Octave 3.8, jcobi/2

Percentage Accurate: 63.8% → 97.5%

Time: 19.1s

Alternatives: 13

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: 63.8% 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: 97.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \frac{\alpha + \beta}{\beta - \alpha}\\ t_2 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_2}}{2 + t\_2} \leq -0.5:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(\frac{2}{\alpha} + 4 \cdot \frac{i}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{t\_0 \cdot t\_1 + i \cdot \left(2 \cdot \left(t\_1 + \frac{t\_0}{\beta - \alpha}\right) + 4 \cdot \frac{i}{\beta - \alpha}\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0)))
        (t_1 (/ (+ alpha beta) (- beta alpha)))
        (t_2 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_2) (+ 2.0 t_2)) -0.5)
     (/ (+ (* 2.0 (/ beta alpha)) (+ (/ 2.0 alpha) (* 4.0 (/ i alpha)))) 2.0)
     (/
      (+
       1.0
       (/
        (+ alpha beta)
        (+
         (* t_0 t_1)
         (*
          i
          (+
           (* 2.0 (+ t_1 (/ t_0 (- beta alpha))))
           (* 4.0 (/ i (- beta alpha))))))))
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (alpha + 2.0);
	double t_1 = (alpha + beta) / (beta - alpha);
	double t_2 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_2) / (2.0 + t_2)) <= -0.5) {
		tmp = ((2.0 * (beta / alpha)) + ((2.0 / alpha) + (4.0 * (i / alpha)))) / 2.0;
	} else {
		tmp = (1.0 + ((alpha + beta) / ((t_0 * t_1) + (i * ((2.0 * (t_1 + (t_0 / (beta - alpha)))) + (4.0 * (i / (beta - 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = beta + (alpha + 2.0d0)
    t_1 = (alpha + beta) / (beta - alpha)
    t_2 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_2) / (2.0d0 + t_2)) <= (-0.5d0)) then
        tmp = ((2.0d0 * (beta / alpha)) + ((2.0d0 / alpha) + (4.0d0 * (i / alpha)))) / 2.0d0
    else
        tmp = (1.0d0 + ((alpha + beta) / ((t_0 * t_1) + (i * ((2.0d0 * (t_1 + (t_0 / (beta - alpha)))) + (4.0d0 * (i / (beta - alpha)))))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = beta + (alpha + 2.0);
	double t_1 = (alpha + beta) / (beta - alpha);
	double t_2 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_2) / (2.0 + t_2)) <= -0.5) {
		tmp = ((2.0 * (beta / alpha)) + ((2.0 / alpha) + (4.0 * (i / alpha)))) / 2.0;
	} else {
		tmp = (1.0 + ((alpha + beta) / ((t_0 * t_1) + (i * ((2.0 * (t_1 + (t_0 / (beta - alpha)))) + (4.0 * (i / (beta - alpha)))))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = beta + (alpha + 2.0)
	t_1 = (alpha + beta) / (beta - alpha)
	t_2 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_2) / (2.0 + t_2)) <= -0.5:
		tmp = ((2.0 * (beta / alpha)) + ((2.0 / alpha) + (4.0 * (i / alpha)))) / 2.0
	else:
		tmp = (1.0 + ((alpha + beta) / ((t_0 * t_1) + (i * ((2.0 * (t_1 + (t_0 / (beta - alpha)))) + (4.0 * (i / (beta - alpha)))))))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	t_1 = Float64(Float64(alpha + beta) / Float64(beta - alpha))
	t_2 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_2) / Float64(2.0 + t_2)) <= -0.5)
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(Float64(2.0 / alpha) + Float64(4.0 * Float64(i / alpha)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(alpha + beta) / Float64(Float64(t_0 * t_1) + Float64(i * Float64(Float64(2.0 * Float64(t_1 + Float64(t_0 / Float64(beta - alpha)))) + Float64(4.0 * Float64(i / Float64(beta - alpha)))))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (alpha + 2.0);
	t_1 = (alpha + beta) / (beta - alpha);
	t_2 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_2) / (2.0 + t_2)) <= -0.5)
		tmp = ((2.0 * (beta / alpha)) + ((2.0 / alpha) + (4.0 * (i / alpha)))) / 2.0;
	else
		tmp = (1.0 + ((alpha + beta) / ((t_0 * t_1) + (i * ((2.0 * (t_1 + (t_0 / (beta - alpha)))) + (4.0 * (i / (beta - alpha)))))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / alpha), $MachinePrecision] + N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(alpha + beta), $MachinePrecision] / N[(N[(t$95$0 * t$95$1), $MachinePrecision] + N[(i * N[(N[(2.0 * N[(t$95$1 + N[(t$95$0 / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(i / N[(beta - alpha), $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 1.8%

