Octave 3.8, jcobi/2

Percentage Accurate: 62.5% → 96.7%

Time: 10.6s

Alternatives: 11

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.5% 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: 96.7% accurate, 0.7× speedup

Math

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

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) -1.0)
     (/ (fma 0.5 (fma 2.0 beta (* i 4.0)) 1.0) alpha)
     (/ (+ 1.0 (/ (- beta alpha) 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) <= -1.0) {
		tmp = fma(0.5, fma(2.0, beta, (i * 4.0)), 1.0) / alpha;
	} else {
		tmp = (1.0 + ((beta - alpha) / 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) <= -1.0)
		tmp = Float64(fma(0.5, fma(2.0, beta, Float64(i * 4.0)), 1.0) / alpha);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / t_1)) / 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]}, 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], -1.0], N[(N[(0.5 * N[(2.0 * beta + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / 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 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -1

   1. Initial program 1.8%

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

   3. Taylor expanded in alpha around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. neg-sub0 N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 86.2

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, 2, \color{blue}{\mathsf{fma}\left(i, 4, 2\right)}\right)}{\alpha}}{2} \]

   5. Simplified 86.2%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\beta, 2, \mathsf{fma}\left(i, 4, 2\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in alpha around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 86.2

         \[\leadsto \frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(2, \beta, \color{blue}{i \cdot 4}\right), 1\right)}{\alpha} \]

   8. Simplified 86.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(2, \beta, i \cdot 4\right), 1\right)}{\alpha}} \]

   if -1 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 83.1%

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

   3. Taylor expanded in i around 0

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

      1. lower--.f64 98.9

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

   5. Simplified 98.9%

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

4. Final simplification 95.3%

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

Alternative 2: 94.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(2, \beta, i \cdot 4\right), 1\right)}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\beta + 2}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -1.0)
     (/ (fma 0.5 (fma 2.0 beta (* i 4.0)) 1.0) alpha)
     (if (<= t_1 5e-24) 0.5 (fma (/ beta (+ beta 2.0)) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = fma(0.5, fma(2.0, beta, (i * 4.0)), 1.0) / alpha;
	} else if (t_1 <= 5e-24) {
		tmp = 0.5;
	} else {
		tmp = fma((beta / (beta + 2.0)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(fma(0.5, fma(2.0, beta, Float64(i * 4.0)), 1.0) / alpha);
	elseif (t_1 <= 5e-24)
		tmp = 0.5;
	else
		tmp = fma(Float64(beta / Float64(beta + 2.0)), 0.5, 0.5);
	end
	return 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[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[(0.5 * N[(2.0 * beta + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 5e-24], 0.5, N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -1

   1. Initial program 1.8%

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

   3. Taylor expanded in alpha around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. neg-sub0 N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 86.2

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, 2, \color{blue}{\mathsf{fma}\left(i, 4, 2\right)}\right)}{\alpha}}{2} \]

   5. Simplified 86.2%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\beta, 2, \mathsf{fma}\left(i, 4, 2\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in alpha around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 86.2

         \[\leadsto \frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(2, \beta, \color{blue}{i \cdot 4}\right), 1\right)}{\alpha} \]

   8. Simplified 86.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(2, \beta, i \cdot 4\right), 1\right)}{\alpha}} \]

   if -1 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 4.9999999999999998e-24

   1. Initial program 100.0%

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

   3. Taylor expanded in i around 0

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

      1. lower--.f64 99.8

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

   5. Simplified 99.8%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 99.8

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

   8. Simplified 99.8%

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

   9. Taylor expanded in beta around 0

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

   11. Simplified 100.0%

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

   if 4.9999999999999998e-24 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 47.4%

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

   3. Taylor expanded in i around 0

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

      1. lower--.f64 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 94.4

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

   8. Simplified 94.4%

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

   9. Taylor expanded in i around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-+.f64 88.7

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{\color{blue}{\beta + 2}}, 0.5, 0.5\right) \]

