Octave 3.8, jcobi/2

Percentage Accurate: 63.0% → 97.7%

Time: 21.2s

Alternatives: 12

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: 63.0% 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: 97.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{2 + t_1} \leq -0.999998:\\ \;\;\;\;\frac{\frac{t_0 + \left(2 + t_0\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}{\frac{\beta - \alpha}{\frac{\alpha + \mathsf{fma}\left(2, i, \beta\right)}{\alpha + \beta}}}}}{2}\\ \end{array} \end{array} \]

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)) -0.999998)
     (/ (/ (+ t_0 (+ 2.0 t_0)) alpha) 2.0)
     (/
      (+
       1.0
       (/
        1.0
        (/
         (+ beta (+ alpha (fma 2.0 i 2.0)))
         (/ (- beta alpha) (/ (+ alpha (fma 2.0 i beta)) (+ alpha beta))))))
      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)) <= -0.999998) {
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (1.0 / ((beta + (alpha + fma(2.0, i, 2.0))) / ((beta - alpha) / ((alpha + fma(2.0, i, beta)) / (alpha + beta)))))) / 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)) <= -0.999998)
		tmp = Float64(Float64(Float64(t_0 + Float64(2.0 + t_0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(1.0 / Float64(Float64(beta + Float64(alpha + fma(2.0, i, 2.0))) / Float64(Float64(beta - alpha) / Float64(Float64(alpha + fma(2.0, i, beta)) / Float64(alpha + beta)))))) / 2.0);
	end
	return 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], -0.999998], N[(N[(N[(t$95$0 + N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(1.0 / N[(N[(beta + N[(alpha + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + beta), $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 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.999998000000000054

   1. Initial program 2.6%

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

   3. Simplified 18.2%

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

   4. Taylor expanded in alpha around inf 88.4%

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

   if -0.999998000000000054 < (/.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 84.6%

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

      1. *-commutative 84.6%

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

      2. *-un-lft-identity 84.6%

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

      3. times-frac 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-udef 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-def 99.9%

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

      6. /-rgt-identity 99.9%

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

      7. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

      2. associate-+l+ 99.9%

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

      3. associate-*r/ 84.6%

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

      4. associate-/l* 99.9%

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

      5. +-commutative 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 97.2%

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

Alternative 2: 97.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{2 + t_1} \leq -0.999998:\\ \;\;\;\;\frac{\frac{t_0 + \left(2 + t_0\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{\alpha + \mathsf{fma}\left(2, i, \beta\right)}{\beta - \alpha} \cdot \frac{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}{\alpha + \beta}}}{2}\\ \end{array} \end{array} \]

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)) -0.999998)
     (/ (/ (+ t_0 (+ 2.0 t_0)) alpha) 2.0)
     (/
      (+
       1.0
       (/
        1.0
        (*
         (/ (+ alpha (fma 2.0 i beta)) (- beta alpha))
         (/ (+ alpha (+ beta (fma 2.0 i 2.0))) (+ alpha beta)))))
      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)) <= -0.999998) {
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (1.0 / (((alpha + fma(2.0, i, beta)) / (beta - alpha)) * ((alpha + (beta + fma(2.0, i, 2.0))) / (alpha + beta))))) / 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)) <= -0.999998)
		tmp = Float64(Float64(Float64(t_0 + Float64(2.0 + t_0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(1.0 / Float64(Float64(Float64(alpha + fma(2.0, i, beta)) / Float64(beta - alpha)) * Float64(Float64(alpha + Float64(beta + fma(2.0, i, 2.0))) / Float64(alpha + beta))))) / 2.0);
	end
	return 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], -0.999998], N[(N[(N[(t$95$0 + N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(1.0 / N[(N[(N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + beta), $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)) < -0.999998000000000054

   1. Initial program 2.6%

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

   3. Simplified 18.2%

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

   4. Taylor expanded in alpha around inf 88.4%

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

   if -0.999998000000000054 < (/.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 84.6%

