Octave 3.8, jcobi/2

Percentage Accurate: 62.4% → 98.0%

Time: 20.5s

Alternatives: 15

Speedup: 4.8×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.4% 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: 98.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.99999998:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\frac{-1}{\frac{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}{\alpha - \beta}}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

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))
        -0.99999998)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (fma
       (+ alpha beta)
       (/
        (/ -1.0 (/ (+ beta (+ alpha (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)) <= -0.99999998) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = fma((alpha + beta), ((-1.0 / ((beta + (alpha + 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)) <= -0.99999998)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(fma(Float64(alpha + beta), Float64(Float64(-1.0 / Float64(Float64(beta + Float64(alpha + fma(2.0, i, 2.0))) / 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], -0.99999998], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(alpha + beta), $MachinePrecision] * N[(N[(-1.0 / N[(N[(beta + N[(alpha + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha - beta), $MachinePrecision]), $MachinePrecision]), $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 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999999980000000011

   1. Initial program 2.7%

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

   3. Simplified 13.2%

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

      1. associate-*r/ 2.7%

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

      2. *-commutative 2.7%

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

      3. fma-undefine 2.7%

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

      4. +-commutative 2.7%

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

      5. fma-define 2.7%

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

      6. associate-+r+ 2.7%

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

      7. associate-/l/ 2.1%

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

      8. +-commutative 2.1%

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

      9. associate-+r+ 2.1%

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

      10. times-frac 13.2%

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

      11. +-commutative 13.2%

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

      12. +-commutative 13.2%

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

      13. fma-define 13.2%

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

   6. Applied egg-rr 13.2%

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

   7. Taylor expanded in beta around 0 13.2%

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

   8. Taylor expanded in alpha around inf 92.8%

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

   if -0.999999980000000011 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 82.2%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. fma-define 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. associate-+l+ 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 98.2%

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

Alternative 2: 98.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.99999998:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))
        -0.99999998)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (fma
       (+ alpha beta)
       (/
        (/ (- beta alpha) (+ alpha (+ beta (fma 2.0 i 2.0))))
        (+ alpha (fma 2.0 i beta)))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.99999998) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = fma((alpha + beta), (((beta - alpha) / (alpha + (beta + fma(2.0, i, 2.0)))) / (alpha + fma(2.0, i, beta))), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.99999998)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(fma(Float64(alpha + beta), Float64(Float64(Float64(beta - alpha) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))) / Float64(alpha + fma(2.0, i, beta))), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.99999998], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(alpha + beta), $MachinePrecision] * N[(N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999999980000000011

   1. Initial program 2.7%

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

   3. Simplified 13.2%

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

      1. associate-*r/ 2.7%

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

      2. *-commutative 2.7%

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

      3. fma-undefine 2.7%

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

      4. +-commutative 2.7%

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

      5. fma-define 2.7%

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

      6. associate-+r+ 2.7%

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

      7. associate-/l/ 2.1%

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

      8. +-commutative 2.1%

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

      9. associate-+r+ 2.1%

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

      10. times-frac 13.2%

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

      11. +-commutative 13.2%

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

      12. +-commutative 13.2%

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

      13. fma-define 13.2%

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

   6. Applied egg-rr 13.2%

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

   7. Taylor expanded in beta around 0 13.2%

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

   8. Taylor expanded in alpha around inf 92.8%

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

   if -0.999999980000000011 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 82.2%

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

   3. Simplified 99.7%

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

4. Final simplification 98.2%

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

Alternative 3: 98.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.99999998:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))
        -0.99999998)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (+
       (/
        (* (- beta alpha) (/ (+ alpha beta) (fma 2.0 i (+ alpha beta))))
        (+ alpha (+ beta (fma 2.0 i 2.0))))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.99999998) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = ((((beta - alpha) * ((alpha + beta) / fma(2.0, i, (alpha + beta)))) / (alpha + (beta + fma(2.0, i, 2.0)))) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.99999998)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) * Float64(Float64(alpha + beta) / fma(2.0, i, Float64(alpha + beta)))) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.99999998], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999999980000000011

   1. Initial program 2.7%

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

   3. Simplified 13.2%

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

      1. associate-*r/ 2.7%

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

      2. *-commutative 2.7%

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

      3. fma-undefine 2.7%

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

      4. +-commutative 2.7%

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

      5. fma-define 2.7%

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

      6. associate-+r+ 2.7%

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

      7. associate-/l/ 2.1%

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

      8. +-commutative 2.1%

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

      9. associate-+r+ 2.1%

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

      10. times-frac 13.2%

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

      11. +-commutative 13.2%

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

      12. +-commutative 13.2%

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

      13. fma-define 13.2%

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

   6. Applied egg-rr 13.2%

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

   7. Taylor expanded in beta around 0 13.2%

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

   8. Taylor expanded in alpha around inf 92.8%

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

   if -0.999999980000000011 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 82.2%