   11. Simplified 88.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\beta}{\beta + 2}, 0.5, 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -1:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(2, \beta, i \cdot 4\right), 1\right)}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 5 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\beta + 2}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, i \cdot 4, 1\right)}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\beta + 2}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -1.0)
     (/ (fma 0.5 (* i 4.0) 1.0) alpha)
     (if (<= t_1 5e-24) 0.5 (fma (/ beta (+ beta 2.0)) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = fma(0.5, (i * 4.0), 1.0) / alpha;
	} else if (t_1 <= 5e-24) {
		tmp = 0.5;
	} else {
		tmp = fma((beta / (beta + 2.0)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(fma(0.5, Float64(i * 4.0), 1.0) / alpha);
	elseif (t_1 <= 5e-24)
		tmp = 0.5;
	else
		tmp = fma(Float64(beta / Float64(beta + 2.0)), 0.5, 0.5);
	end
	return 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[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[(0.5 * N[(i * 4.0), $MachinePrecision] + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 5e-24], 0.5, N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -1

   1. Initial program 1.8%

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

   3. Taylor expanded in alpha around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. neg-sub0 N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 86.2

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, 2, \color{blue}{\mathsf{fma}\left(i, 4, 2\right)}\right)}{\alpha}}{2} \]

   5. Simplified 86.2%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\beta, 2, \mathsf{fma}\left(i, 4, 2\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in beta around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, 4 \cdot i, 1\right)}}{\alpha} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{i \cdot 4}, 1\right)}{\alpha} \]

      8. lower-*.f64 75.8

         \[\leadsto \frac{\mathsf{fma}\left(0.5, \color{blue}{i \cdot 4}, 1\right)}{\alpha} \]

   8. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, i \cdot 4, 1\right)}{\alpha}} \]

   if -1 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 4.9999999999999998e-24

   1. Initial program 100.0%

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

   3. Taylor expanded in i around 0

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

      1. lower--.f64 99.8

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

   5. Simplified 99.8%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 99.8

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

   8. Simplified 99.8%

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

   9. Taylor expanded in beta around 0

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

   11. Simplified 100.0%

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

   if 4.9999999999999998e-24 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 47.4%

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

   3. Taylor expanded in i around 0

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

      1. lower--.f64 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 94.4

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

   8. Simplified 94.4%

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

   9. Taylor expanded in i around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-+.f64 88.7

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{\color{blue}{\beta + 2}}, 0.5, 0.5\right) \]

   11. Simplified 88.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\beta}{\beta + 2}, 0.5, 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -1:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, i \cdot 4, 1\right)}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 5 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\beta + 2}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\beta + 2}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -1.0)
     (/ (+ beta 1.0) alpha)
     (if (<= t_1 5e-24) 0.5 (fma (/ beta (+ beta 2.0)) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = (beta + 1.0) / alpha;
	} else if (t_1 <= 5e-24) {
		tmp = 0.5;
	} else {
		tmp = fma((beta / (beta + 2.0)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	elseif (t_1 <= 5e-24)
		tmp = 0.5;
	else
		tmp = fma(Float64(beta / Float64(beta + 2.0)), 0.5, 0.5);
	end
	return 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[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 5e-24], 0.5, N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -1

   1. Initial program 1.8%

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

   3. Taylor expanded in alpha around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. neg-sub0 N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 86.2

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, 2, \color{blue}{\mathsf{fma}\left(i, 4, 2\right)}\right)}{\alpha}}{2} \]

   5. Simplified 86.2%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\beta, 2, \mathsf{fma}\left(i, 4, 2\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 61.1

         \[\leadsto \frac{\mathsf{fma}\left(0.5, \color{blue}{2 \cdot \beta}, 1\right)}{\alpha} \]

   8. Simplified 61.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, 2 \cdot \beta, 1\right)}{\alpha}} \]

   9. Taylor expanded in alpha around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 61.1

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

   11. Simplified 61.1%

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

   if -1 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 4.9999999999999998e-24