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

      1. associate-/l/ 84.1%

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

      2. associate-+l+ 84.1%

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

      3. associate-+l+ 84.1%

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

   3. Simplified 84.1%

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

      1. associate-+r+ 84.1%

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

      2. associate-/l/ 84.6%

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

      3. associate-+r+ 84.6%

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

      4. clear-num 84.6%

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

      5. inv-pow 84.6%

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

   5. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

      2. associate-/r/ 99.9%

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

      3. *-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 97.2%

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

Alternative 3: 97.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := 2 + t_1\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{t_2} \leq -0.999998:\\ \;\;\;\;\frac{\frac{t_0 + \left(2 + t_0\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}{t_2}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (+ 2.0 t_1)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) t_2) -0.999998)
     (/ (/ (+ t_0 (+ 2.0 t_0)) alpha) 2.0)
     (/
      (+
       1.0
       (/
        (* (- beta alpha) (/ (+ alpha beta) (+ alpha (fma 2.0 i beta))))
        t_2))
      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 t_2 = 2.0 + t_1;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= -0.999998) {
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((beta - alpha) * ((alpha + beta) / (alpha + fma(2.0, i, beta)))) / t_2)) / 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))
	t_2 = Float64(2.0 + t_1)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / t_2) <= -0.999998)
		tmp = Float64(Float64(Float64(t_0 + Float64(2.0 + t_0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(beta - alpha) * Float64(Float64(alpha + beta) / Float64(alpha + fma(2.0, i, beta)))) / t_2)) / 2.0);
	end
	return 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]}, Block[{t$95$2 = N[(2.0 + t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], -0.999998], N[(N[(N[(t$95$0 + N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $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)) < -0.999998000000000054

   1. Initial program 2.6%

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

   3. Simplified 18.2%

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

   4. Taylor expanded in alpha around inf 88.4%

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

   if -0.999998000000000054 < (/.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 84.6%

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

      1. *-commutative 84.6%

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

      2. *-un-lft-identity 84.6%

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

      3. times-frac 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-udef 99.8%

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

   3. Applied egg-rr 99.8%

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

4. Final simplification 97.2%

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

Alternative 4: 97.1% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (+ 2.0 t_1)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) t_2) -0.5)
     (/ (/ (+ t_0 (+ 2.0 t_0)) alpha) 2.0)
     (/ (+ 1.0 (/ (* (- beta alpha) (/ beta t_0)) t_2)) 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 t_2 = 2.0 + t_1;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= -0.5) {
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((beta - alpha) * (beta / t_0)) / t_2)) / 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 + (2.0d0 * i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = 2.0d0 + t_1
    if (((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= (-0.5d0)) then
        tmp = ((t_0 + (2.0d0 + t_0)) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + (((beta - alpha) * (beta / t_0)) / t_2)) / 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 t_2 = 2.0 + t_1;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= -0.5) {
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((beta - alpha) * (beta / t_0)) / t_2)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = beta + (2.0 * i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = 2.0 + t_1
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= -0.5:
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0
	else:
		tmp = (1.0 + (((beta - alpha) * (beta / t_0)) / t_2)) / 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))
	t_2 = Float64(2.0 + t_1)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / t_2) <= -0.5)
		tmp = Float64(Float64(Float64(t_0 + Float64(2.0 + t_0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(beta - alpha) * Float64(beta / t_0)) / t_2)) / 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);
	t_2 = 2.0 + t_1;
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= -0.5)
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	else
		tmp = (1.0 + (((beta - alpha) * (beta / t_0)) / t_2)) / 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]}, Block[{t$95$2 = N[(2.0 + t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], -0.5], N[(N[(N[(t$95$0 + N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$2), $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. Step-by-step derivation

   3. Simplified 19.1%

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

   4. Taylor expanded in alpha around inf 87.7%

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

   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 84.6%

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

      1. *-commutative 84.6%

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

      2. *-un-lft-identity 84.6%

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

      3. times-frac 100.0%

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

      4. associate-+r+ 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-udef 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in alpha around 0 100.0%

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

4. Final simplification 97.1%

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

Alternative 5: 88.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ \mathbf{if}\;\alpha \leq 4.2 \cdot 10^{+55}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0 + \left(2 + t_0\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i))))
   (if (<= alpha 4.2e+55)
     (/ (+ 1.0 (/ beta (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
     (/ (/ (+ t_0 (+ 2.0 t_0)) alpha) 2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double tmp;
	if (alpha <= 4.2e+55) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((t_0 + (2.0 + t_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) :: t_0
    real(8) :: tmp
    t_0 = beta + (2.0d0 * i)
    if (alpha <= 4.2d+55) then
        tmp = (1.0d0 + (beta / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = ((t_0 + (2.0d0 + t_0)) / 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 tmp;
	if (alpha <= 4.2e+55) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = beta + (2.0 * i)
	tmp = 0
	if alpha <= 4.2e+55:
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	tmp = 0.0
	if (alpha <= 4.2e+55)
		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(t_0 + Float64(2.0 + t_0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (2.0 * i);
	tmp = 0.0;
	if (alpha <= 4.2e+55)
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = ((t_0 + (2.0 + t_0)) / 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]}, If[LessEqual[alpha, 4.2e+55], 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[(t$95$0 + N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.2000000000000001e55

