Octave 3.8, jcobi/2

Percentage Accurate: 63.3% → 97.9%

Time: 17.1s

Alternatives: 10

Speedup: 9.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 63.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.995:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}{\alpha + \beta} \cdot \frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}}{2}\\ \end{array} \end{array} \]

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)) -0.995)
     (/ (/ (+ (* i 4.0) (+ 2.0 (* beta 2.0))) alpha) 2.0)
     (/
      (+
       1.0
       (/
        1.0
        (*
         (/ (+ (+ alpha beta) (fma 2.0 i 2.0)) (+ alpha beta))
         (/ (+ beta (fma 2.0 i alpha)) (- beta alpha)))))
      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)) <= -0.995) {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (1.0 / ((((alpha + beta) + fma(2.0, i, 2.0)) / (alpha + beta)) * ((beta + fma(2.0, i, alpha)) / (beta - alpha))))) / 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)) <= -0.995)
		tmp = Float64(Float64(Float64(Float64(i * 4.0) + Float64(2.0 + Float64(beta * 2.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(1.0 / Float64(Float64(Float64(Float64(alpha + beta) + fma(2.0, i, 2.0)) / Float64(alpha + beta)) * Float64(Float64(beta + fma(2.0, i, alpha)) / Float64(beta - alpha))))) / 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], -0.995], N[(N[(N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(1.0 / N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision] / N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision] / N[(beta - alpha), $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)) < -0.994999999999999996

   1. Initial program 3.3%

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

   3. Simplified 20.9%

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

      1. frac-times 2.7%

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

      2. fma-udef 2.7%

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

      3. +-commutative 2.7%

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

      4. frac-times 20.9%

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

      5. clear-num 20.9%

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

      6. frac-times 20.9%

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

      7. *-un-lft-identity 20.9%

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

      8. +-commutative 20.9%

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

      9. associate-+r+ 20.9%

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

      10. +-commutative 20.9%

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

      11. fma-def 20.9%

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

   5. Applied egg-rr 20.9%

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

   6. Taylor expanded in beta around inf 20.4%

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

   7. Taylor expanded in alpha around inf 85.8%

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

   if -0.994999999999999996 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 76.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. clear-num 100.0%

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

      3. fma-udef 100.0%

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

      4. +-commutative 100.0%

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

      5. frac-times 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

      8. associate-+r+ 100.0%

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

      9. +-commutative 100.0%

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

      10. fma-def 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.995:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}{\alpha + \beta} \cdot \frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}}{2}\\ \end{array} \]

Alternative 2: 97.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.995:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{2}\\ \end{array} \end{array} \]

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)) -0.995)
     (/ (/ (+ (* i 4.0) (+ 2.0 (* beta 2.0))) 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)) <= -0.995) {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / 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)) <= -0.995)
		tmp = Float64(Float64(Float64(Float64(i * 4.0) + Float64(2.0 + Float64(beta * 2.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(alpha + beta) / Float64(Float64(alpha + 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], -0.995], N[(N[(N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(alpha + beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $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)) < -0.994999999999999996

   1. Initial program 3.3%

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

   3. Simplified 20.9%

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

      1. frac-times 2.7%

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

      2. fma-udef 2.7%

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

      3. +-commutative 2.7%

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

      4. frac-times 20.9%

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

      5. clear-num 20.9%

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

      6. frac-times 20.9%

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

      7. *-un-lft-identity 20.9%

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

      8. +-commutative 20.9%

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

      9. associate-+r+ 20.9%

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

      10. +-commutative 20.9%

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

      11. fma-def 20.9%

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

   5. Applied egg-rr 20.9%

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

   6. Taylor expanded in beta around inf 20.4%

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

   7. Taylor expanded in alpha around inf 85.8%

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

   if -0.994999999999999996 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 76.9%

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

   3. Simplified 99.9%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.995:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{2}\\ \end{array} \]