   1. Initial program 100.0%

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

   3. Taylor expanded in i around 0

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

      1. lower--.f64 99.8

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

   5. Simplified 99.8%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 99.8

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

   8. Simplified 99.8%

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

   9. Taylor expanded in beta around 0

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

   11. Simplified 100.0%

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

   if 4.9999999999999998e-24 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 47.4%

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

   3. Taylor expanded in i around 0

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

      1. lower--.f64 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 94.4

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

   8. Simplified 94.4%

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

   9. Taylor expanded in i around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-+.f64 88.7

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{\color{blue}{\beta + 2}}, 0.5, 0.5\right) \]

   11. Simplified 88.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\beta}{\beta + 2}, 0.5, 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -1:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 5 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\beta + 2}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 10^{-5}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta + -1\right) - \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -1.0)
     (/ (+ beta 1.0) alpha)
     (if (<= t_1 1e-5) 0.5 (/ (- (+ beta -1.0) alpha) beta)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = (beta + 1.0) / alpha;
	} else if (t_1 <= 1e-5) {
		tmp = 0.5;
	} else {
		tmp = ((beta + -1.0) - alpha) / beta;
	}
	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 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)
    if (t_1 <= (-1.0d0)) then
        tmp = (beta + 1.0d0) / alpha
    else if (t_1 <= 1d-5) then
        tmp = 0.5d0
    else
        tmp = ((beta + (-1.0d0)) - alpha) / beta
    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 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = (beta + 1.0) / alpha;
	} else if (t_1 <= 1e-5) {
		tmp = 0.5;
	} else {
		tmp = ((beta + -1.0) - alpha) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)
	tmp = 0
	if t_1 <= -1.0:
		tmp = (beta + 1.0) / alpha
	elif t_1 <= 1e-5:
		tmp = 0.5
	else:
		tmp = ((beta + -1.0) - alpha) / beta
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	elseif (t_1 <= 1e-5)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(beta + -1.0) - alpha) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	tmp = 0.0;
	if (t_1 <= -1.0)
		tmp = (beta + 1.0) / alpha;
	elseif (t_1 <= 1e-5)
		tmp = 0.5;
	else
		tmp = ((beta + -1.0) - alpha) / beta;
	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[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 1e-5], 0.5, N[(N[(N[(beta + -1.0), $MachinePrecision] - alpha), $MachinePrecision] / beta), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -1

   1. Initial program 1.8%

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

   3. Taylor expanded in alpha around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. neg-sub0 N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 86.2

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, 2, \color{blue}{\mathsf{fma}\left(i, 4, 2\right)}\right)}{\alpha}}{2} \]

   5. Simplified 86.2%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\beta, 2, \mathsf{fma}\left(i, 4, 2\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 61.1

         \[\leadsto \frac{\mathsf{fma}\left(0.5, \color{blue}{2 \cdot \beta}, 1\right)}{\alpha} \]

   8. Simplified 61.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, 2 \cdot \beta, 1\right)}{\alpha}} \]

   9. Taylor expanded in alpha around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 61.1

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

   11. Simplified 61.1%

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

   if -1 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 1.00000000000000008e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in i around 0

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

      1. lower--.f64 99.4

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

   5. Simplified 99.4%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 98.8

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

   8. Simplified 98.8%

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

   9. Taylor expanded in beta around 0

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

   11. Simplified 98.4%

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

   if 1.00000000000000008e-5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 40.4%

