Octave 3.8, jcobi/2

Percentage Accurate: 63.4% → 97.4%

Time: 15.9s

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: 63.4% 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: 97.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 2.0%

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

      1. expm1-log1p-u 0.0%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-udef 0.0%

         \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. *-commutative 0.0%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 0.0%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \color{blue}{\left(\beta + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      5. associate-+r+ 0.0%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\color{blue}{\alpha + \left(\beta + 2 \cdot i\right)}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      6. +-commutative 0.0%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \color{blue}{\left(2 \cdot i + \beta\right)}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      7. fma-udef 0.0%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \color{blue}{\mathsf{fma}\left(2, i, \beta\right)}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 0.0%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. expm1-def 0.0%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-log1p 2.0%

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

      3. associate-*r/ 16.6%

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

      4. +-commutative 16.6%

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

   5. Simplified 16.6%

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

   6. Taylor expanded in beta around 0 16.4%

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

   7. Taylor expanded in alpha around inf 89.3%

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

   8. Taylor expanded in i around 0 89.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha} + 2 \cdot \frac{i}{\alpha}} \]

   if -1 < (/.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 84.1%

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

      1. associate-/l/ 83.6%

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

      2. *-commutative 83.6%

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

      3. times-frac 99.7%

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

      4. fma-def 99.7%

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

      5. associate-+l+ 99.7%

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

      6. fma-def 99.7%

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

      7. associate-+l+ 99.7%

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

      8. +-commutative 99.7%

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

      9. fma-def 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 97.7%

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

Alternative 2: 97.4% accurate, 0.2× speedup

Math

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

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) -1.0)
     (+ (* 0.5 (/ (+ 2.0 (* beta 2.0)) alpha)) (* 2.0 (/ i alpha)))
     (/
      (+
       1.0
       (/
        (* (- beta alpha) (/ (+ alpha beta) (+ alpha (fma 2.0 i 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) <= -1.0) {
		tmp = (0.5 * ((2.0 + (beta * 2.0)) / alpha)) + (2.0 * (i / alpha));
	} else {
		tmp = (1.0 + (((beta - alpha) * ((alpha + beta) / (alpha + fma(2.0, i, 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) <= -1.0)
		tmp = Float64(Float64(0.5 * Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha)) + Float64(2.0 * Float64(i / alpha)));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(beta - alpha) * Float64(Float64(alpha + beta) / Float64(alpha + fma(2.0, i, beta)))) / t_1)) / 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]}, 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], -1.0], N[(N[(0.5 * N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -1

   1. Initial program 2.0%

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

      1. expm1-log1p-u 0.0%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-udef 0.0%

         \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. *-commutative 0.0%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 0.0%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \color{blue}{\left(\beta + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      5. associate-+r+ 0.0%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\color{blue}{\alpha + \left(\beta + 2 \cdot i\right)}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      6. +-commutative 0.0%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \color{blue}{\left(2 \cdot i + \beta\right)}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      7. fma-udef 0.0%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \color{blue}{\mathsf{fma}\left(2, i, \beta\right)}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 0.0%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. expm1-def 0.0%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-log1p 2.0%

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

      3. associate-*r/ 16.6%

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

      4. +-commutative 16.6%

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

   5. Simplified 16.6%

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

   6. Taylor expanded in beta around 0 16.4%

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

   7. Taylor expanded in alpha around inf 89.3%

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

   8. Taylor expanded in i around 0 89.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha} + 2 \cdot \frac{i}{\alpha}} \]

   if -1 < (/.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 84.1%

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

      1. expm1-log1p-u 80.3%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-udef 80.3%

         \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. *-commutative 80.3%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 80.3%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \color{blue}{\left(\beta + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      5. associate-+r+ 80.3%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\color{blue}{\alpha + \left(\beta + 2 \cdot i\right)}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      6. +-commutative 80.3%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \color{blue}{\left(2 \cdot i + \beta\right)}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      7. fma-udef 80.3%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \color{blue}{\mathsf{fma}\left(2, i, \beta\right)}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 80.3%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. expm1-def 80.3%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-log1p 84.1%