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

   3. Simplified 99.7%

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

4. Final simplification 98.2%

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

Alternative 4: 98.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.99999998:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))
        -0.99999998)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (+
       (*
        (/ (+ alpha beta) (+ (+ alpha beta) (fma 2.0 i 2.0)))
        (/ (- beta alpha) (+ beta (fma 2.0 i alpha))))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.99999998) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = ((((alpha + beta) / ((alpha + beta) + fma(2.0, i, 2.0))) * ((beta - alpha) / (beta + fma(2.0, i, alpha)))) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.99999998)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha + beta) / Float64(Float64(alpha + beta) + fma(2.0, i, 2.0))) * Float64(Float64(beta - alpha) / Float64(beta + fma(2.0, i, alpha)))) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.99999998], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(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[(beta + N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999999980000000011

   1. Initial program 2.7%

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

   3. Simplified 13.2%

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

      1. associate-*r/ 2.7%

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

      2. *-commutative 2.7%

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

      3. fma-undefine 2.7%

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

      4. +-commutative 2.7%

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

      5. fma-define 2.7%

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

      6. associate-+r+ 2.7%

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

      7. associate-/l/ 2.1%

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

      8. +-commutative 2.1%

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

      9. associate-+r+ 2.1%

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

      10. times-frac 13.2%

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

      11. +-commutative 13.2%

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

      12. +-commutative 13.2%

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

      13. fma-define 13.2%

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

   6. Applied egg-rr 13.2%

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

   7. Taylor expanded in beta around 0 13.2%

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

   8. Taylor expanded in alpha around inf 92.8%

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

   if -0.999999980000000011 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 82.2%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 82.2%

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

      2. *-commutative 82.2%

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

      3. fma-undefine 82.2%

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

      4. +-commutative 82.2%

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

      5. fma-define 82.2%

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

      6. associate-+r+ 82.2%

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

      7. associate-/l/ 81.7%

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

      8. +-commutative 81.7%

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

      9. associate-+r+ 81.7%

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

      10. times-frac 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

      13. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

4. Final simplification 98.1%

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

Alternative 5: 97.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.999998:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999998000000000054

   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

   3. Simplified 14.0%

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

      1. associate-*r/ 3.8%

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

      2. *-commutative 3.8%

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

      3. fma-undefine 3.8%

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

      4. +-commutative 3.8%

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

      5. fma-define 3.8%

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

      6. associate-+r+ 3.8%

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

      7. associate-/l/ 3.1%

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

      8. +-commutative 3.1%

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

      9. associate-+r+ 3.1%

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

      10. times-frac 14.1%

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

      11. +-commutative 14.1%

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

      12. +-commutative 14.1%

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

      13. fma-define 14.1%

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

   6. Applied egg-rr 14.1%

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

   7. Taylor expanded in beta around 0 14.1%

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

   8. Taylor expanded in alpha around inf 92.1%

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

   if -0.999998000000000054 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 82.3%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 98.7%

