Octave 3.8, jcobi/2

Percentage Accurate: 63.9% → 97.6%

Time: 17.3s

Alternatives: 8

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: 63.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.5:\\ \;\;\;\;0.5 \cdot \frac{2 + i \cdot 4}{\alpha} + \frac{\beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{2 + \left(\alpha + \left(\beta + i \cdot \left(2 + \left(2 \cdot \frac{\left(\alpha + \beta\right) + 2}{\alpha + \beta} + 4 \cdot \frac{i}{\alpha + \beta}\right)\right)\right)\right)} + 1}{2}\\ \end{array} \end{array} \]

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)) -0.5)
     (+ (* 0.5 (/ (+ 2.0 (* i 4.0)) alpha)) (/ beta alpha))
     (/
      (+
       (/
        (- beta alpha)
        (+
         2.0
         (+
          alpha
          (+
           beta
           (*
            i
            (+
             2.0
             (+
              (* 2.0 (/ (+ (+ alpha beta) 2.0) (+ alpha beta)))
              (* 4.0 (/ i (+ alpha 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)) <= -0.5) {
		tmp = (0.5 * ((2.0 + (i * 4.0)) / alpha)) + (beta / alpha);
	} else {
		tmp = (((beta - alpha) / (2.0 + (alpha + (beta + (i * (2.0 + ((2.0 * (((alpha + beta) + 2.0) / (alpha + beta))) + (4.0 * (i / (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) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)) <= (-0.5d0)) then
        tmp = (0.5d0 * ((2.0d0 + (i * 4.0d0)) / alpha)) + (beta / alpha)
    else
        tmp = (((beta - alpha) / (2.0d0 + (alpha + (beta + (i * (2.0d0 + ((2.0d0 * (((alpha + beta) + 2.0d0) / (alpha + beta))) + (4.0d0 * (i / (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 tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.5) {
		tmp = (0.5 * ((2.0 + (i * 4.0)) / alpha)) + (beta / alpha);
	} else {
		tmp = (((beta - alpha) / (2.0 + (alpha + (beta + (i * (2.0 + ((2.0 * (((alpha + beta) + 2.0) / (alpha + beta))) + (4.0 * (i / (alpha + beta)))))))))) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.5:
		tmp = (0.5 * ((2.0 + (i * 4.0)) / alpha)) + (beta / alpha)
	else:
		tmp = (((beta - alpha) / (2.0 + (alpha + (beta + (i * (2.0 + ((2.0 * (((alpha + beta) + 2.0) / (alpha + beta))) + (4.0 * (i / (alpha + 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)) <= -0.5)
		tmp = Float64(Float64(0.5 * Float64(Float64(2.0 + Float64(i * 4.0)) / alpha)) + Float64(beta / alpha));
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(2.0 + Float64(alpha + Float64(beta + Float64(i * Float64(2.0 + Float64(Float64(2.0 * Float64(Float64(Float64(alpha + beta) + 2.0) / Float64(alpha + beta))) + Float64(4.0 * Float64(i / 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);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.5)
		tmp = (0.5 * ((2.0 + (i * 4.0)) / alpha)) + (beta / alpha);
	else
		tmp = (((beta - alpha) / (2.0 + (alpha + (beta + (i * (2.0 + ((2.0 * (((alpha + beta) + 2.0) / (alpha + beta))) + (4.0 * (i / (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]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(0.5 * N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(alpha + N[(beta + N[(i * N[(2.0 + N[(N[(2.0 * N[(N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision] / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(i / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $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))) < -0.5

   1. Initial program 3.6%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. 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 alpha around inf 94.4%

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

   6. Taylor expanded in beta around 0 94.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 4 \cdot i}{\alpha} + \frac{\beta}{\alpha}} \]

   if -0.5 < (/.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 81.4%

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

   3. Simplified 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%

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

      2. inv-pow 100.0%

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

      3. fma-undefine 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-+r+ 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-define 100.0%

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

      9. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. fma-undefine 100.0%

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

      3. +-commutative 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

      7. +-commutative 100.0%

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

   8. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. un-div-inv 100.0%

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

      3. +-commutative 100.0%

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

      4. +-commutative 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. *-lft-identity 100.0%

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

       2. associate-/l/ 100.0%

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

       3. +-commutative 100.0%

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

       4. +-commutative 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in i around 0 100.0%

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

4. Final simplification 98.8%

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

Alternative 2: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.5:\\ \;\;\;\;0.5 \cdot \frac{2 + i \cdot 4}{\alpha} + \frac{\beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\beta + i \cdot \left(2 + \left(2 \cdot \frac{\beta + 2}{\beta} + 4 \cdot \frac{i}{\beta}\right)\right)\right)}}{2}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.5

   1. Initial program 3.6%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. 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 alpha around inf 94.4%