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

   3. Taylor expanded in alpha around inf 96.6%

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

      1. cancel-sign-sub-inv 96.6%

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

      2. distribute-rgt1-in 96.6%

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

      3. metadata-eval 96.6%

         \[\leadsto \frac{\frac{\color{blue}{0} \cdot \beta + \left(--1\right) \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      4. metadata-eval 96.6%

         \[\leadsto \frac{\frac{0 \cdot \beta + \color{blue}{1} \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      5. *-commutative 96.6%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\color{blue}{\beta \cdot 2} + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      6. *-commutative 96.6%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + \color{blue}{i \cdot 4}\right)\right)}{\alpha}}{2} \]

   5. Simplified 96.6%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in beta around 0 96.6%

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

      1. +-commutative 96.6%

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

      2. associate-*r/ 96.6%

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

      3. metadata-eval 96.6%

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

   8. Simplified 96.6%

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

   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 79.3%

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

      1. associate-/l/ 78.7%

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

      2. +-commutative 78.7%

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

      3. +-commutative 78.7%

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

      4. associate-+r+ 78.7%

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

      5. fma-undefine 78.7%

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

      6. associate-+l+ 78.7%

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

      7. +-commutative 78.7%

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

      8. +-commutative 78.7%

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

      9. +-commutative 78.7%

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

      10. associate-+r+ 78.7%

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

      11. fma-undefine 78.7%

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

      12. +-commutative 78.7%

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

      13. frac-times 100.0%

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

      14. clear-num 100.0%

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

      15. frac-times 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in i around 0 84.3%

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

      1. +-commutative 84.3%

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

      2. associate-/l* 100.0%

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

      3. associate-+r+ 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+r+ 100.0%

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

      6. distribute-lft-out 100.0%

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

      7. associate-+r+ 100.0%

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

      8. +-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.2%

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

Alternative 2: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_1}}{2 + t\_1} \leq -0.5:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(\frac{2}{\alpha} + 4 \cdot \frac{i}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\left(\alpha + \left(2 + t\_0\right)\right) \cdot \frac{\alpha + t\_0}{\beta - \alpha}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i))) (t_1 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ 2.0 t_1)) -0.5)
     (/ (+ (* 2.0 (/ beta alpha)) (+ (/ 2.0 alpha) (* 4.0 (/ i alpha)))) 2.0)
     (/
      (+
       1.0
       (/
        (+ alpha beta)
        (* (+ alpha (+ 2.0 t_0)) (/ (+ alpha t_0) (- beta alpha)))))
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.5) {
		tmp = ((2.0 * (beta / alpha)) + ((2.0 / alpha) + (4.0 * (i / alpha)))) / 2.0;
	} else {
		tmp = (1.0 + ((alpha + beta) / ((alpha + (2.0 + t_0)) * ((alpha + t_0) / (beta - 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = beta + (2.0d0 * i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0d0 + t_1)) <= (-0.5d0)) then
        tmp = ((2.0d0 * (beta / alpha)) + ((2.0d0 / alpha) + (4.0d0 * (i / alpha)))) / 2.0d0
    else
        tmp = (1.0d0 + ((alpha + beta) / ((alpha + (2.0d0 + t_0)) * ((alpha + t_0) / (beta - alpha))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.5) {
		tmp = ((2.0 * (beta / alpha)) + ((2.0 / alpha) + (4.0 * (i / alpha)))) / 2.0;
	} else {
		tmp = (1.0 + ((alpha + beta) / ((alpha + (2.0 + t_0)) * ((alpha + t_0) / (beta - alpha))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = beta + (2.0 * i)
	t_1 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.5:
		tmp = ((2.0 * (beta / alpha)) + ((2.0 / alpha) + (4.0 * (i / alpha)))) / 2.0
	else:
		tmp = (1.0 + ((alpha + beta) / ((alpha + (2.0 + t_0)) * ((alpha + t_0) / (beta - alpha))))) / 2.0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (2.0 * i);
	t_1 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.5)
		tmp = ((2.0 * (beta / alpha)) + ((2.0 / alpha) + (4.0 * (i / alpha)))) / 2.0;
	else
		tmp = (1.0 + ((alpha + beta) / ((alpha + (2.0 + t_0)) * ((alpha + t_0) / (beta - alpha))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #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 1.8%