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

4. Final simplification 97.2%

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

Alternative 6: 97.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t\_0\\ t_2 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_1}\\ \mathbf{if}\;t\_2 \leq -0.99999998:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{elif}\;t\_2 \leq 0.9:\\ \;\;\;\;\frac{1 - \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\alpha - \beta\right)}{t\_0}}{t\_1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(\left(2 \cdot \frac{\alpha}{\beta} + \left(4 \cdot \frac{i}{\beta} + 2 \cdot \frac{1}{\beta}\right)\right) + 1\right) + \left(\frac{\alpha}{\beta} - \frac{\alpha}{\beta}\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (+ 2.0 t_0))
        (t_2 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1)))
   (if (<= t_2 -0.99999998)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (if (<= t_2 0.9)
       (/ (- 1.0 (/ (/ (* (+ alpha beta) (- alpha beta)) t_0) t_1)) 2.0)
       (/
        (+
         (/
          1.0
          (+
           (+
            (+
             (* 2.0 (/ alpha beta))
             (+ (* 4.0 (/ i beta)) (* 2.0 (/ 1.0 beta))))
            1.0)
           (- (/ alpha beta) (/ alpha beta))))
         1.0)
        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 t_2 = (((alpha + beta) * (beta - alpha)) / t_0) / t_1;
	double tmp;
	if (t_2 <= -0.99999998) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else if (t_2 <= 0.9) {
		tmp = (1.0 - ((((alpha + beta) * (alpha - beta)) / t_0) / t_1)) / 2.0;
	} else {
		tmp = ((1.0 / ((((2.0 * (alpha / beta)) + ((4.0 * (i / beta)) + (2.0 * (1.0 / beta)))) + 1.0) + ((alpha / beta) - (alpha / beta)))) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = 2.0d0 + t_0
    t_2 = (((alpha + beta) * (beta - alpha)) / t_0) / t_1
    if (t_2 <= (-0.99999998d0)) then
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    else if (t_2 <= 0.9d0) then
        tmp = (1.0d0 - ((((alpha + beta) * (alpha - beta)) / t_0) / t_1)) / 2.0d0
    else
        tmp = ((1.0d0 / ((((2.0d0 * (alpha / beta)) + ((4.0d0 * (i / beta)) + (2.0d0 * (1.0d0 / beta)))) + 1.0d0) + ((alpha / beta) - (alpha / beta)))) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double t_2 = (((alpha + beta) * (beta - alpha)) / t_0) / t_1;
	double tmp;
	if (t_2 <= -0.99999998) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else if (t_2 <= 0.9) {
		tmp = (1.0 - ((((alpha + beta) * (alpha - beta)) / t_0) / t_1)) / 2.0;
	} else {
		tmp = ((1.0 / ((((2.0 * (alpha / beta)) + ((4.0 * (i / beta)) + (2.0 * (1.0 / beta)))) + 1.0) + ((alpha / beta) - (alpha / beta)))) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = 2.0 + t_0
	t_2 = (((alpha + beta) * (beta - alpha)) / t_0) / t_1
	tmp = 0
	if t_2 <= -0.99999998:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	elif t_2 <= 0.9:
		tmp = (1.0 - ((((alpha + beta) * (alpha - beta)) / t_0) / t_1)) / 2.0
	else:
		tmp = ((1.0 / ((((2.0 * (alpha / beta)) + ((4.0 * (i / beta)) + (2.0 * (1.0 / beta)))) + 1.0) + ((alpha / beta) - (alpha / beta)))) + 1.0) / 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)
	t_2 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / t_1)
	tmp = 0.0
	if (t_2 <= -0.99999998)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	elseif (t_2 <= 0.9)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(alpha + beta) * Float64(alpha - beta)) / t_0) / t_1)) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(Float64(Float64(2.0 * Float64(alpha / beta)) + Float64(Float64(4.0 * Float64(i / beta)) + Float64(2.0 * Float64(1.0 / beta)))) + 1.0) + Float64(Float64(alpha / beta) - Float64(alpha / beta)))) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = 2.0 + t_0;
	t_2 = (((alpha + beta) * (beta - alpha)) / t_0) / t_1;
	tmp = 0.0;
	if (t_2 <= -0.99999998)
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	elseif (t_2 <= 0.9)
		tmp = (1.0 - ((((alpha + beta) * (alpha - beta)) / t_0) / t_1)) / 2.0;
	else
		tmp = ((1.0 / ((((2.0 * (alpha / beta)) + ((4.0 * (i / beta)) + (2.0 * (1.0 / beta)))) + 1.0) + ((alpha / beta) - (alpha / beta)))) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -0.99999998], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$2, 0.9], N[(N[(1.0 - N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(alpha - beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 / N[(N[(N[(N[(2.0 * N[(alpha / beta), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(i / beta), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(alpha / beta), $MachinePrecision] - N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999999980000000011

   1. Initial program 2.7%

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

   3. Simplified 13.2%

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

      1. associate-*r/ 2.7%

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

      2. *-commutative 2.7%

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

      3. fma-undefine 2.7%

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

      4. +-commutative 2.7%

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

      5. fma-define 2.7%

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

      6. associate-+r+ 2.7%

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

      7. associate-/l/ 2.1%

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

      8. +-commutative 2.1%

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

      9. associate-+r+ 2.1%

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

      10. times-frac 13.2%

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

      11. +-commutative 13.2%

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

      12. +-commutative 13.2%

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

      13. fma-define 13.2%

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

   6. Applied egg-rr 13.2%

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

   7. Taylor expanded in beta around 0 13.2%

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

   8. Taylor expanded in alpha around inf 92.8%

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

   if -0.999999980000000011 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 0.900000000000000022

   1. Initial program 99.6%

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

   if 0.900000000000000022 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 28.8%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. fma-define 100.0%

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

      3. associate-+r+ 100.0%

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

      4. associate-+r+ 100.0%

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

      5. associate-*r/ 28.8%

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

      6. *-commutative 28.8%

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

      7. fma-undefine 28.8%

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

      8. +-commutative 28.8%

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

      9. inv-pow 28.8%

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

   6. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. associate-+l+ 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l/ 28.8%