   1. Initial program 86.0%

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

   2. Taylor expanded in beta around inf 96.8%

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

   if 4.2000000000000001e55 < alpha

   1. Initial program 10.7%

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

   3. Simplified 35.3%

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

   4. Taylor expanded in alpha around inf 71.3%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.2 \cdot 10^{+55}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + 2 \cdot i\right) + \left(2 + \left(\beta + 2 \cdot i\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 6: 85.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + 2 \cdot i\right) - i \cdot -2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 5.5e+55)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (/ (/ (- (+ 2.0 (* 2.0 i)) (* i -2.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5.5e+55) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = (((2.0 + (2.0 * i)) - (i * -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 <= 5.5d+55) then
        tmp = (1.0d0 + (beta / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = (((2.0d0 + (2.0d0 * i)) - (i * (-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 <= 5.5e+55) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = (((2.0 + (2.0 * i)) - (i * -2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 5.5e+55:
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = (((2.0 + (2.0 * i)) - (i * -2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 5.5e+55)
		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(Float64(2.0 + Float64(2.0 * i)) - Float64(i * -2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 5.5e+55)
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = (((2.0 + (2.0 * i)) - (i * -2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 5.5e+55], 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[(N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] - N[(i * -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 5.5000000000000004e55

   1. Initial program 86.0%

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

   2. Taylor expanded in beta around inf 96.8%

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

   if 5.5000000000000004e55 < alpha

   1. Initial program 10.7%

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

   3. Simplified 35.3%

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

   4. Taylor expanded in alpha around inf 71.3%

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

   5. Taylor expanded in beta around 0 60.8%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + 2 \cdot i\right) - i \cdot -2}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 79.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.55 \cdot 10^{+20}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + 2 \cdot i\right) - i \cdot -2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.55e+20)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (- (+ 2.0 (* 2.0 i)) (* i -2.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.55e+20) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (((2.0 + (2.0 * i)) - (i * -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 <= 1.55d+20) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (((2.0d0 + (2.0d0 * i)) - (i * (-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 <= 1.55e+20) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (((2.0 + (2.0 * i)) - (i * -2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.55e+20:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (((2.0 + (2.0 * i)) - (i * -2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.55e+20)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(2.0 * i)) - Float64(i * -2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.55e+20)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (((2.0 + (2.0 * i)) - (i * -2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.55e+20], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] - N[(i * -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.55e20

   1. Initial program 87.1%

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

      1. associate-/l/ 86.7%

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

      2. associate-+l+ 86.7%

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

      3. associate-+l+ 86.7%

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

   3. Simplified 86.7%

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

   4. Taylor expanded in i around 0 93.6%

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

      1. associate-+r+ 93.6%

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

   6. Simplified 93.6%

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

   7. Taylor expanded in alpha around 0 94.7%

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

   if 1.55e20 < alpha

   1. Initial program 14.0%

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

   3. Simplified 36.6%

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

   4. Taylor expanded in alpha around inf 69.8%

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

   5. Taylor expanded in beta around 0 59.5%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.55 \cdot 10^{+20}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + 2 \cdot i\right) - i \cdot -2}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 75.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 6.5 \cdot 10^{+148}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 6.5e+148) (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 6.5e+148) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 6.5d+148) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 6.5e+148) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if i <= 6.5e+148:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 6.5e+148)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 6.5e+148)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 6.5e+148], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if i < 6.49999999999999947e148

   1. Initial program 62.1%

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

      1. associate-/l/ 61.5%

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

      2. associate-+l+ 61.5%

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

      3. associate-+l+ 61.5%

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

   3. Simplified 61.5%

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

   4. Taylor expanded in i around 0 73.7%

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

      1. associate-+r+ 73.7%

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

   6. Simplified 73.7%

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

   7. Taylor expanded in alpha around 0 74.1%

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

   if 6.49999999999999947e148 < i

   1. Initial program 77.9%

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

      1. *-commutative 77.9%

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

      2. *-un-lft-identity 77.9%

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

      3. times-frac 97.0%

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

      4. associate-+r+ 97.0%

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

      5. +-commutative 97.0%

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

      6. fma-udef 97.0%

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

   3. Applied egg-rr 97.0%

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

   4. Taylor expanded in i around inf 92.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 6.5 \cdot 10^{+148}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 9: 76.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 9.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 9.5e+19)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (+ beta beta)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 9.5e+19) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta + 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) :: tmp
    if (alpha <= 9.5d+19) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta + beta)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 9.5e+19) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta + beta)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 9.5e+19:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta + beta)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 9.5e+19)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta + beta)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 9.5e+19)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta + beta)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 9.5e+19], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 9.5e19

