Octave 3.8, jcobi/2

Percentage Accurate: 63.5% → 97.5%

Time: 16.4s

Alternatives: 9

Speedup: 9.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 63.5% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.9:\\ \;\;\;\;\frac{\frac{\left(\beta + 2\right) + \left(\beta - i \cdot -4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}\\ \end{array} \end{array} \]

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)) -0.9)
     (/ (/ (+ (+ beta 2.0) (- beta (* i -4.0))) alpha) 2.0)
     (/
      (+
       (*
        (/ (- beta alpha) (+ (+ alpha beta) (fma 2.0 i 2.0)))
        (/ (+ alpha beta) (fma 2.0 i (+ alpha beta))))
       1.0)
      2.0))))

C

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

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.9)
		tmp = Float64(Float64(Float64(Float64(beta + 2.0) + Float64(beta - Float64(i * -4.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + fma(2.0, i, 2.0))) * Float64(Float64(alpha + beta) / fma(2.0, i, Float64(alpha + beta)))) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.900000000000000022

   1. Initial program 2.1%

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

   2. Taylor expanded in alpha around inf 9.6%

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

   3. Taylor expanded in alpha around -inf 83.4%

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

   4. Taylor expanded in alpha around inf 83.4%

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

      1. neg-mul-1 83.4%

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

      2. associate--r+ 83.4%

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

      3. cancel-sign-sub-inv 83.4%

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

      4. metadata-eval 83.4%

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

      5. +-commutative 83.4%

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

      6. unsub-neg 83.4%

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

      7. *-commutative 83.4%

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

   6. Simplified 83.4%

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

   if -0.900000000000000022 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 84.4%

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

      1. associate-/l/ 83.9%

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

      2. *-commutative 83.9%

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

      3. times-frac 99.7%

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

      4. associate-+l+ 99.7%

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

      5. fma-def 99.7%

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

      6. +-commutative 99.7%

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

      7. fma-def 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9:\\ \;\;\;\;\frac{\frac{\left(\beta + 2\right) + \left(\beta - i \cdot -4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}\\ \end{array} \]

Alternative 2: 97.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 3.8%

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

   2. Taylor expanded in alpha around inf 11.1%

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

   3. Taylor expanded in alpha around -inf 82.4%

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

   4. Taylor expanded in alpha around inf 82.4%

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

      1. neg-mul-1 82.4%

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

      2. associate--r+ 82.4%

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

      3. cancel-sign-sub-inv 82.4%

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

      4. metadata-eval 82.4%

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

      5. +-commutative 82.4%

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

      6. unsub-neg 82.4%

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

      7. *-commutative 82.4%

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

   6. Simplified 82.4%

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

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

   1. Initial program 84.3%

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

      1. associate-/l/ 83.8%

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

      2. *-commutative 83.8%

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

      3. times-frac 99.7%

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

      4. associate-+l+ 99.7%

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

      5. fma-def 99.7%

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

      6. +-commutative 99.7%

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

      7. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in alpha around 0 99.4%

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

4. Final simplification 95.6%

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

Alternative 3: 96.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 3.8%

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

   2. Taylor expanded in alpha around inf 11.1%

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

   3. Taylor expanded in alpha around -inf 82.4%

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

   4. Taylor expanded in alpha around inf 82.4%

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

      1. neg-mul-1 82.4%

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

      2. associate--r+ 82.4%

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

      3. cancel-sign-sub-inv 82.4%

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

      4. metadata-eval 82.4%

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

      5. +-commutative 82.4%

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

      6. unsub-neg 82.4%

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

      7. *-commutative 82.4%

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

   6. Simplified 82.4%

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

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

   1. Initial program 84.3%

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

      1. associate-/l/ 83.8%

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

      2. *-commutative 83.8%

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

      3. times-frac 99.7%

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

      4. associate-+l+ 99.7%

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

      5. fma-def 99.7%

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

      6. +-commutative 99.7%

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

      7. fma-def 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. fma-udef 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. fma-udef 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in alpha around 0 98.8%