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

      5. associate-*r/ 100.0%

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

      6. +-commutative 100.0%

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

      7. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in beta around inf 97.6%

      \[\leadsto \frac{\frac{1}{\color{blue}{\left(1 + \left(2 \cdot \frac{\alpha}{\beta} + \left(4 \cdot \frac{i}{\beta} + 2 \cdot \frac{1}{\beta}\right)\right)\right) - \left(-1 \cdot \frac{\alpha}{\beta} + \frac{\alpha}{\beta}\right)}} + 1}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999998:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 0.9:\\ \;\;\;\;\frac{1 - \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\alpha - \beta\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(\left(2 \cdot \frac{\alpha}{\beta} + \left(4 \cdot \frac{i}{\beta} + 2 \cdot \frac{1}{\beta}\right)\right) + 1\right) + \left(\frac{\alpha}{\beta} - \frac{\alpha}{\beta}\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t\_0\\ t_2 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_1}\\ \mathbf{if}\;t\_2 \leq -0.99999998:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{elif}\;t\_2 \leq 0.9:\\ \;\;\;\;\frac{1 - \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\alpha - \beta\right)}{t\_0}}{t\_1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (+ 2.0 t_0))
        (t_2 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1)))
   (if (<= t_2 -0.99999998)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (if (<= t_2 0.9)
       (/ (- 1.0 (/ (/ (* (+ alpha beta) (- alpha beta)) t_0) t_1)) 2.0)
       (/ (+ (/ beta (+ beta 2.0)) 1.0) 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 t_2 = (((alpha + beta) * (beta - alpha)) / t_0) / t_1;
	double tmp;
	if (t_2 <= -0.99999998) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else if (t_2 <= 0.9) {
		tmp = (1.0 - ((((alpha + beta) * (alpha - beta)) / t_0) / t_1)) / 2.0;
	} else {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = 2.0d0 + t_0
    t_2 = (((alpha + beta) * (beta - alpha)) / t_0) / t_1
    if (t_2 <= (-0.99999998d0)) then
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    else if (t_2 <= 0.9d0) then
        tmp = (1.0d0 - ((((alpha + beta) * (alpha - beta)) / t_0) / t_1)) / 2.0d0
    else
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double t_2 = (((alpha + beta) * (beta - alpha)) / t_0) / t_1;
	double tmp;
	if (t_2 <= -0.99999998) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else if (t_2 <= 0.9) {
		tmp = (1.0 - ((((alpha + beta) * (alpha - beta)) / t_0) / t_1)) / 2.0;
	} else {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = 2.0 + t_0
	t_2 = (((alpha + beta) * (beta - alpha)) / t_0) / t_1
	tmp = 0
	if t_2 <= -0.99999998:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	elif t_2 <= 0.9:
		tmp = (1.0 - ((((alpha + beta) * (alpha - beta)) / t_0) / t_1)) / 2.0
	else:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 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)
	t_2 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / t_1)
	tmp = 0.0
	if (t_2 <= -0.99999998)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	elseif (t_2 <= 0.9)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(alpha + beta) * Float64(alpha - beta)) / t_0) / t_1)) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = 2.0 + t_0;
	t_2 = (((alpha + beta) * (beta - alpha)) / t_0) / t_1;
	tmp = 0.0;
	if (t_2 <= -0.99999998)
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	elseif (t_2 <= 0.9)
		tmp = (1.0 - ((((alpha + beta) * (alpha - beta)) / t_0) / t_1)) / 2.0;
	else
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -0.99999998], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$2, 0.9], N[(N[(1.0 - N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(alpha - beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999999980000000011

   1. Initial program 2.7%

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

   3. Simplified 13.2%

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

      1. associate-*r/ 2.7%

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

      2. *-commutative 2.7%

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

      3. fma-undefine 2.7%

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

      4. +-commutative 2.7%

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

      5. fma-define 2.7%

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

      6. associate-+r+ 2.7%

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

      7. associate-/l/ 2.1%

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

      8. +-commutative 2.1%

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

      9. associate-+r+ 2.1%

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

      10. times-frac 13.2%

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

      11. +-commutative 13.2%

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

      12. +-commutative 13.2%

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

      13. fma-define 13.2%

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

   6. Applied egg-rr 13.2%

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

   7. Taylor expanded in beta around 0 13.2%

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

   8. Taylor expanded in alpha around inf 92.8%

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

   if -0.999999980000000011 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 0.900000000000000022

   1. Initial program 99.6%

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

   if 0.900000000000000022 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 28.8%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in i around 0 51.9%