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

   3. Taylor expanded in alpha around inf 96.6%

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

      1. cancel-sign-sub-inv 96.6%

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

      2. distribute-rgt1-in 96.6%

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

      3. metadata-eval 96.6%

         \[\leadsto \frac{\frac{\color{blue}{0} \cdot \beta + \left(--1\right) \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      4. metadata-eval 96.6%

         \[\leadsto \frac{\frac{0 \cdot \beta + \color{blue}{1} \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      5. *-commutative 96.6%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\color{blue}{\beta \cdot 2} + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      6. *-commutative 96.6%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + \color{blue}{i \cdot 4}\right)\right)}{\alpha}}{2} \]

   5. Simplified 96.6%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in beta around 0 96.6%

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

      1. +-commutative 96.6%

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

      2. associate-*r/ 96.6%

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

      3. metadata-eval 96.6%

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

   8. Simplified 96.6%

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

   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 79.3%

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

      1. associate-/l/ 78.7%

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

      2. +-commutative 78.7%

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

      3. +-commutative 78.7%

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

      4. associate-+r+ 78.7%

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

      5. fma-undefine 78.7%

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

      6. associate-+l+ 78.7%

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

      7. +-commutative 78.7%

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

      8. +-commutative 78.7%

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

      9. +-commutative 78.7%

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

      10. associate-+r+ 78.7%

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

      11. fma-undefine 78.7%

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

      12. +-commutative 78.7%

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

      13. frac-times 100.0%

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

      14. clear-num 100.0%

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

      15. frac-times 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 99.2%

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

Alternative 3: 96.8% accurate, 0.6× speedup

Math

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

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)
     (/ (+ (* 2.0 (/ beta alpha)) (+ (/ 2.0 alpha) (* 4.0 (/ i alpha)))) 2.0)
     (/
      (+
       1.0
       (* (/ beta (+ beta (* 2.0 i))) (/ beta (+ (* 2.0 i) (+ beta 2.0)))))
      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 = ((2.0 * (beta / alpha)) + ((2.0 / alpha) + (4.0 * (i / alpha)))) / 2.0;
	} else {
		tmp = (1.0 + ((beta / (beta + (2.0 * i))) * (beta / ((2.0 * i) + (beta + 2.0))))) / 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 = ((2.0d0 * (beta / alpha)) + ((2.0d0 / alpha) + (4.0d0 * (i / alpha)))) / 2.0d0
    else
        tmp = (1.0d0 + ((beta / (beta + (2.0d0 * i))) * (beta / ((2.0d0 * i) + (beta + 2.0d0))))) / 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 = ((2.0 * (beta / alpha)) + ((2.0 / alpha) + (4.0 * (i / alpha)))) / 2.0;
	} else {
		tmp = (1.0 + ((beta / (beta + (2.0 * i))) * (beta / ((2.0 * i) + (beta + 2.0))))) / 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 = ((2.0 * (beta / alpha)) + ((2.0 / alpha) + (4.0 * (i / alpha)))) / 2.0
	else:
		tmp = (1.0 + ((beta / (beta + (2.0 * i))) * (beta / ((2.0 * i) + (beta + 2.0))))) / 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(Float64(2.0 * Float64(beta / alpha)) + Float64(Float64(2.0 / alpha) + Float64(4.0 * Float64(i / alpha)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta / Float64(beta + Float64(2.0 * i))) * Float64(beta / Float64(Float64(2.0 * i) + Float64(beta + 2.0))))) / 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 = ((2.0 * (beta / alpha)) + ((2.0 / alpha) + (4.0 * (i / alpha)))) / 2.0;
	else
		tmp = (1.0 + ((beta / (beta + (2.0 * i))) * (beta / ((2.0 * i) + (beta + 2.0))))) / 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[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / alpha), $MachinePrecision] + N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta / N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta / N[(N[(2.0 * i), $MachinePrecision] + N[(beta + 2.0), $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 1.8%

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

   3. Taylor expanded in alpha around inf 96.6%

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

      1. cancel-sign-sub-inv 96.6%

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

      2. distribute-rgt1-in 96.6%

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

      3. metadata-eval 96.6%

         \[\leadsto \frac{\frac{\color{blue}{0} \cdot \beta + \left(--1\right) \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      4. metadata-eval 96.6%

         \[\leadsto \frac{\frac{0 \cdot \beta + \color{blue}{1} \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      5. *-commutative 96.6%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\color{blue}{\beta \cdot 2} + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      6. *-commutative 96.6%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + \color{blue}{i \cdot 4}\right)\right)}{\alpha}}{2} \]