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

   3. Taylor expanded in i around 0

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

      1. lower--.f64 97.6

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

   5. Simplified 97.6%

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

   6. Taylor expanded in i around 0

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

      1. lower-+.f64 91.3

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

   8. Simplified 91.3%

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

   9. Taylor expanded in beta around inf

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 90.5

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

   11. Simplified 90.5%

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

   12. Taylor expanded in beta around 0

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

       1. lower-/.f64 N/A

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

       2. distribute-lft-in N/A

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

       3. metadata-eval N/A

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

       4. associate-+r+ N/A

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

       5. associate-*r* N/A

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

       6. metadata-eval N/A

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

       7. mul-1-neg N/A

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

       8. unsub-neg N/A

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

       9. lower--.f64 N/A

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

       10. lower-+.f64 90.5

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

   14. Simplified 90.5%

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

4. Final simplification 86.2%

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

Alternative 6: 87.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 10^{-5}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -1.0)
     (/ (+ beta 1.0) alpha)
     (if (<= t_1 1e-5) 0.5 (+ 1.0 (/ -1.0 beta))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = (beta + 1.0) / alpha;
	} else if (t_1 <= 1e-5) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	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 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)
    if (t_1 <= (-1.0d0)) then
        tmp = (beta + 1.0d0) / alpha
    else if (t_1 <= 1d-5) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 + ((-1.0d0) / beta)
    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 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = (beta + 1.0) / alpha;
	} else if (t_1 <= 1e-5) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)
	tmp = 0
	if t_1 <= -1.0:
		tmp = (beta + 1.0) / alpha
	elif t_1 <= 1e-5:
		tmp = 0.5
	else:
		tmp = 1.0 + (-1.0 / beta)
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	elseif (t_1 <= 1e-5)
		tmp = 0.5;
	else
		tmp = Float64(1.0 + Float64(-1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	tmp = 0.0;
	if (t_1 <= -1.0)
		tmp = (beta + 1.0) / alpha;
	elseif (t_1 <= 1e-5)
		tmp = 0.5;
	else
		tmp = 1.0 + (-1.0 / beta);
	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[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 1e-5], 0.5, N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -1

   1. Initial program 1.8%

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

   3. Taylor expanded in alpha around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. neg-sub0 N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 86.2

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, 2, \color{blue}{\mathsf{fma}\left(i, 4, 2\right)}\right)}{\alpha}}{2} \]

   5. Simplified 86.2%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\beta, 2, \mathsf{fma}\left(i, 4, 2\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 61.1

         \[\leadsto \frac{\mathsf{fma}\left(0.5, \color{blue}{2 \cdot \beta}, 1\right)}{\alpha} \]

   8. Simplified 61.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, 2 \cdot \beta, 1\right)}{\alpha}} \]

   9. Taylor expanded in alpha around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 61.1

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

   11. Simplified 61.1%

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

   if -1 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 1.00000000000000008e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in i around 0

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

      1. lower--.f64 99.4

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

   5. Simplified 99.4%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 98.8

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

   8. Simplified 98.8%

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

   9. Taylor expanded in beta around 0

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

   11. Simplified 98.4%

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

   if 1.00000000000000008e-5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 40.4%

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

   3. Taylor expanded in i around 0

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

      1. lower--.f64 97.6

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

   5. Simplified 97.6%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 96.4

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

   8. Simplified 96.4%

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

   9. Taylor expanded in i around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-+.f64 90.1

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{\color{blue}{\beta + 2}}, 0.5, 0.5\right) \]