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

      1. *-commutative 51.9%

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

   7. Simplified 51.9%

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

   8. Taylor expanded in alpha around 0 85.7%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999998:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 0.9:\\ \;\;\;\;\frac{1 - \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\alpha - \beta\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \frac{1}{\alpha - \beta}\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_1}}{2 + t\_1} \leq -0.99999998:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\left(\left(\frac{\alpha}{\alpha - \beta} + \frac{\beta}{\alpha - \beta}\right) + t\_0\right) + i \cdot \left(t\_0 + 2 \cdot \frac{\left(\alpha + \beta\right) + 2}{\left(\alpha + \beta\right) \cdot \left(\alpha - \beta\right)}\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* 2.0 (/ 1.0 (- alpha beta))))
        (t_1 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ 2.0 t_1))
        -0.99999998)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (+
       (/
        -1.0
        (+
         (+ (+ (/ alpha (- alpha beta)) (/ beta (- alpha beta))) t_0)
         (*
          i
          (+
           t_0
           (*
            2.0
            (/ (+ (+ alpha beta) 2.0) (* (+ alpha beta) (- alpha beta))))))))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = 2.0 * (1.0 / (alpha - beta));
	double t_1 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.99999998) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = ((-1.0 / ((((alpha / (alpha - beta)) + (beta / (alpha - beta))) + t_0) + (i * (t_0 + (2.0 * (((alpha + beta) + 2.0) / ((alpha + beta) * (alpha - beta)))))))) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 * (1.0d0 / (alpha - beta))
    t_1 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0d0 + t_1)) <= (-0.99999998d0)) then
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    else
        tmp = (((-1.0d0) / ((((alpha / (alpha - beta)) + (beta / (alpha - beta))) + t_0) + (i * (t_0 + (2.0d0 * (((alpha + beta) + 2.0d0) / ((alpha + beta) * (alpha - beta)))))))) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = 2.0 * (1.0 / (alpha - beta));
	double t_1 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.99999998) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = ((-1.0 / ((((alpha / (alpha - beta)) + (beta / (alpha - beta))) + t_0) + (i * (t_0 + (2.0 * (((alpha + beta) + 2.0) / ((alpha + beta) * (alpha - beta)))))))) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = 2.0 * (1.0 / (alpha - beta))
	t_1 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.99999998:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	else:
		tmp = ((-1.0 / ((((alpha / (alpha - beta)) + (beta / (alpha - beta))) + t_0) + (i * (t_0 + (2.0 * (((alpha + beta) + 2.0) / ((alpha + beta) * (alpha - beta)))))))) + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(2.0 * Float64(1.0 / Float64(alpha - beta)))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(2.0 + t_1)) <= -0.99999998)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(-1.0 / Float64(Float64(Float64(Float64(alpha / Float64(alpha - beta)) + Float64(beta / Float64(alpha - beta))) + t_0) + Float64(i * Float64(t_0 + Float64(2.0 * Float64(Float64(Float64(alpha + beta) + 2.0) / Float64(Float64(alpha + beta) * Float64(alpha - beta)))))))) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = 2.0 * (1.0 / (alpha - beta));
	t_1 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.99999998)
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	else
		tmp = ((-1.0 / ((((alpha / (alpha - beta)) + (beta / (alpha - beta))) + t_0) + (i * (t_0 + (2.0 * (((alpha + beta) + 2.0) / ((alpha + beta) * (alpha - beta)))))))) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * N[(1.0 / N[(alpha - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], -0.99999998], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(-1.0 / N[(N[(N[(N[(alpha / N[(alpha - beta), $MachinePrecision]), $MachinePrecision] + N[(beta / N[(alpha - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + N[(i * N[(t$95$0 + N[(2.0 * N[(N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] * N[(alpha - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999999980000000011

   1. Initial program 2.7%

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

   3. Simplified 13.2%

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

      1. associate-*r/ 2.7%

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

      2. *-commutative 2.7%

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

      3. fma-undefine 2.7%

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

      4. +-commutative 2.7%

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

      5. fma-define 2.7%

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

      6. associate-+r+ 2.7%

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

      7. associate-/l/ 2.1%

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

      8. +-commutative 2.1%

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

      9. associate-+r+ 2.1%

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

      10. times-frac 13.2%

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

      11. +-commutative 13.2%

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

      12. +-commutative 13.2%

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

      13. fma-define 13.2%

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

   6. Applied egg-rr 13.2%

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

   7. Taylor expanded in beta around 0 13.2%

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

   8. Taylor expanded in alpha around inf 92.8%

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

   if -0.999999980000000011 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 82.2%