Alternative 3: 97.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.995:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\frac{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}{\alpha + \beta} \cdot \left(\beta + \mathsf{fma}\left(2, i, \alpha\right)\right)}}{2}\\ \end{array} \end{array} \]

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)) -0.995)
     (/ (/ (+ (* i 4.0) (+ 2.0 (* beta 2.0))) alpha) 2.0)
     (/
      (+
       1.0
       (/
        (- beta alpha)
        (*
         (/ (+ (+ alpha beta) (fma 2.0 i 2.0)) (+ alpha beta))
         (+ beta (fma 2.0 i alpha)))))
      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)) <= -0.995) {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / ((((alpha + beta) + fma(2.0, i, 2.0)) / (alpha + beta)) * (beta + fma(2.0, i, alpha))))) / 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)) <= -0.995)
		tmp = Float64(Float64(Float64(Float64(i * 4.0) + Float64(2.0 + Float64(beta * 2.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(Float64(Float64(Float64(alpha + beta) + fma(2.0, i, 2.0)) / Float64(alpha + beta)) * Float64(beta + fma(2.0, i, alpha))))) / 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], -0.995], N[(N[(N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision] / N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * N[(beta + N[(2.0 * i + alpha), $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)) < -0.994999999999999996

   1. Initial program 3.3%

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

   3. Simplified 20.9%

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

      1. frac-times 2.7%

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

      2. fma-udef 2.7%

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

      3. +-commutative 2.7%

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

      4. frac-times 20.9%

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

      5. clear-num 20.9%

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

      6. frac-times 20.9%

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

      7. *-un-lft-identity 20.9%

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

      8. +-commutative 20.9%

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

      9. associate-+r+ 20.9%

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

      10. +-commutative 20.9%

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

      11. fma-def 20.9%

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

   5. Applied egg-rr 20.9%

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

   6. Taylor expanded in beta around inf 20.4%

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

   7. Taylor expanded in alpha around inf 85.8%

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

   if -0.994999999999999996 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 76.9%

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

   3. Simplified 99.9%

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

      1. frac-times 76.2%

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

      2. fma-udef 76.2%

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

      3. +-commutative 76.2%

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

      4. frac-times 99.9%

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

      5. clear-num 99.9%

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

      6. frac-times 100.0%

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

      7. *-un-lft-identity 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+r+ 100.0%

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

      10. +-commutative 100.0%

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

      11. fma-def 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.995:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\frac{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}{\alpha + \beta} \cdot \left(\beta + \mathsf{fma}\left(2, i, \alpha\right)\right)}}{2}\\ \end{array} \]

Alternative 4: 97.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)) <= (-0.5d0)) then
        tmp = (((i * 4.0d0) + (2.0d0 + (beta * 2.0d0))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + ((beta / (beta + (2.0d0 + (2.0d0 * i)))) * (beta / (beta + (2.0d0 * i))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.5) {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta / (beta + (2.0 + (2.0 * i)))) * (beta / (beta + (2.0 * i))))) / 2.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 5.0%

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

   3. Simplified 22.2%

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

      1. frac-times 4.4%

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

      2. fma-udef 4.4%

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

      3. +-commutative 4.4%

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

      4. frac-times 22.2%

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

      5. clear-num 22.3%

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

      6. frac-times 22.3%

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

      7. *-un-lft-identity 22.3%

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

      8. +-commutative 22.3%

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

      9. associate-+r+ 22.3%

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

      10. +-commutative 22.3%

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

      11. fma-def 22.3%

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

   5. Applied egg-rr 22.3%

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

   6. Taylor expanded in beta around inf 20.6%

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

   7. Taylor expanded in alpha around inf 84.7%

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

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

   1. Initial program 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

   3. Simplified 100.0%

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

   4. Taylor expanded in alpha around 0 99.1%

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

   5. Taylor expanded in alpha around 0 99.1%

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

4. Final simplification 96.2%

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

Alternative 5: 83.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3 \cdot 10^{+123}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 3e+123)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ (* i 4.0) (+ 2.0 (* beta 2.0))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3e+123) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 3d+123) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (((i * 4.0d0) + (2.0d0 + (beta * 2.0d0))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3e+123) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 3e+123:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 3e+123)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(i * 4.0) + Float64(2.0 + Float64(beta * 2.0))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 3e+123)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 3e+123], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3.00000000000000008e123