   5. Simplified 96.6%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in beta around 0 96.6%

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

      1. +-commutative 96.6%

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

      2. associate-*r/ 96.6%

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

      3. metadata-eval 96.6%

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

   8. Simplified 96.6%

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

   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 79.3%

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

   3. Taylor expanded in alpha around 0 77.4%

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

      1. unpow2 77.4%

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

      2. *-commutative 77.4%

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

      3. times-frac 98.5%

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

      4. associate-+r+ 98.5%

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

   5. Simplified 98.5%

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

4. Final simplification 98.1%

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

Alternative 4: 78.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 10^{+29}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 1.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 1.35 \cdot 10^{+166}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1e+29)
   (/ (+ 1.0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))) 2.0)
   (if (<= alpha 1.5e+77)
     (/ (+ (* 2.0 (/ beta alpha)) (/ 2.0 alpha)) 2.0)
     (if (<= alpha 1.35e+166)
       (/ (+ 1.0 (/ beta (+ beta (+ alpha 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 <= 1e+29) {
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	} else if (alpha <= 1.5e+77) {
		tmp = ((2.0 * (beta / alpha)) + (2.0 / alpha)) / 2.0;
	} else if (alpha <= 1.35e+166) {
		tmp = (1.0 + (beta / (beta + (alpha + 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 <= 1d+29) then
        tmp = (1.0d0 + ((beta - alpha) / ((alpha + beta) + 2.0d0))) / 2.0d0
    else if (alpha <= 1.5d+77) then
        tmp = ((2.0d0 * (beta / alpha)) + (2.0d0 / alpha)) / 2.0d0
    else if (alpha <= 1.35d+166) then
        tmp = (1.0d0 + (beta / (beta + (alpha + 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 <= 1e+29) {
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	} else if (alpha <= 1.5e+77) {
		tmp = ((2.0 * (beta / alpha)) + (2.0 / alpha)) / 2.0;
	} else if (alpha <= 1.35e+166) {
		tmp = (1.0 + (beta / (beta + (alpha + 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 <= 1e+29:
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0
	elif alpha <= 1.5e+77:
		tmp = ((2.0 * (beta / alpha)) + (2.0 / alpha)) / 2.0
	elif alpha <= 1.35e+166:
		tmp = (1.0 + (beta / (beta + (alpha + 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 <= 1e+29)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0))) / 2.0);
	elseif (alpha <= 1.5e+77)
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(2.0 / alpha)) / 2.0);
	elseif (alpha <= 1.35e+166)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + Float64(alpha + 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 <= 1e+29)
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	elseif (alpha <= 1.5e+77)
		tmp = ((2.0 * (beta / alpha)) + (2.0 / alpha)) / 2.0;
	elseif (alpha <= 1.35e+166)
		tmp = (1.0 + (beta / (beta + (alpha + 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, 1e+29], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 1.5e+77], N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 1.35e+166], N[(N[(1.0 + N[(beta / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if alpha < 9.99999999999999914e28

   1. Initial program 82.6%

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

   3. Taylor expanded in i around 0 93.1%

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

      1. +-commutative 93.1%

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

   5. Simplified 93.1%

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

   if 9.99999999999999914e28 < alpha < 1.4999999999999999e77

   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. Add Preprocessing

   3. Taylor expanded in alpha around inf 71.8%

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

      1. cancel-sign-sub-inv 71.8%

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

      2. distribute-rgt1-in 71.8%

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

      3. metadata-eval 71.8%

         \[\leadsto \frac{\frac{\color{blue}{0} \cdot \beta + \left(--1\right) \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      4. metadata-eval 71.8%

         \[\leadsto \frac{\frac{0 \cdot \beta + \color{blue}{1} \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      5. *-commutative 71.8%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\color{blue}{\beta \cdot 2} + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      6. *-commutative 71.8%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + \color{blue}{i \cdot 4}\right)\right)}{\alpha}}{2} \]