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

   3. Simplified 99.7%

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

      1. clear-num 99.6%

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

      2. fma-define 99.6%

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

      3. associate-+r+ 99.6%

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

      4. associate-+r+ 99.6%

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

      5. associate-*r/ 82.2%

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

      6. *-commutative 82.2%

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

      7. fma-undefine 82.2%

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

      8. +-commutative 82.2%

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

      9. inv-pow 82.2%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-+l+ 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*l/ 82.2%

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

      5. associate-*r/ 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in i around 0 98.2%

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

4. Final simplification 97.0%

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

Alternative 9: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.85 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 4.8 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{\alpha}{2 + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.85e-22)
   (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)
   (if (<= alpha 4.8e+126)
     (/
      (+
       (*
        (/ alpha (+ 2.0 (+ alpha (* 2.0 i))))
        (/ (- beta alpha) (+ (+ alpha beta) (* 2.0 i))))
       1.0)
      2.0)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.85e-22) {
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	} else if (alpha <= 4.8e+126) {
		tmp = (((alpha / (2.0 + (alpha + (2.0 * i)))) * ((beta - alpha) / ((alpha + beta) + (2.0 * i)))) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 2.85d-22) then
        tmp = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
    else if (alpha <= 4.8d+126) then
        tmp = (((alpha / (2.0d0 + (alpha + (2.0d0 * i)))) * ((beta - alpha) / ((alpha + beta) + (2.0d0 * i)))) + 1.0d0) / 2.0d0
    else
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.85e-22) {
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	} else if (alpha <= 4.8e+126) {
		tmp = (((alpha / (2.0 + (alpha + (2.0 * i)))) * ((beta - alpha) / ((alpha + beta) + (2.0 * i)))) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.85e-22:
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0
	elif alpha <= 4.8e+126:
		tmp = (((alpha / (2.0 + (alpha + (2.0 * i)))) * ((beta - alpha) / ((alpha + beta) + (2.0 * i)))) + 1.0) / 2.0
	else:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.85e-22)
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0);
	elseif (alpha <= 4.8e+126)
		tmp = Float64(Float64(Float64(Float64(alpha / Float64(2.0 + Float64(alpha + Float64(2.0 * i)))) * Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + Float64(2.0 * i)))) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.85e-22)
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	elseif (alpha <= 4.8e+126)
		tmp = (((alpha / (2.0 + (alpha + (2.0 * i)))) * ((beta - alpha) / ((alpha + beta) + (2.0 * i)))) + 1.0) / 2.0;
	else
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.85e-22], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 4.8e+126], N[(N[(N[(N[(alpha / N[(2.0 + N[(alpha + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 2.8499999999999998e-22