   1. Initial program 76.9%

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

   3. Simplified 97.1%

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

      1. frac-times 76.2%

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

      2. fma-udef 76.2%

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

      3. +-commutative 76.2%

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

      4. frac-times 97.1%

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

      5. clear-num 97.1%

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

      6. frac-times 97.1%

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

      7. *-un-lft-identity 97.1%

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

      8. +-commutative 97.1%

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

      9. associate-+r+ 97.1%

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

      10. +-commutative 97.1%

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

      11. fma-def 97.1%

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

   5. Applied egg-rr 97.1%

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

   6. Taylor expanded in i around 0 82.5%

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

      1. +-commutative 82.5%

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

   8. Simplified 82.5%

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

   9. Taylor expanded in alpha around 0 88.0%

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

   if 3.00000000000000008e123 < alpha

   1. Initial program 3.5%

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

   3. Simplified 32.7%

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

      1. frac-times 2.4%

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

      2. fma-udef 2.4%

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

      3. +-commutative 2.4%

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

      4. frac-times 32.7%

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

      5. clear-num 32.8%

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

      6. frac-times 32.8%

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

      7. *-un-lft-identity 32.8%

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

      8. +-commutative 32.8%

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

      9. associate-+r+ 32.8%

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

      10. +-commutative 32.8%

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

      11. fma-def 32.8%

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

   5. Applied egg-rr 32.8%

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

   6. Taylor expanded in beta around inf 30.7%

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

   7. Taylor expanded in alpha around inf 73.7%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3 \cdot 10^{+123}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 6: 81.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.75 \cdot 10^{+126}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.75e+126)
   (/ (+ 1.0 (/ beta (+ beta 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 <= 1.75e+126) {
		tmp = (1.0 + (beta / (beta + 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 <= 1.75d+126) then
        tmp = (1.0d0 + (beta / (beta + 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 <= 1.75e+126) {
		tmp = (1.0 + (beta / (beta + 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 <= 1.75e+126:
		tmp = (1.0 + (beta / (beta + 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 <= 1.75e+126)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta + 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 <= 1.75e+126)
		tmp = (1.0 + (beta / (beta + 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, 1.75e+126], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta + 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 < 1.7500000000000001e126

   1. Initial program 76.9%

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

   3. Simplified 97.1%

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

      1. frac-times 76.2%

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

      2. fma-udef 76.2%

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

      3. +-commutative 76.2%

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

      4. frac-times 97.1%

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

      5. clear-num 97.1%

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

      6. frac-times 97.1%

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

      7. *-un-lft-identity 97.1%

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

      8. +-commutative 97.1%

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

      9. associate-+r+ 97.1%

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

      10. +-commutative 97.1%

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

      11. fma-def 97.1%

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

   5. Applied egg-rr 97.1%

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

   6. Taylor expanded in i around 0 82.5%

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

      1. +-commutative 82.5%

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

   8. Simplified 82.5%

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

   9. Taylor expanded in alpha around 0 88.0%

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

   if 1.7500000000000001e126 < alpha

   1. Initial program 3.5%

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

   3. Simplified 32.7%

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

      1. frac-times 2.4%

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

      2. fma-udef 2.4%

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

      3. +-commutative 2.4%

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

      4. frac-times 32.7%

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

      5. clear-num 32.8%

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

      6. frac-times 32.8%

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

      7. *-un-lft-identity 32.8%

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

      8. +-commutative 32.8%

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

      9. associate-+r+ 32.8%

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

      10. +-commutative 32.8%

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

      11. fma-def 32.8%

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

   5. Applied egg-rr 32.8%

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

   6. Taylor expanded in beta around 0 20.2%

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

   7. Taylor expanded in alpha around inf 54.8%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.75 \cdot 10^{+126}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 76.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.4 \cdot 10^{+202}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 1.4e+202) (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1.4e+202) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 1.4d+202) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1.4e+202) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if i <= 1.4e+202:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 1.4e+202)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 1.4e+202)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 1.4e+202], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if i < 1.40000000000000008e202