   5. Simplified 71.8%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in beta around 0 72.0%

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

      1. +-commutative 72.0%

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

      2. associate-*r/ 72.0%

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

      3. metadata-eval 72.0%

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

   8. Simplified 72.0%

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

   9. Taylor expanded in i around 0 72.2%

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

       1. associate-*r/ 72.2%

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

       2. metadata-eval 72.2%

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

       3. +-commutative 72.2%

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

   11. Simplified 72.2%

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

   if 1.4999999999999999e77 < alpha < 1.35000000000000006e166

   1. Initial program 29.8%

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

   3. Taylor expanded in beta around inf 70.0%

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

   4. Taylor expanded in i around 0 66.3%

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

      1. associate-+r+ 66.3%

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

   6. Simplified 66.3%

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

   if 1.35000000000000006e166 < 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. Add Preprocessing

   3. Taylor expanded in alpha around inf 91.7%

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

      1. cancel-sign-sub-inv 91.7%

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

      2. distribute-rgt1-in 91.7%

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

      3. metadata-eval 91.7%

         \[\leadsto \frac{\frac{\color{blue}{0} \cdot \beta + \left(--1\right) \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      4. metadata-eval 91.7%

         \[\leadsto \frac{\frac{0 \cdot \beta + \color{blue}{1} \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      5. *-commutative 91.7%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\color{blue}{\beta \cdot 2} + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      6. *-commutative 91.7%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + \color{blue}{i \cdot 4}\right)\right)}{\alpha}}{2} \]

   5. Simplified 91.7%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in beta around 0 67.4%

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

      1. *-commutative 67.4%

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

   8. Simplified 67.4%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 10^{+29}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 1.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 1.35 \cdot 10^{+166}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 9 \cdot 10^{+165}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(\frac{2}{\alpha} + 4 \cdot \frac{i}{\alpha}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 9e+165)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (/ (+ (* 2.0 (/ beta alpha)) (+ (/ 2.0 alpha) (* 4.0 (/ i alpha)))) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 9e+165) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 * (beta / alpha)) + ((2.0 / alpha) + (4.0 * (i / 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 <= 9d+165) then
        tmp = (1.0d0 + (beta / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = ((2.0d0 * (beta / alpha)) + ((2.0d0 / alpha) + (4.0d0 * (i / alpha)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 9e+165) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 * (beta / alpha)) + ((2.0 / alpha) + (4.0 * (i / alpha)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 9e+165:
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = ((2.0 * (beta / alpha)) + ((2.0 / alpha) + (4.0 * (i / alpha)))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 9e+165)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(Float64(2.0 / alpha) + Float64(4.0 * Float64(i / alpha)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 9e+165)
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = ((2.0 * (beta / alpha)) + ((2.0 / alpha) + (4.0 * (i / alpha)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 9e+165], N[(N[(1.0 + N[(beta / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / alpha), $MachinePrecision] + N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 8.9999999999999993e165

   1. Initial program 74.7%

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

   3. Taylor expanded in beta around inf 91.4%

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

   if 8.9999999999999993e165 < 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. Add Preprocessing

   3. Taylor expanded in alpha around inf 91.7%

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

      1. cancel-sign-sub-inv 91.7%

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

      2. distribute-rgt1-in 91.7%

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

      3. metadata-eval 91.7%

         \[\leadsto \frac{\frac{\color{blue}{0} \cdot \beta + \left(--1\right) \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      4. metadata-eval 91.7%

         \[\leadsto \frac{\frac{0 \cdot \beta + \color{blue}{1} \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      5. *-commutative 91.7%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\color{blue}{\beta \cdot 2} + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      6. *-commutative 91.7%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + \color{blue}{i \cdot 4}\right)\right)}{\alpha}}{2} \]

   5. Simplified 91.7%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in beta around 0 91.7%

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

      1. +-commutative 91.7%

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

      2. associate-*r/ 91.7%

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

      3. metadata-eval 91.7%

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

   8. Simplified 91.7%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 9 \cdot 10^{+165}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(\frac{2}{\alpha} + 4 \cdot \frac{i}{\alpha}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 8.8 \cdot 10^{+165}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 8.8e+165)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 8.8e+165) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 8.8d+165) then
        tmp = (1.0d0 + (beta / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 8.8e+165) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 8.8e+165:
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 8.8e+165)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 8.8e+165)
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 8.8e+165], N[(N[(1.0 + N[(beta / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 8.7999999999999996e165

   1. Initial program 74.7%

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

   3. Taylor expanded in beta around inf 91.4%

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

   if 8.7999999999999996e165 < 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. Add Preprocessing