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

      3. associate-*r/ 99.7%

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

      4. +-commutative 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 97.7%

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

Alternative 3: 97.1% accurate, 0.6× speedup

Math

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

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)
     (+ (* 0.5 (/ (+ 2.0 (* beta 2.0)) alpha)) (* 2.0 (/ i alpha)))
     (/
      (+
       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 = (0.5 * ((2.0 + (beta * 2.0)) / alpha)) + (2.0 * (i / alpha));
	} 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 = (0.5d0 * ((2.0d0 + (beta * 2.0d0)) / alpha)) + (2.0d0 * (i / alpha))
    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 = (0.5 * ((2.0 + (beta * 2.0)) / alpha)) + (2.0 * (i / alpha));
	} 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 = (0.5 * ((2.0 + (beta * 2.0)) / alpha)) + (2.0 * (i / alpha))
	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(0.5 * Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha)) + Float64(2.0 * Float64(i / alpha)));
	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 = (0.5 * ((2.0 + (beta * 2.0)) / alpha)) + (2.0 * (i / alpha));
	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[(0.5 * N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(beta / N[(beta + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta / N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.7%

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

      1. expm1-log1p-u 0.0%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-udef 0.0%

         \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. *-commutative 0.0%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 0.0%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \color{blue}{\left(\beta + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      5. associate-+r+ 0.0%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\color{blue}{\alpha + \left(\beta + 2 \cdot i\right)}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      6. +-commutative 0.0%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \color{blue}{\left(2 \cdot i + \beta\right)}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      7. fma-udef 0.0%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \color{blue}{\mathsf{fma}\left(2, i, \beta\right)}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 0.0%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. expm1-def 0.0%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-log1p 4.7%

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

      3. associate-*r/ 18.8%

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

      4. +-commutative 18.8%

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

   5. Simplified 18.8%

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

   6. Taylor expanded in beta around 0 18.5%

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

   7. Taylor expanded in alpha around inf 87.8%

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

   8. Taylor expanded in i around 0 87.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha} + 2 \cdot \frac{i}{\alpha}} \]

   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 84.3%

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

      1. associate-/l/ 83.7%

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

      2. *-commutative 83.7%

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

      3. times-frac 100.0%

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

      4. associate-+l+ 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in alpha around 0 99.7%

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

   5. Taylor expanded in alpha around 0 99.7%

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

4. Final simplification 97.2%

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

Alternative 4: 89.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 6.5 \cdot 10^{+136}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 6.5e+136)
   (/ (+ 1.0 (/ (- beta alpha) (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 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 <= 6.5e+136) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 6.5d+136) then
        tmp = (1.0d0 + ((beta - alpha) / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = (((2.0d0 + (beta * 2.0d0)) + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 6.5e+136) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 6.5e+136:
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 6.5e+136)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(2.0 + 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 <= 6.5e+136)
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 6.5e+136], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 6.4999999999999998e136

   1. Initial program 80.6%

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

   2. Taylor expanded in i around 0 92.3%

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

   if 6.4999999999999998e136 < alpha

   1. Initial program 6.0%

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

      1. expm1-log1p-u 2.6%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-udef 2.6%

         \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. *-commutative 2.6%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 2.6%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \color{blue}{\left(\beta + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      5. associate-+r+ 2.6%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\color{blue}{\alpha + \left(\beta + 2 \cdot i\right)}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      6. +-commutative 2.6%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \color{blue}{\left(2 \cdot i + \beta\right)}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      7. fma-udef 2.6%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \color{blue}{\mathsf{fma}\left(2, i, \beta\right)}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 2.6%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. expm1-def 2.6%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-log1p 6.0%

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

      3. associate-*r/ 34.0%

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

      4. +-commutative 34.0%

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

   5. Simplified 34.0%

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

   6. Taylor expanded in beta around 0 31.9%

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

   7. Taylor expanded in alpha around inf 72.9%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 6.5 \cdot 10^{+136}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 5: 75.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot i \leq 5 \cdot 10^{+116}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= (* 2.0 i) 5e+116)
   (/ (+ 1.0 (/ (- beta alpha) (+ beta (+ alpha 2.0)))) 2.0)
   0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if ((2.0 * i) <= 5e+116) {
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 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 ((2.0d0 * i) <= 5d+116) then
        tmp = (1.0d0 + ((beta - alpha) / (beta + (alpha + 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 ((2.0 * i) <= 5e+116) {
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if (2.0 * i) <= 5e+116:
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (Float64(2.0 * i) <= 5e+116)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0)))) / 2.0);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if ((2.0 * i) <= 5e+116)
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[N[(2.0 * i), $MachinePrecision], 5e+116], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if (*.f64 2 i) < 5.00000000000000025e116