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

   7. Taylor expanded in alpha around 0 83.1%

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

      1. unpow2 83.1%

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

      2. *-commutative 83.1%

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

      3. times-frac 98.8%

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

      4. associate-+r+ 98.8%

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

      5. +-commutative 98.8%

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

      6. fma-udef 98.8%

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

   9. Simplified 98.8%

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

4. Final simplification 95.1%

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

Alternative 4: 96.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 3.8%

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

   2. Taylor expanded in alpha around inf 11.1%

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

   3. Taylor expanded in alpha around -inf 82.4%

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

   4. Taylor expanded in alpha around inf 82.4%

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

      1. neg-mul-1 82.4%

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

      2. associate--r+ 82.4%

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

      3. cancel-sign-sub-inv 82.4%

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

      4. metadata-eval 82.4%

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

      5. +-commutative 82.4%

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

      6. unsub-neg 82.4%

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

      7. *-commutative 82.4%

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

   6. Simplified 82.4%

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

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

   1. Initial program 84.3%

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

   2. Taylor expanded in beta around inf 98.2%

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

4. Final simplification 94.6%

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

Alternative 5: 82.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + 2\right) + \left(\beta - i \cdot -4\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.5e+131)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ (+ beta 2.0) (- beta (* i -4.0))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.5e+131) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (((beta + 2.0) + (beta - (i * -4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 2.5d+131) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (((beta + 2.0d0) + (beta - (i * (-4.0d0)))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.5e+131) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (((beta + 2.0) + (beta - (i * -4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.5e+131:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (((beta + 2.0) + (beta - (i * -4.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.5e+131)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(beta + 2.0) + Float64(beta - Float64(i * -4.0))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.5e+131)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (((beta + 2.0) + (beta - (i * -4.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.5e+131], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta + 2.0), $MachinePrecision] + N[(beta - N[(i * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.49999999999999998e131

   1. Initial program 80.6%

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

      1. associate-/l/ 80.2%

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

      2. *-commutative 80.2%

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

      3. times-frac 94.1%

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

      4. associate-+l+ 94.1%

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

      5. fma-def 94.1%

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

      6. +-commutative 94.1%

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

      7. fma-def 94.1%

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

   3. Simplified 94.1%

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

      1. associate-*r/ 94.1%

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

      2. +-commutative 94.1%

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

      3. +-commutative 94.1%

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

      4. fma-udef 94.1%

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

      5. +-commutative 94.1%

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

      6. associate-+r+ 94.1%

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

      7. +-commutative 94.1%

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

      8. fma-udef 94.1%

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

   5. Applied egg-rr 94.1%

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

   6. Taylor expanded in i around 0 81.6%

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

      1. +-commutative 81.6%

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

   8. Simplified 81.6%

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

   9. Taylor expanded in alpha around 0 85.1%

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

   if 2.49999999999999998e131 < alpha

   1. Initial program 1.3%

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

   2. Taylor expanded in alpha around inf 14.4%

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

   3. Taylor expanded in alpha around -inf 73.9%

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

   4. Taylor expanded in alpha around inf 73.9%

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

      1. neg-mul-1 73.9%

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

      2. associate--r+ 73.9%

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

      3. cancel-sign-sub-inv 73.9%

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

      4. metadata-eval 73.9%

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

      5. +-commutative 73.9%

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

      6. unsub-neg 73.9%

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

      7. *-commutative 73.9%

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

   6. Simplified 73.9%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + 2\right) + \left(\beta - i \cdot -4\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 6: 78.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.7 \cdot 10^{+181}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 4.7e+181)
   (/ (+ 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 <= 4.7e+181) {
		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 <= 4.7d+181) 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 <= 4.7e+181) {
		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 <= 4.7e+181:
		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 <= 4.7e+181)
		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 <= 4.7e+181)
		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, 4.7e+181], 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 < 4.70000000000000027e181