   3. Taylor expanded in alpha around inf 91.7%

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

      1. cancel-sign-sub-inv 91.7%

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

      2. distribute-rgt1-in 91.7%

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

      3. metadata-eval 91.7%

         \[\leadsto \frac{\frac{\color{blue}{0} \cdot \beta + \left(--1\right) \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      4. metadata-eval 91.7%

         \[\leadsto \frac{\frac{0 \cdot \beta + \color{blue}{1} \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      5. *-commutative 91.7%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\color{blue}{\beta \cdot 2} + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      6. *-commutative 91.7%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + \color{blue}{i \cdot 4}\right)\right)}{\alpha}}{2} \]

   5. Simplified 91.7%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in beta around 0 67.4%

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

      1. *-commutative 67.4%

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

   8. Simplified 67.4%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 8.8 \cdot 10^{+165}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 54000000000000:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 2.65 \cdot 10^{+26}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 7.8 \cdot 10^{+36}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 54000000000000.0)
   0.5
   (if (<= beta 2.65e+26)
     (/ (* 2.0 (/ beta alpha)) 2.0)
     (if (<= beta 7.8e+36) 0.5 1.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 54000000000000.0) {
		tmp = 0.5;
	} else if (beta <= 2.65e+26) {
		tmp = (2.0 * (beta / alpha)) / 2.0;
	} else if (beta <= 7.8e+36) {
		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 <= 54000000000000.0d0) then
        tmp = 0.5d0
    else if (beta <= 2.65d+26) then
        tmp = (2.0d0 * (beta / alpha)) / 2.0d0
    else if (beta <= 7.8d+36) 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 <= 54000000000000.0) {
		tmp = 0.5;
	} else if (beta <= 2.65e+26) {
		tmp = (2.0 * (beta / alpha)) / 2.0;
	} else if (beta <= 7.8e+36) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 54000000000000.0:
		tmp = 0.5
	elif beta <= 2.65e+26:
		tmp = (2.0 * (beta / alpha)) / 2.0
	elif beta <= 7.8e+36:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 54000000000000.0)
		tmp = 0.5;
	elseif (beta <= 2.65e+26)
		tmp = Float64(Float64(2.0 * Float64(beta / alpha)) / 2.0);
	elseif (beta <= 7.8e+36)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 54000000000000.0)
		tmp = 0.5;
	elseif (beta <= 2.65e+26)
		tmp = (2.0 * (beta / alpha)) / 2.0;
	elseif (beta <= 7.8e+36)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 54000000000000.0], 0.5, If[LessEqual[beta, 2.65e+26], N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 7.8e+36], 0.5, 1.0]]]

Derivation

1. Split input into 3 regimes

2. if beta < 5.4e13 or 2.64999999999999984e26 < beta < 7.80000000000000042e36

   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. Add Preprocessing

   3. Taylor expanded in i around inf 75.6%

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

   if 5.4e13 < beta < 2.64999999999999984e26

   1. Initial program 18.5%

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

   3. Taylor expanded in alpha around inf 83.5%

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

      1. cancel-sign-sub-inv 83.5%

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

      2. distribute-rgt1-in 83.5%

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

      3. metadata-eval 83.5%

         \[\leadsto \frac{\frac{\color{blue}{0} \cdot \beta + \left(--1\right) \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      4. metadata-eval 83.5%

         \[\leadsto \frac{\frac{0 \cdot \beta + \color{blue}{1} \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      5. *-commutative 83.5%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\color{blue}{\beta \cdot 2} + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      6. *-commutative 83.5%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + \color{blue}{i \cdot 4}\right)\right)}{\alpha}}{2} \]

   5. Simplified 83.5%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in beta around inf 81.2%

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

   if 7.80000000000000042e36 < beta

   1. Initial program 33.5%

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

   3. Taylor expanded in beta around inf 75.3%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 54000000000000:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 2.65 \cdot 10^{+26}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 7.8 \cdot 10^{+36}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.3 \cdot 10^{+166}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.3e+166)
   (/ (+ 1.0 (/ beta (+ beta (+ alpha 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 <= 1.3e+166) {
		tmp = (1.0 + (beta / (beta + (alpha + 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 <= 1.3d+166) then
        tmp = (1.0d0 + (beta / (beta + (alpha + 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 <= 1.3e+166) {
		tmp = (1.0 + (beta / (beta + (alpha + 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 <= 1.3e+166:
		tmp = (1.0 + (beta / (beta + (alpha + 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 <= 1.3e+166)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + Float64(alpha + 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 <= 1.3e+166)
		tmp = (1.0 + (beta / (beta + (alpha + 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, 1.3e+166], N[(N[(1.0 + N[(beta / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.3e166