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

   6. Taylor expanded in beta around 0 94.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 4 \cdot i}{\alpha} + \frac{\beta}{\alpha}} \]

   if -0.5 < (/.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 81.4%

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

   3. Simplified 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%

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

      2. inv-pow 100.0%

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

      3. fma-undefine 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-+r+ 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-define 100.0%

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

      9. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. fma-undefine 100.0%

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

      3. +-commutative 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

      7. +-commutative 100.0%

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

   8. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. un-div-inv 100.0%

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

      3. +-commutative 100.0%

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

      4. +-commutative 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. *-lft-identity 100.0%

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

       2. associate-/l/ 100.0%

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

       3. +-commutative 100.0%

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

       4. +-commutative 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in i around 0 100.0%

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

   14. Taylor expanded in alpha around 0 99.7%

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

4. Final simplification 98.6%

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

Alternative 3: 89.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.55 \cdot 10^{+100}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\alpha + \left(\beta + i \cdot 4\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{2 + i \cdot 4}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.55e+100)
   (/ (+ 1.0 (/ (- beta alpha) (+ 2.0 (+ alpha (+ beta (* i 4.0)))))) 2.0)
   (+ (* 0.5 (/ (+ 2.0 (* i 4.0)) alpha)) (/ beta alpha))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.55e+100) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + (alpha + (beta + (i * 4.0)))))) / 2.0;
	} else {
		tmp = (0.5 * ((2.0 + (i * 4.0)) / alpha)) + (beta / 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 <= 1.55d+100) then
        tmp = (1.0d0 + ((beta - alpha) / (2.0d0 + (alpha + (beta + (i * 4.0d0)))))) / 2.0d0
    else
        tmp = (0.5d0 * ((2.0d0 + (i * 4.0d0)) / alpha)) + (beta / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.55e+100) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + (alpha + (beta + (i * 4.0)))))) / 2.0;
	} else {
		tmp = (0.5 * ((2.0 + (i * 4.0)) / alpha)) + (beta / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.55e+100:
		tmp = (1.0 + ((beta - alpha) / (2.0 + (alpha + (beta + (i * 4.0)))))) / 2.0
	else:
		tmp = (0.5 * ((2.0 + (i * 4.0)) / alpha)) + (beta / alpha)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.55e+100)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(2.0 + Float64(alpha + Float64(beta + Float64(i * 4.0)))))) / 2.0);
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(2.0 + Float64(i * 4.0)) / alpha)) + Float64(beta / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.55e+100)
		tmp = (1.0 + ((beta - alpha) / (2.0 + (alpha + (beta + (i * 4.0)))))) / 2.0;
	else
		tmp = (0.5 * ((2.0 + (i * 4.0)) / alpha)) + (beta / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.55e+100], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(alpha + N[(beta + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.5 * N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.55000000000000003e100

   1. Initial program 79.9%

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

   3. Simplified 96.4%

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

      1. clear-num 96.4%

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

      2. inv-pow 96.4%

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

      3. fma-undefine 96.4%

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

      4. +-commutative 96.4%

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

      5. +-commutative 96.4%

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

      6. associate-+r+ 96.4%

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

      7. +-commutative 96.4%

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

      8. fma-define 96.4%

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

      9. +-commutative 96.4%

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

   6. Applied egg-rr 96.4%

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

      1. unpow-1 96.4%

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

      2. fma-undefine 96.4%

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

      3. +-commutative 96.4%

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

      4. +-commutative 96.4%

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

      5. +-commutative 96.4%

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

      6. fma-undefine 96.4%

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

      7. +-commutative 96.4%

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

   8. Simplified 96.4%

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

      1. *-un-lft-identity 96.4%

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

      2. un-div-inv 96.4%

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

      3. +-commutative 96.4%

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

      4. +-commutative 96.4%

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

   10. Applied egg-rr 96.4%

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

       1. *-lft-identity 96.4%

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

       2. associate-/l/ 96.4%

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

       3. +-commutative 96.4%

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

       4. +-commutative 96.4%

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

   12. Simplified 96.4%

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

   13. Taylor expanded in i around 0 96.4%

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

   14. Taylor expanded in alpha around inf 95.9%

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

   if 1.55000000000000003e100 < alpha

   1. Initial program 6.6%

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

   3. Simplified 17.7%

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

   5. Taylor expanded in alpha around inf 83.6%

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

   6. Taylor expanded in beta around 0 83.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 4 \cdot i}{\alpha} + \frac{\beta}{\alpha}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.55 \cdot 10^{+100}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\alpha + \left(\beta + i \cdot 4\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{2 + i \cdot 4}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.02 \cdot 10^{+100}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{2 + i \cdot 4}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.02e+100)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (+ (* 0.5 (/ (+ 2.0 (* i 4.0)) alpha)) (/ beta alpha))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.02e+100) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (0.5 * ((2.0 + (i * 4.0)) / alpha)) + (beta / 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 <= 1.02d+100) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (0.5d0 * ((2.0d0 + (i * 4.0d0)) / alpha)) + (beta / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.02e+100) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (0.5 * ((2.0 + (i * 4.0)) / alpha)) + (beta / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.02e+100:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (0.5 * ((2.0 + (i * 4.0)) / alpha)) + (beta / alpha)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.02e+100)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(2.0 + Float64(i * 4.0)) / alpha)) + Float64(beta / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.02e+100)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (0.5 * ((2.0 + (i * 4.0)) / alpha)) + (beta / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.02e+100], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.5 * N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.0199999999999999e100