   11. Simplified 90.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\beta}{\beta + 2}, 0.5, 0.5\right)} \]

   12. Taylor expanded in beta around inf

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

       1. sub-neg N/A

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

       2. lower-+.f64 N/A

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

       3. distribute-neg-frac N/A

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

       4. metadata-eval N/A

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

       5. lower-/.f64 90.1

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

   14. Simplified 90.1%

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

4. Final simplification 86.1%

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

Alternative 7: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 10^{-5}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -1.0)
     (/ 1.0 alpha)
     (if (<= t_1 1e-5) 0.5 (+ 1.0 (/ -1.0 beta))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = 1.0 / alpha;
	} else if (t_1 <= 1e-5) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	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 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)
    if (t_1 <= (-1.0d0)) then
        tmp = 1.0d0 / alpha
    else if (t_1 <= 1d-5) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 + ((-1.0d0) / beta)
    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 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = 1.0 / alpha;
	} else if (t_1 <= 1e-5) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)
	tmp = 0
	if t_1 <= -1.0:
		tmp = 1.0 / alpha
	elif t_1 <= 1e-5:
		tmp = 0.5
	else:
		tmp = 1.0 + (-1.0 / beta)
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(1.0 / alpha);
	elseif (t_1 <= 1e-5)
		tmp = 0.5;
	else
		tmp = Float64(1.0 + Float64(-1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	tmp = 0.0;
	if (t_1 <= -1.0)
		tmp = 1.0 / alpha;
	elseif (t_1 <= 1e-5)
		tmp = 0.5;
	else
		tmp = 1.0 + (-1.0 / beta);
	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[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$1, 1e-5], 0.5, N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -1

   1. Initial program 1.8%

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

   3. Taylor expanded in alpha around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. neg-sub0 N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 86.2

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, 2, \color{blue}{\mathsf{fma}\left(i, 4, 2\right)}\right)}{\alpha}}{2} \]

   5. Simplified 86.2%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\beta, 2, \mathsf{fma}\left(i, 4, 2\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 61.1

         \[\leadsto \frac{\mathsf{fma}\left(0.5, \color{blue}{2 \cdot \beta}, 1\right)}{\alpha} \]

   8. Simplified 61.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, 2 \cdot \beta, 1\right)}{\alpha}} \]

   9. Taylor expanded in beta around 0

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

       1. lower-/.f64 50.7

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

   11. Simplified 50.7%

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

   if -1 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 1.00000000000000008e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in i around 0

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

      1. lower--.f64 99.4

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

   5. Simplified 99.4%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 98.8

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

   8. Simplified 98.8%

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

   9. Taylor expanded in beta around 0

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

   11. Simplified 98.4%

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

   if 1.00000000000000008e-5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 40.4%

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

   3. Taylor expanded in i around 0

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

      1. lower--.f64 97.6

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

   5. Simplified 97.6%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 96.4

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

   8. Simplified 96.4%

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

   9. Taylor expanded in i around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-+.f64 90.1

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{\color{blue}{\beta + 2}}, 0.5, 0.5\right) \]

   11. Simplified 90.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\beta}{\beta + 2}, 0.5, 0.5\right)} \]

   12. Taylor expanded in beta around inf

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

       1. sub-neg N/A

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

       2. lower-+.f64 N/A

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

       3. distribute-neg-frac N/A

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

       4. metadata-eval N/A

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

       5. lower-/.f64 90.1

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

   14. Simplified 90.1%

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

4. Final simplification 83.1%

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

Alternative 8: 84.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 10^{-5}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -1.0) (/ 1.0 alpha) (if (<= t_1 1e-5) 0.5 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = 1.0 / alpha;
	} else if (t_1 <= 1e-5) {
		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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)
    if (t_1 <= (-1.0d0)) then
        tmp = 1.0d0 / alpha
    else if (t_1 <= 1d-5) 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 t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = 1.0 / alpha;
	} else if (t_1 <= 1e-5) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)
	tmp = 0
	if t_1 <= -1.0:
		tmp = 1.0 / alpha
	elif t_1 <= 1e-5:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(1.0 / alpha);
	elseif (t_1 <= 1e-5)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	tmp = 0.0;
	if (t_1 <= -1.0)
		tmp = 1.0 / alpha;
	elseif (t_1 <= 1e-5)
		tmp = 0.5;
	else
		tmp = 1.0;
	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[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$1, 1e-5], 0.5, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -1

   1. Initial program 1.8%

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

   3. Taylor expanded in alpha around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. neg-sub0 N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 86.2

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, 2, \color{blue}{\mathsf{fma}\left(i, 4, 2\right)}\right)}{\alpha}}{2} \]

   5. Simplified 86.2%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\beta, 2, \mathsf{fma}\left(i, 4, 2\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 61.1

         \[\leadsto \frac{\mathsf{fma}\left(0.5, \color{blue}{2 \cdot \beta}, 1\right)}{\alpha} \]