   1. Initial program 60.8%

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

   3. Simplified 81.3%

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

      1. frac-times 60.1%

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

      2. fma-udef 60.1%

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

      3. +-commutative 60.1%

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

      4. frac-times 81.3%

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

      5. clear-num 81.3%

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

      6. frac-times 81.3%

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

      7. *-un-lft-identity 81.3%

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

      8. +-commutative 81.3%

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

      9. associate-+r+ 81.3%

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

      10. +-commutative 81.3%

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

      11. fma-def 81.3%

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

   5. Applied egg-rr 81.3%

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

   6. Taylor expanded in i around 0 75.3%

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

      1. +-commutative 75.3%

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

   8. Simplified 75.3%

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

   9. Taylor expanded in alpha around 0 76.8%

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

   if 1.40000000000000008e202 < i

   1. Initial program 69.3%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in i around inf 95.0%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.4 \cdot 10^{+202}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 8: 78.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.7 \cdot 10^{+160}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.7e+160)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.7e+160) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.7d+160) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.7e+160) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.7e+160:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.7e+160)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.7e+160)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.7e+160], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.70000000000000015e160

   1. Initial program 74.6%

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

   3. Simplified 94.9%

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

      1. frac-times 73.9%

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

      2. fma-udef 73.9%

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

      3. +-commutative 73.9%

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

      4. frac-times 94.9%

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

      5. clear-num 94.9%

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

      6. frac-times 95.0%

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

      7. *-un-lft-identity 95.0%

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

      8. +-commutative 95.0%

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

      9. associate-+r+ 95.0%

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

      10. +-commutative 95.0%

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

      11. fma-def 95.0%

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

   5. Applied egg-rr 95.0%

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

   6. Taylor expanded in i around 0 80.5%

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

      1. +-commutative 80.5%

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

   8. Simplified 80.5%

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

   9. Taylor expanded in alpha around 0 86.3%

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

   if 1.70000000000000015e160 < alpha

   1. Initial program 1.2%

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

   3. Simplified 31.4%

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

      1. frac-times 0.0%

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

      2. fma-udef 0.0%

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

      3. +-commutative 0.0%

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

      4. frac-times 31.4%

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

      5. clear-num 31.4%

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

      6. frac-times 31.4%

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

      7. *-un-lft-identity 31.4%

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

      8. +-commutative 31.4%

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

      9. associate-+r+ 31.4%

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

      10. +-commutative 31.4%

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

      11. fma-def 31.4%

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

   5. Applied egg-rr 31.4%

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

   6. Taylor expanded in i around 0 14.2%

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

      1. +-commutative 14.2%

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

   8. Simplified 14.2%

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

   9. Taylor expanded in alpha around inf 51.4%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.7 \cdot 10^{+160}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 9: 72.8% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+21}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 8e+21) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8e+21) {
		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 <= 8d+21) 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 <= 8e+21) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 8e+21:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 8e+21)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 8e+21)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 8e+21], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 8e21

   1. Initial program 79.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in i around inf 80.4%

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

   if 8e21 < beta

   1. Initial program 32.5%

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

   3. Simplified 85.5%

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

   4. Taylor expanded in beta around inf 69.5%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+21}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 61.9% 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 62.5%

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

3. Simplified 84.5%

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

4. Taylor expanded in i around inf 61.8%

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

5. Final simplification 61.8%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023264 
(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))