   1. Initial program 79.9%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in i around 0 85.1%

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

   6. Taylor expanded in alpha around 0 89.3%

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

   if 1.0199999999999999e100 < alpha

   1. Initial program 6.6%

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

   3. Simplified 17.7%

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

   5. Taylor expanded in alpha around inf 83.6%

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

   6. Taylor expanded in beta around 0 83.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 4 \cdot i}{\alpha} + \frac{\beta}{\alpha}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.02 \cdot 10^{+100}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{2 + i \cdot 4}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.5 \cdot 10^{+100}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.5e+100)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.5e+100) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 2.5d+100) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.5e+100) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.5e+100:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.5e+100)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.5e+100)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.5e+100], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.4999999999999999e100

   1. Initial program 79.9%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in i around 0 85.1%

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

   6. Taylor expanded in alpha around 0 89.3%

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

   if 2.4999999999999999e100 < alpha

   1. Initial program 6.6%

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

   3. Simplified 17.7%

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

   5. Taylor expanded in alpha around inf 83.6%

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

   6. Taylor expanded in beta around 0 64.2%

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

      1. *-commutative 64.2%

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

   8. Simplified 64.2%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.5 \cdot 10^{+100}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3.9 \cdot 10^{+100}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 3.9e+100)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (+ (/ beta alpha) (/ 1.0 alpha))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.9e+100) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta / alpha) + (1.0 / 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 <= 3.9d+100) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (beta / alpha) + (1.0d0 / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.9e+100) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta / alpha) + (1.0 / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 3.9e+100:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (beta / alpha) + (1.0 / alpha)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 3.9e+100)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(beta / alpha) + Float64(1.0 / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 3.9e+100)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (beta / alpha) + (1.0 / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 3.9e+100], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta / alpha), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3.9e100

   1. Initial program 79.9%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in i around 0 85.1%

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

   6. Taylor expanded in alpha around 0 89.3%

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

   if 3.9e100 < alpha

   1. Initial program 6.6%

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

   3. Simplified 22.8%

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

   5. Taylor expanded in i around 0 8.6%

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

   6. Taylor expanded in alpha around inf 55.4%

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

   7. Taylor expanded in beta around 0 55.4%

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

      1. +-commutative 55.4%

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

   9. Simplified 55.4%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.9 \cdot 10^{+100}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.35 \cdot 10^{+63}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 2.35e+63) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.35e+63) {
		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 <= 2.35d+63) 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 <= 2.35e+63) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.35e+63:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.35e+63)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.35e+63)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 2.35e+63], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2.3500000000000001e63

   1. Initial program 77.7%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in i around inf 76.1%

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

   if 2.3500000000000001e63 < beta

   1. Initial program 32.8%

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

   3. Simplified 87.5%

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

      1. clear-num 87.6%

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

      2. inv-pow 87.6%

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

      3. fma-undefine 87.6%

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

      4. +-commutative 87.6%

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

      5. +-commutative 87.6%

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

      6. associate-+r+ 87.6%

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

      7. +-commutative 87.6%

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

      8. fma-define 87.6%

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

      9. +-commutative 87.6%

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

   6. Applied egg-rr 87.6%

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

      1. unpow-1 87.6%

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

      2. fma-undefine 87.6%

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

      3. +-commutative 87.6%

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

      4. +-commutative 87.6%

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

      5. +-commutative 87.6%

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

      6. fma-undefine 87.6%

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

      7. +-commutative 87.6%

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

   8. Simplified 87.6%

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

      1. *-un-lft-identity 87.6%

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

      2. un-div-inv 87.6%

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

      3. +-commutative 87.6%

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

      4. +-commutative 87.6%

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

   10. Applied egg-rr 87.6%

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

       1. *-lft-identity 87.6%

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

       2. associate-/l/ 87.6%

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

       3. +-commutative 87.6%

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

       4. +-commutative 87.6%

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

   12. Simplified 87.6%

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

   13. Taylor expanded in beta around inf 73.4%

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

Alternative 8: 62.1% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 65.3%

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

3. Simplified 70.4%

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

5. Taylor expanded in i around inf 62.8%

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

Reproduce

herbie shell --seed 2024166 
(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))