   1. Initial program 74.7%

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

   3. Taylor expanded in beta around inf 91.4%

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

   4. Taylor expanded in i around 0 87.3%

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

      1. associate-+r+ 87.3%

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

   6. Simplified 87.3%

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

   if 1.3e166 < 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. Add Preprocessing

   3. Taylor expanded in alpha around inf 91.7%

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

      1. cancel-sign-sub-inv 91.7%

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

      2. distribute-rgt1-in 91.7%

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

      3. metadata-eval 91.7%

         \[\leadsto \frac{\frac{\color{blue}{0} \cdot \beta + \left(--1\right) \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      4. metadata-eval 91.7%

         \[\leadsto \frac{\frac{0 \cdot \beta + \color{blue}{1} \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      5. *-commutative 91.7%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\color{blue}{\beta \cdot 2} + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      6. *-commutative 91.7%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + \color{blue}{i \cdot 4}\right)\right)}{\alpha}}{2} \]

   5. Simplified 91.7%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in beta around 0 67.4%

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

      1. *-commutative 67.4%

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

   8. Simplified 67.4%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.3 \cdot 10^{+166}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 8.8 \cdot 10^{+165}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 8.8e+165)
   (/ (+ 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 <= 8.8e+165) {
		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 <= 8.8d+165) 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 <= 8.8e+165) {
		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 <= 8.8e+165:
		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 <= 8.8e+165)
		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 <= 8.8e+165)
		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, 8.8e+165], 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 < 8.7999999999999996e165

   1. Initial program 74.7%

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

      1. associate-/l/ 74.1%

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

      2. +-commutative 74.1%

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

      3. +-commutative 74.1%

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

      4. associate-+r+ 74.1%

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

      5. fma-undefine 74.1%

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

      6. associate-+l+ 74.1%

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

      7. +-commutative 74.1%

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

      8. +-commutative 74.1%

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

      9. +-commutative 74.1%

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

      10. associate-+r+ 74.1%

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

      11. fma-undefine 74.1%

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

      12. +-commutative 74.1%

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

      13. frac-times 93.1%

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

      14. clear-num 93.1%

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

      15. frac-times 93.1%

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

   4. Applied egg-rr 93.1%

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

   5. Taylor expanded in i around 0 70.8%

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

      1. associate-/l* 83.7%

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

      2. associate-+r+ 83.7%

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

      3. +-commutative 83.7%

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

   7. Simplified 83.7%

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

   8. Taylor expanded in alpha around 0 86.5%

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

      1. +-commutative 86.5%

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

   10. Simplified 86.5%

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

   if 8.7999999999999996e165 < 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. Add Preprocessing

   3. Taylor expanded in alpha around inf 91.7%

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

      1. cancel-sign-sub-inv 91.7%

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

      2. distribute-rgt1-in 91.7%

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

      3. metadata-eval 91.7%

         \[\leadsto \frac{\frac{\color{blue}{0} \cdot \beta + \left(--1\right) \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      4. metadata-eval 91.7%

         \[\leadsto \frac{\frac{0 \cdot \beta + \color{blue}{1} \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      5. *-commutative 91.7%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\color{blue}{\beta \cdot 2} + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      6. *-commutative 91.7%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + \color{blue}{i \cdot 4}\right)\right)}{\alpha}}{2} \]

   5. Simplified 91.7%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in beta around 0 67.4%

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

      1. *-commutative 67.4%

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

   8. Simplified 67.4%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 8.8 \cdot 10^{+165}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3.1 \cdot 10^{+181}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 3.1e+181)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.1e+181) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 3.1d+181) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.1e+181) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 3.1e+181:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 3.1e+181)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 3.1e+181)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 3.1e+181], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3.09999999999999989e181

   1. Initial program 72.6%

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

      1. associate-/l/ 72.0%

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

      2. +-commutative 72.0%

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

      3. +-commutative 72.0%

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

      4. associate-+r+ 72.0%

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

      5. fma-undefine 72.0%

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

      6. associate-+l+ 72.0%

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

      7. +-commutative 72.0%

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

      8. +-commutative 72.0%

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

      9. +-commutative 72.0%

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

      10. associate-+r+ 72.0%

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

      11. fma-undefine 72.0%

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

      12. +-commutative 72.0%

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

      13. frac-times 91.5%

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

      14. clear-num 91.5%

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

      15. frac-times 91.5%

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

   4. Applied egg-rr 91.5%

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

   5. Taylor expanded in i around 0 69.9%

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

      1. associate-/l* 81.5%

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

      2. associate-+r+ 81.5%

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

      3. +-commutative 81.5%

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

   7. Simplified 81.5%

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

   8. Taylor expanded in alpha around 0 84.7%

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

      1. +-commutative 84.7%

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

   10. Simplified 84.7%

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

   if 3.09999999999999989e181 < 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. Add Preprocessing