   1. Initial program 77.4%

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

      1. associate-/l/ 77.0%

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

      2. *-commutative 77.0%

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

      3. times-frac 92.1%

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

      4. associate-+l+ 92.1%

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

      5. fma-def 92.1%

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

      6. +-commutative 92.1%

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

      7. fma-def 92.1%

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

   3. Simplified 92.1%

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

      1. associate-*r/ 92.1%

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

      2. +-commutative 92.1%

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

      3. +-commutative 92.1%

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

      4. fma-udef 92.1%

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

      5. +-commutative 92.1%

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

      6. associate-+r+ 92.1%

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

      7. +-commutative 92.1%

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

      8. fma-udef 92.1%

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

   5. Applied egg-rr 92.1%

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

   6. Taylor expanded in i around 0 79.3%

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

      1. +-commutative 79.3%

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

   8. Simplified 79.3%

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

   9. Taylor expanded in alpha around 0 83.6%

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

   if 4.70000000000000027e181 < alpha

   1. Initial program 1.0%

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

      1. associate-/l/ 0.0%

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

      2. *-commutative 0.0%

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

      3. times-frac 27.5%

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

      4. associate-+l+ 27.5%

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

      5. fma-def 27.5%

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

      6. +-commutative 27.5%

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

      7. fma-def 27.5%

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

   3. Simplified 27.5%

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

   4. Taylor expanded in beta around 0 0.0%

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

      1. mul-1-neg 0.0%

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

      2. unsub-neg 0.0%

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

      3. unpow2 0.0%

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

      4. associate-+r+ 0.0%

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

      5. +-commutative 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in alpha around inf 60.1%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.7 \cdot 10^{+181}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 72.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{1 + \frac{\beta}{\beta + 2}}{2} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0))

C

double code(double alpha, double beta, double i) {
	return (1.0 + (beta / (beta + 2.0))) / 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
    code = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
end function

Java

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

Python

def code(alpha, beta, i):
	return (1.0 + (beta / (beta + 2.0))) / 2.0

Julia

function code(alpha, beta, i)
	return Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0)
end

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 66.1%

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

   1. associate-/l/ 65.5%

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

   2. *-commutative 65.5%

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

   3. times-frac 82.5%

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

   4. associate-+l+ 82.5%

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

   5. fma-def 82.5%

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

   6. +-commutative 82.5%

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

   7. fma-def 82.5%

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

3. Simplified 82.5%

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

   1. associate-*r/ 82.5%

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

   2. +-commutative 82.5%

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

   3. +-commutative 82.5%

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

   4. fma-udef 82.5%

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

   5. +-commutative 82.5%

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

   6. associate-+r+ 82.5%

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

   7. +-commutative 82.5%

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

   8. fma-udef 82.5%

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

5. Applied egg-rr 82.5%

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

6. Taylor expanded in i around 0 68.6%

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

   1. +-commutative 68.6%

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

8. Simplified 68.6%

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

9. Taylor expanded in alpha around 0 73.9%

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

10. Final simplification 73.9%

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

Alternative 8: 73.2% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+39}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 2e+39) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2e+39) {
		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 <= 2d+39) 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 <= 2e+39) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 2e+39:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2e+39)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2e+39)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 2e+39], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 1.99999999999999988e39

   1. Initial program 74.9%

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

      1. associate-/l/ 74.7%

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

      2. *-commutative 74.7%

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

      3. times-frac 79.9%

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

      4. associate-+l+ 79.9%

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

      5. fma-def 79.9%

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

      6. +-commutative 79.9%

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

      7. fma-def 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in i around inf 76.0%

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

   if 1.99999999999999988e39 < beta

   1. Initial program 43.9%

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

      1. associate-/l/ 42.5%

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

      2. *-commutative 42.5%

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

      3. times-frac 89.1%

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

      4. associate-+l+ 89.1%

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

      5. fma-def 89.1%

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

      6. +-commutative 89.1%

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

      7. fma-def 89.1%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in beta around inf 72.5%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+39}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 62.3% 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 66.1%

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

   1. associate-/l/ 65.5%

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

   2. *-commutative 65.5%

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

   3. times-frac 82.5%

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

   4. associate-+l+ 82.5%

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

   5. fma-def 82.5%

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

   6. +-commutative 82.5%

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

   7. fma-def 82.5%

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

3. Simplified 82.5%

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

4. Taylor expanded in i around inf 63.4%

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

5. Final simplification 63.4%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023201 
(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))