   1. Initial program 83.2%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 93.2%

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

   if 2.8499999999999998e-22 < alpha < 4.80000000000000024e126

   1. Initial program 56.9%

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

   3. Simplified 68.1%

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

      1. associate-*r/ 56.9%

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

      2. *-commutative 56.9%

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

      3. fma-undefine 56.9%

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

      4. +-commutative 56.9%

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

      5. fma-define 56.9%

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

      6. associate-+r+ 56.9%

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

      7. associate-/l/ 56.6%

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

      8. +-commutative 56.6%

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

      9. associate-+r+ 56.6%

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

      10. times-frac 68.1%

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

      11. +-commutative 68.1%

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

      12. +-commutative 68.1%

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

      13. fma-define 68.1%

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

   6. Applied egg-rr 68.1%

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

   7. Taylor expanded in beta around 0 66.2%

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

   8. Taylor expanded in beta around 0 66.2%

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

      1. associate-+r+ 66.2%

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

   10. Simplified 66.2%

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

   if 4.80000000000000024e126 < alpha

   1. Initial program 5.7%

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

   3. Simplified 22.6%

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

      1. associate-*r/ 5.7%

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

      2. *-commutative 5.7%

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

      3. fma-undefine 5.7%

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

      4. +-commutative 5.7%

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

      5. fma-define 5.7%

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

      6. associate-+r+ 5.7%

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

      7. associate-/l/ 4.7%

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

      8. +-commutative 4.7%

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

      9. associate-+r+ 4.7%

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

      10. times-frac 22.7%

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

      11. +-commutative 22.7%

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

      12. +-commutative 22.7%

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

      13. fma-define 22.7%

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

   6. Applied egg-rr 22.7%

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

   7. Taylor expanded in beta around 0 20.9%

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

   8. Taylor expanded in alpha around inf 83.6%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.85 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 4.8 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{\alpha}{2 + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.62 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 1.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{\alpha}{2 + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\alpha + \beta} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.62e-21)
   (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)
   (if (<= alpha 1.5e+61)
     (/
      (+
       (*
        (/ alpha (+ 2.0 (+ alpha (* 2.0 i))))
        (/ (- beta alpha) (+ alpha beta)))
       1.0)
      2.0)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.62e-21) {
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	} else if (alpha <= 1.5e+61) {
		tmp = (((alpha / (2.0 + (alpha + (2.0 * i)))) * ((beta - alpha) / (alpha + beta))) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.62d-21) then
        tmp = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
    else if (alpha <= 1.5d+61) then
        tmp = (((alpha / (2.0d0 + (alpha + (2.0d0 * i)))) * ((beta - alpha) / (alpha + beta))) + 1.0d0) / 2.0d0
    else
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.62e-21) {
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	} else if (alpha <= 1.5e+61) {
		tmp = (((alpha / (2.0 + (alpha + (2.0 * i)))) * ((beta - alpha) / (alpha + beta))) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.62e-21:
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0
	elif alpha <= 1.5e+61:
		tmp = (((alpha / (2.0 + (alpha + (2.0 * i)))) * ((beta - alpha) / (alpha + beta))) + 1.0) / 2.0
	else:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.62e-21)
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0);
	elseif (alpha <= 1.5e+61)
		tmp = Float64(Float64(Float64(Float64(alpha / Float64(2.0 + Float64(alpha + Float64(2.0 * i)))) * Float64(Float64(beta - alpha) / Float64(alpha + beta))) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.62e-21)
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	elseif (alpha <= 1.5e+61)
		tmp = (((alpha / (2.0 + (alpha + (2.0 * i)))) * ((beta - alpha) / (alpha + beta))) + 1.0) / 2.0;
	else
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.62e-21], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 1.5e+61], N[(N[(N[(N[(alpha / N[(2.0 + N[(alpha + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 1.62000000000000003e-21

   1. Initial program 83.2%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 93.2%

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

   if 1.62000000000000003e-21 < alpha < 1.5e61

   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

   3. Simplified 80.5%

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

      1. associate-*r/ 67.8%

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

      2. *-commutative 67.8%

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

      3. fma-undefine 67.8%

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

      4. +-commutative 67.8%

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

      5. fma-define 67.8%

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

      6. associate-+r+ 67.8%

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

      7. associate-/l/ 67.3%

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

      8. +-commutative 67.3%

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

      9. associate-+r+ 67.3%

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

      10. times-frac 80.5%

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

      11. +-commutative 80.5%

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

      12. +-commutative 80.5%

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

      13. fma-define 80.5%

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

   6. Applied egg-rr 80.5%

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

   7. Taylor expanded in beta around 0 76.9%

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

   8. Taylor expanded in i around 0 76.9%

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

   if 1.5e61 < alpha

   1. Initial program 17.4%

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

   3. Simplified 32.1%

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

      1. associate-*r/ 17.4%

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

      2. *-commutative 17.4%

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

      3. fma-undefine 17.4%

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

      4. +-commutative 17.4%

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

      5. fma-define 17.4%

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

      6. associate-+r+ 17.4%

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

      7. associate-/l/ 16.7%

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

      8. +-commutative 16.7%

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

      9. associate-+r+ 16.7%

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

      10. times-frac 32.1%

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

      11. +-commutative 32.1%

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

      12. +-commutative 32.1%

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

      13. fma-define 32.1%

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

   6. Applied egg-rr 32.1%

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

   7. Taylor expanded in beta around 0 30.9%

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

   8. Taylor expanded in alpha around inf 74.2%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.62 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 1.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{\alpha}{2 + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\alpha + \beta} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.3 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.3e+61)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.3e+61) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.3d+61) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.3e+61) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.3e+61:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.3e+61)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.3e+61)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.3e+61], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.29999999999999986e61

   1. Initial program 81.3%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in i around 0 79.5%

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

      1. *-commutative 79.5%

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

   7. Simplified 79.5%

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

   8. Taylor expanded in alpha around 0 88.6%

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

   if 1.29999999999999986e61 < alpha

   1. Initial program 17.4%

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

   3. Simplified 32.1%

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

      1. associate-*r/ 17.4%

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

      2. *-commutative 17.4%

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

      3. fma-undefine 17.4%

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

      4. +-commutative 17.4%

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

      5. fma-define 17.4%

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

      6. associate-+r+ 17.4%

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

      7. associate-/l/ 16.7%

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

      8. +-commutative 16.7%

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

      9. associate-+r+ 16.7%

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

      10. times-frac 32.1%

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

      11. +-commutative 32.1%

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

      12. +-commutative 32.1%

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

      13. fma-define 32.1%

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

   6. Applied egg-rr 32.1%

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

   7. Taylor expanded in beta around 0 30.9%

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

   8. Taylor expanded in alpha around inf 74.2%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.3 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 8 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 8e+151)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 8e+151) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 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 <= 8d+151) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 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 <= 8e+151) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 8e+151:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 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 <= 8e+151)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 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 <= 8e+151)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 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, 8e+151], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $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 < 8.00000000000000014e151