   3. Taylor expanded in alpha around inf 95.1%

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

      1. cancel-sign-sub-inv 95.1%

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

      2. distribute-rgt1-in 95.1%

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

      3. metadata-eval 95.1%

         \[\leadsto \frac{\frac{\color{blue}{0} \cdot \beta + \left(--1\right) \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      4. metadata-eval 95.1%

         \[\leadsto \frac{\frac{0 \cdot \beta + \color{blue}{1} \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      5. *-commutative 95.1%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\color{blue}{\beta \cdot 2} + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      6. *-commutative 95.1%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + \color{blue}{i \cdot 4}\right)\right)}{\alpha}}{2} \]

   5. Simplified 95.1%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in i around 0 60.2%

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

      1. *-commutative 60.2%

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

   8. Simplified 60.2%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.1 \cdot 10^{+181}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.85 \cdot 10^{+167}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.85e+167)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (* 4.0 (/ i alpha)) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.85e+167) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (4.0 * (i / alpha)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.85d+167) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (4.0d0 * (i / alpha)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.85e+167) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (4.0 * (i / alpha)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.85e+167:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (4.0 * (i / alpha)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.85e+167)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(4.0 * Float64(i / alpha)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.85e+167)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (4.0 * (i / alpha)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.85e+167], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.85e167

   1. Initial program 74.7%

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

      1. associate-/l/ 74.1%

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

      2. +-commutative 74.1%

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

      3. +-commutative 74.1%

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

      4. associate-+r+ 74.1%

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

      5. fma-undefine 74.1%

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

      6. associate-+l+ 74.1%

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

      7. +-commutative 74.1%

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

      8. +-commutative 74.1%

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

      9. +-commutative 74.1%

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

      10. associate-+r+ 74.1%

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

      11. fma-undefine 74.1%

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

      12. +-commutative 74.1%

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

      13. frac-times 93.1%

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

      14. clear-num 93.1%

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

      15. frac-times 93.1%

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

   4. Applied egg-rr 93.1%

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

   5. Taylor expanded in i around 0 70.8%

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

      1. associate-/l* 83.7%

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

      2. associate-+r+ 83.7%

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

      3. +-commutative 83.7%

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

   7. Simplified 83.7%

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

   8. Taylor expanded in alpha around 0 86.5%

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

      1. +-commutative 86.5%

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

   10. Simplified 86.5%

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

   if 1.85e167 < 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. Add Preprocessing

   3. Taylor expanded in alpha around inf 91.7%

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

      1. cancel-sign-sub-inv 91.7%

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

      2. distribute-rgt1-in 91.7%

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

      3. metadata-eval 91.7%

         \[\leadsto \frac{\frac{\color{blue}{0} \cdot \beta + \left(--1\right) \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      4. metadata-eval 91.7%

         \[\leadsto \frac{\frac{0 \cdot \beta + \color{blue}{1} \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      5. *-commutative 91.7%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\color{blue}{\beta \cdot 2} + 4 \cdot i\right)\right)}{\alpha}}{2} \]

      6. *-commutative 91.7%

         \[\leadsto \frac{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + \color{blue}{i \cdot 4}\right)\right)}{\alpha}}{2} \]

   5. Simplified 91.7%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot \beta + 1 \cdot \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in i around inf 42.0%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.85 \cdot 10^{+167}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+37}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 2.2e+37) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.2e+37) {
		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 <= 2.2d+37) 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 <= 2.2e+37) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.2e+37:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.2e+37)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.2e+37)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 2.2e+37], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2000000000000001e37

   1. Initial program 74.0%

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

   3. Taylor expanded in i around inf 73.3%

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

   if 2.2000000000000001e37 < beta

   1. Initial program 33.5%

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

   3. Taylor expanded in beta around inf 75.3%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+37}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.8% 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 61.5%

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

3. Taylor expanded in i around inf 58.4%

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

4. Final simplification 58.4%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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))