   1. Initial program 59.0%

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

      1. associate-/l/ 58.4%

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

      2. *-commutative 58.4%

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

      3. times-frac 75.8%

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

      4. fma-def 75.8%

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

      5. associate-+l+ 75.8%

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

      6. fma-def 75.8%

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

      7. associate-+l+ 75.8%

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

      8. +-commutative 75.8%

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

      9. fma-def 75.8%

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

   3. Simplified 75.8%

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

   4. Taylor expanded in i around 0 74.3%

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

      1. associate--l+ 74.3%

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

      2. div-sub 74.3%

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

      3. +-commutative 74.3%

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

   6. Simplified 74.3%

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

   if 5.00000000000000025e116 < (*.f64 2 i)

   1. Initial program 83.1%

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

      1. associate-/l/ 82.8%

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

      2. *-commutative 82.8%

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

      3. times-frac 96.1%

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

      4. fma-def 96.1%

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

      5. associate-+l+ 96.1%

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

      6. fma-def 96.1%

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

      7. associate-+l+ 96.1%

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

      8. +-commutative 96.1%

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

      9. fma-def 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in i around inf 90.7%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot i \leq 5 \cdot 10^{+116}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 6: 85.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.8 \cdot 10^{+145}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.8e+145)
   (/ (+ 1.0 (/ beta (+ beta (+ 2.0 (* 2.0 i))))) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.8e+145) {
		tmp = (1.0 + (beta / (beta + (2.0 + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.8d+145) then
        tmp = (1.0d0 + (beta / (beta + (2.0d0 + (2.0d0 * i))))) / 2.0d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.8e+145) {
		tmp = (1.0 + (beta / (beta + (2.0 + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.8e+145:
		tmp = (1.0 + (beta / (beta + (2.0 + (2.0 * i))))) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.8e+145)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + Float64(2.0 + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.8e+145)
		tmp = (1.0 + (beta / (beta + (2.0 + (2.0 * i))))) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.8e+145], N[(N[(1.0 + N[(beta / N[(beta + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.79999999999999987e145

   1. Initial program 80.3%

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

   2. Taylor expanded in i around 0 91.9%

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

   3. Taylor expanded in alpha around 0 91.5%

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

   if 1.79999999999999987e145 < alpha

   1. Initial program 3.8%

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

      1. expm1-log1p-u 0.4%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-udef 0.4%

         \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. *-commutative 0.4%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 0.4%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \color{blue}{\left(\beta + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      5. associate-+r+ 0.4%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\color{blue}{\alpha + \left(\beta + 2 \cdot i\right)}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      6. +-commutative 0.4%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \color{blue}{\left(2 \cdot i + \beta\right)}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      7. fma-udef 0.4%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \color{blue}{\mathsf{fma}\left(2, i, \beta\right)}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 0.4%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. expm1-def 0.4%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-log1p 3.8%

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

      3. associate-*r/ 33.1%

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

      4. +-commutative 33.1%

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

   5. Simplified 33.1%

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

   6. Taylor expanded in beta around 0 30.9%

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

   7. Taylor expanded in alpha around inf 73.9%

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

   8. Taylor expanded in beta around 0 60.0%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.8 \cdot 10^{+145}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 88.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.75 \cdot 10^{+144}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.75e+144)
   (/ (+ 1.0 (/ beta (+ beta (+ 2.0 (* 2.0 i))))) 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 <= 2.75e+144) {
		tmp = (1.0 + (beta / (beta + (2.0 + (2.0 * i))))) / 2.0;
	} else {
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 2.75d+144) then
        tmp = (1.0d0 + (beta / (beta + (2.0d0 + (2.0d0 * i))))) / 2.0d0
    else
        tmp = (((2.0d0 + (beta * 2.0d0)) + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.75e+144) {
		tmp = (1.0 + (beta / (beta + (2.0 + (2.0 * i))))) / 2.0;
	} else {
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.75e+144:
		tmp = (1.0 + (beta / (beta + (2.0 + (2.0 * i))))) / 2.0
	else:
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.75e+144)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + Float64(2.0 + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(2.0 + 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 <= 2.75e+144)
		tmp = (1.0 + (beta / (beta + (2.0 + (2.0 * i))))) / 2.0;
	else
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.75e+144], N[(N[(1.0 + N[(beta / N[(beta + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.75000000000000011e144