   1. Initial program 87.1%

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

      1. associate-/l/ 86.7%

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

      2. associate-+l+ 86.7%

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

      3. associate-+l+ 86.7%

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

   3. Simplified 86.7%

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

   4. Taylor expanded in i around 0 93.6%

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

      1. associate-+r+ 93.6%

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

   6. Simplified 93.6%

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

   7. Taylor expanded in alpha around 0 94.7%

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

   if 9.5e19 < alpha

   1. Initial program 14.0%

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

   3. Simplified 36.6%

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

   4. Taylor expanded in alpha around inf 69.8%

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

   5. Taylor expanded in i around 0 48.3%

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

      1. associate--l+ 48.3%

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

      2. sub-neg 48.3%

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

      3. mul-1-neg 48.3%

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

      4. remove-double-neg 48.3%

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

   7. Simplified 48.3%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 9.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 10: 71.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 680000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 680000.0) 0.5 (/ (- 2.0 (/ 2.0 beta)) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 680000.0) {
		tmp = 0.5;
	} else {
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 680000.0d0) then
        tmp = 0.5d0
    else
        tmp = (2.0d0 - (2.0d0 / beta)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 680000.0) {
		tmp = 0.5;
	} else {
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 680000.0:
		tmp = 0.5
	else:
		tmp = (2.0 - (2.0 / beta)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 680000.0)
		tmp = 0.5;
	else
		tmp = Float64(Float64(2.0 - Float64(2.0 / beta)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 680000.0)
		tmp = 0.5;
	else
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 680000.0], 0.5, N[(N[(2.0 - N[(2.0 / beta), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.8e5

   1. Initial program 74.4%

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

      1. *-commutative 74.4%

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

      2. *-un-lft-identity 74.4%

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

      3. times-frac 77.4%

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

      4. associate-+r+ 77.4%

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

      5. +-commutative 77.4%

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

      6. fma-udef 77.4%

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

   3. Applied egg-rr 77.4%

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

   4. Taylor expanded in i around inf 76.4%

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

   if 6.8e5 < 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/ 42.4%

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

      2. associate-+l+ 42.4%

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

      3. associate-+l+ 42.4%

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

   3. Simplified 42.4%

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

   4. Taylor expanded in i around 0 73.1%

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

      1. associate-+r+ 73.1%

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

   6. Simplified 73.1%

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

   7. Taylor expanded in beta around inf 70.3%

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

      1. mul-1-neg 70.3%

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

      2. unsub-neg 70.3%

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

      3. *-commutative 70.3%

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

   9. Simplified 70.3%

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

   10. Taylor expanded in alpha around 0 71.5%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 680000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \]

Alternative 11: 71.8% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1200000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 1200000.0) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1200000.0) {
		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 <= 1200000.0d0) 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 <= 1200000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1200000.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1200000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1200000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1200000.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 1.2e6

   1. Initial program 74.4%

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

      1. *-commutative 74.4%

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

      2. *-un-lft-identity 74.4%

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

      3. times-frac 77.4%

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

      4. associate-+r+ 77.4%

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

      5. +-commutative 77.4%

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

      6. fma-udef 77.4%

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

   3. Applied egg-rr 77.4%

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

   4. Taylor expanded in i around inf 76.4%

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

   if 1.2e6 < 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/ 42.4%

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

      2. associate-+l+ 42.4%

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

      3. associate-+l+ 42.4%

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

   3. Simplified 42.4%

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

   4. Taylor expanded in beta around inf 70.3%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1200000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12: 61.1% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 65.7%

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

   1. *-commutative 65.7%

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

   2. *-un-lft-identity 65.7%

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

   3. times-frac 81.0%

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

   4. associate-+r+ 81.0%

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

   5. +-commutative 81.0%

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

   6. fma-udef 81.0%

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

3. Applied egg-rr 81.0%

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

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 2023333 
(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))