. Applied egg-rr 0.2%

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

   6. Taylor expanded in alpha around inf 78.6%

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

   7. Taylor expanded in i around inf 49.7%

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

      1. *-commutative 49.7%

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

   9. Simplified 49.7%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4 \cdot 10^{+88}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\beta + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 5.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot i}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 1.8 \cdot 10^{+140}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta \cdot 2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 88.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.85 \cdot 10^{+87}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\beta + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta \cdot 2 + \left(2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.85e+87)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ beta (* 2.0 i))))) 2.0)
   (/ (/ (+ (* beta 2.0) (+ 2.0 (* i 4.0))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.85e+87) {
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 2.0;
	} else {
		tmp = (((beta * 2.0) + (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 <= 1.85d+87) then
        tmp = (1.0d0 + (beta / (2.0d0 + (beta + (2.0d0 * i))))) / 2.0d0
    else
        tmp = (((beta * 2.0d0) + (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 <= 1.85e+87) {
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 2.0;
	} else {
		tmp = (((beta * 2.0) + (2.0 + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.85e+87:
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 2.0
	else:
		tmp = (((beta * 2.0) + (2.0 + (i * 4.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.85e+87)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + Float64(beta + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(beta * 2.0) + Float64(2.0 + Float64(i * 4.0))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.85e+87)
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 2.0;
	else
		tmp = (((beta * 2.0) + (2.0 + (i * 4.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.85e+87], N[(N[(1.0 + N[(beta / N[(2.0 + N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta * 2.0), $MachinePrecision] + N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.85000000000000001e87

   1. Initial program 81.7%

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

   2. Taylor expanded in beta around inf 94.3%

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

   3. Taylor expanded in alpha around 0 94.3%

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

   if 1.85000000000000001e87 < alpha

   1. Initial program 6.6%

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

   3. Simplified 30.5%

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

   5. Applied egg-rr 5.5%

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

   6. Taylor expanded in alpha around inf 75.4%

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

   7. Taylor expanded in i around 0 75.4%

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

      1. *-commutative 75.4%

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

   9. Simplified 75.4%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.85 \cdot 10^{+87}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\beta + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta \cdot 2 + \left(2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 72.4% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 480000000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 480000000000.0) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 480000000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 480000000000.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 480000000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 480000000000.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 480000000000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 480000000000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 480000000000.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 4.8e11

   1. Initial program 73.9%

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

   3. Simplified 76.4%

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

   4. Taylor expanded in i around inf 75.2%

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

   if 4.8e11 < beta

   1. Initial program 41.6%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in beta around inf 68.7%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 480000000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 62.1% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 63.8%

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

3. Simplified 79.7%

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

4. Taylor expanded in i around inf 62.4%

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

5. Final simplification 62.4%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023297 
(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))