   1. Initial program 80.3%

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

   2. Taylor expanded in i around 0 91.9%

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

   3. Taylor expanded in alpha around 0 91.5%

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

   if 2.75000000000000011e144 < alpha

   1. Initial program 3.8%

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

      1. expm1-log1p-u 0.4%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-udef 0.4%

         \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. *-commutative 0.4%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 0.4%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \color{blue}{\left(\beta + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      5. associate-+r+ 0.4%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\color{blue}{\alpha + \left(\beta + 2 \cdot i\right)}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      6. +-commutative 0.4%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \color{blue}{\left(2 \cdot i + \beta\right)}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      7. fma-udef 0.4%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \color{blue}{\mathsf{fma}\left(2, i, \beta\right)}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 0.4%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. expm1-def 0.4%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-log1p 3.8%

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

      3. associate-*r/ 33.1%

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

      4. +-commutative 33.1%

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

   5. Simplified 33.1%

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

   6. Taylor expanded in beta around 0 30.9%

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

   7. Taylor expanded in alpha around inf 73.9%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.75 \cdot 10^{+144}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 75.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot i \leq 5 \cdot 10^{+108}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= (* 2.0 i) 5e+108) (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if ((2.0 * i) <= 5e+108) {
		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 ((2.0d0 * i) <= 5d+108) 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 ((2.0 * i) <= 5e+108) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if (2.0 * i) <= 5e+108:
		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 (Float64(2.0 * i) <= 5e+108)
		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 ((2.0 * i) <= 5e+108)
		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[N[(2.0 * i), $MachinePrecision], 5e+108], 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 (*.f64 2 i) < 4.99999999999999991e108

   1. Initial program 59.3%

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

      1. associate-/l/ 58.6%

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

      2. *-commutative 58.6%

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

      3. times-frac 76.3%

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

      4. associate-+l+ 76.3%

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

      5. fma-def 76.3%

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

      6. +-commutative 76.3%

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

      7. fma-def 76.3%

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

   3. Simplified 76.3%

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

   4. Taylor expanded in alpha around 0 75.3%

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

   5. Taylor expanded in i around 0 74.8%

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

   if 4.99999999999999991e108 < (*.f64 2 i)

   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. Step-by-step derivation

      1. associate-/l/ 81.4%

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

      2. *-commutative 81.4%

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

      3. times-frac 94.3%

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

      4. fma-def 94.3%

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

      5. associate-+l+ 94.3%

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

      6. fma-def 94.3%

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

      7. associate-+l+ 94.3%

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

      8. +-commutative 94.3%

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

      9. fma-def 94.3%

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

   3. Simplified 94.3%

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

   4. Taylor expanded in i around inf 89.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot i \leq 5 \cdot 10^{+108}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 9: 72.3% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+89}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 2.6e+89) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.6e+89) {
		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 <= 2.6d+89) 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 <= 2.6e+89) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.6e+89:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.6e+89)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.6e+89)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 2.6e+89], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2.6000000000000001e89

   1. Initial program 78.8%

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

      1. associate-/l/ 78.7%

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

      2. *-commutative 78.7%

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

      3. times-frac 81.4%

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

      4. fma-def 81.4%

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

      5. associate-+l+ 81.4%

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

      6. fma-def 81.4%

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

      7. associate-+l+ 81.4%

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

      8. +-commutative 81.4%

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

      9. fma-def 81.4%

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

   3. Simplified 81.4%

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

   4. Taylor expanded in i around inf 76.2%

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

   if 2.6000000000000001e89 < beta

   1. Initial program 27.4%

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

      1. associate-/l/ 25.5%

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

      2. *-commutative 25.5%

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

      3. times-frac 89.3%

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

      4. fma-def 89.3%

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

      5. associate-+l+ 89.3%

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

      6. fma-def 89.3%

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

      7. associate-+l+ 89.4%

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

      8. +-commutative 89.4%

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

      9. fma-def 89.4%

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

   3. Simplified 89.4%

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

   4. Taylor expanded in beta around inf 77.2%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+89}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 61.5% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 67.8%

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

   1. associate-/l/ 67.2%

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

   2. *-commutative 67.2%

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

   3. times-frac 83.1%

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

   4. fma-def 83.1%

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

   5. associate-+l+ 83.1%

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

   6. fma-def 83.1%

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

   7. associate-+l+ 83.2%

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

   8. +-commutative 83.2%

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

   9. fma-def 83.2%

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

3. Simplified 83.2%

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

4. Taylor expanded in i around inf 65.4%

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

5. Final simplification 65.4%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023196 
(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))