   1. Initial program 76.0%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in i around 0 70.4%

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

      1. *-commutative 70.4%

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

   7. Simplified 70.4%

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

   8. Taylor expanded in alpha around 0 82.1%

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

   if 8.00000000000000014e151 < alpha

   1. Initial program 1.1%

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

   3. Simplified 10.5%

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

   5. Taylor expanded in i around 0 10.5%

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

      1. *-commutative 10.5%

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

   7. Simplified 10.5%

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

   8. Taylor expanded in alpha around inf 64.4%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 8 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.5 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(2 - \frac{4}{\alpha}\right)}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 4.5e+160)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (/ (* 0.5 (- 2.0 (/ 4.0 alpha))) alpha)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.5e+160) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = (0.5 * (2.0 - (4.0 / alpha))) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 4.5d+160) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else
        tmp = (0.5d0 * (2.0d0 - (4.0d0 / alpha))) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.5e+160) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = (0.5 * (2.0 - (4.0 / alpha))) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 4.5e+160:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = (0.5 * (2.0 - (4.0 / alpha))) / alpha
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 4.5e+160)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(0.5 * Float64(2.0 - Float64(4.0 / alpha))) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 4.5e+160)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = (0.5 * (2.0 - (4.0 / alpha))) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 4.5e+160], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.5 * N[(2.0 - N[(4.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.4999999999999998e160

   1. Initial program 75.6%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in i around 0 70.1%

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

      1. *-commutative 70.1%

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

   7. Simplified 70.1%

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

   8. Taylor expanded in alpha around 0 81.7%

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

   if 4.4999999999999998e160 < alpha

   1. Initial program 1.1%

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

   3. Simplified 10.6%

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

   5. Taylor expanded in i around 0 10.6%

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

      1. *-commutative 10.6%

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

   7. Simplified 10.6%

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

   8. Taylor expanded in alpha around inf 53.2%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(2 + 2 \cdot \beta\right) + 0.5 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha}}{\alpha}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 53.2%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\left(2 + 2 \cdot \beta\right) + \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha}\right)}}{\alpha} \]

      2. mul-1-neg 53.2%

         \[\leadsto \frac{0.5 \cdot \left(\left(2 + 2 \cdot \beta\right) + \frac{\color{blue}{\left(-{\left(2 + \beta\right)}^{2}\right)} - \beta \cdot \left(2 + \beta\right)}{\alpha}\right)}{\alpha} \]

   10. Simplified 53.2%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\left(2 + 2 \cdot \beta\right) + \frac{\left(-{\left(2 + \beta\right)}^{2}\right) - \beta \cdot \left(2 + \beta\right)}{\alpha}\right)}{\alpha}} \]

   11. Taylor expanded in beta around 0 49.8%

       \[\leadsto \frac{\color{blue}{0.5 \cdot \left(2 - 4 \cdot \frac{1}{\alpha}\right)}}{\alpha} \]
   12. Step-by-step derivation

       1. associate-*r/ 49.8%

          \[\leadsto \frac{0.5 \cdot \left(2 - \color{blue}{\frac{4 \cdot 1}{\alpha}}\right)}{\alpha} \]

       2. metadata-eval 49.8%

          \[\leadsto \frac{0.5 \cdot \left(2 - \frac{\color{blue}{4}}{\alpha}\right)}{\alpha} \]

   13. Simplified 49.8%

       \[\leadsto \frac{\color{blue}{0.5 \cdot \left(2 - \frac{4}{\alpha}\right)}}{\alpha} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.5 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(2 - \frac{4}{\alpha}\right)}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.75 \cdot 10^{+45}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 2.75e+45) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.75e+45) {
		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 <= 2.75d+45) 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 <= 2.75e+45) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.75e+45:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.75e+45)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.75e+45)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 2.75e+45], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2.75e45

   1. Initial program 75.9%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in i around inf 73.2%

      \[\leadsto \color{blue}{0.5} \]

   if 2.75e45 < beta

   1. Initial program 25.4%

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

   3. Simplified 90.6%

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

   5. Taylor expanded in beta around inf 70.2%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.75 \cdot 10^{+45}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.1% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 64.8%

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

3. Simplified 70.8%

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

5. Taylor expanded in i around inf 64.2%

   \[\leadsto \color{blue}{0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024117 
(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))