   8. Simplified 61.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, 2 \cdot \beta, 1\right)}{\alpha}} \]

   9. Taylor expanded in beta around 0

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

       1. lower-/.f64 50.7

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

   11. Simplified 50.7%

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

   if -1 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 1.00000000000000008e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in i around 0

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

      1. lower--.f64 99.4

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

   5. Simplified 99.4%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 98.8

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

   8. Simplified 98.8%

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

   9. Taylor expanded in beta around 0

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

   11. Simplified 98.4%

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

   if 1.00000000000000008e-5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 40.4%

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

   3. Taylor expanded in i around 0

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

      1. lower--.f64 97.6

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

   5. Simplified 97.6%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 96.4

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

   8. Simplified 96.4%

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

   9. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1} \]
   10. Step-by-step derivation

   11. Simplified 89.2%

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

4. Final simplification 82.9%

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

Alternative 9: 95.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -1.0)
		tmp = Float64(fma(0.5, fma(2.0, beta, Float64(i * 4.0)), 1.0) / alpha);
	else
		tmp = Float64(Float64(1.0 + Float64(beta / fma(2.0, i, Float64(beta + 2.0)))) / 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], -1.0], N[(N[(0.5 * N[(2.0 * beta + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(1.0 + N[(beta / N[(2.0 * i + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -1

   1. Initial program 1.8%

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

   3. Taylor expanded in alpha around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. neg-sub0 N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 86.2

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, 2, \color{blue}{\mathsf{fma}\left(i, 4, 2\right)}\right)}{\alpha}}{2} \]

   5. Simplified 86.2%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\beta, 2, \mathsf{fma}\left(i, 4, 2\right)\right)}{\alpha}}}{2} \]

   6. Taylor expanded in alpha around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 86.2

         \[\leadsto \frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(2, \beta, \color{blue}{i \cdot 4}\right), 1\right)}{\alpha} \]

   8. Simplified 86.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(2, \beta, i \cdot 4\right), 1\right)}{\alpha}} \]

   if -1 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 83.1%

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

   3. Taylor expanded in i around 0

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

      1. lower--.f64 98.9

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

   5. Simplified 98.9%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 98.1

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

   8. Simplified 98.1%

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

4. Final simplification 94.7%

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

Alternative 10: 76.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq 10^{-5}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 1e-5)
     0.5
     1.0)))

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)) <= 1e-5) {
		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) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)) <= 1d-5) 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 t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= 1e-5) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= 1e-5:
		tmp = 0.5
	else:
		tmp = 1.0
	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)) <= 1e-5)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= 1e-5)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], 1e-5], 0.5, 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 1.00000000000000008e-5

   1. Initial program 64.8%

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

   3. Taylor expanded in i around 0

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

      1. lower--.f64 70.2

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

   5. Simplified 70.2%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 70.1

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

   8. Simplified 70.1%

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

   9. Taylor expanded in beta around 0

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

   11. Simplified 69.9%

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

   if 1.00000000000000008e-5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 40.4%

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

   3. Taylor expanded in i around 0

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

      1. lower--.f64 97.6

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

   5. Simplified 97.6%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 96.4

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

   8. Simplified 96.4%

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

   9. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1} \]
   10. Step-by-step derivation

   11. Simplified 89.2%

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

4. Final simplification 73.8%

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

Alternative 11: 60.3% accurate, 73.0× 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 59.9%

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

3. Taylor expanded in i around 0

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

   1. lower--.f64 75.8

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

5. Simplified 75.8%

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

6. Taylor expanded in alpha around 0

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

   1. lower-/.f64 N/A

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

   2. associate-+r+ N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-+.f64 75.5

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

8. Simplified 75.5%

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

9. Taylor expanded in beta around 0

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

11. Simplified 60.8%

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

Reproduce

herbie shell --seed 2024215 
(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))