Octave 3.8, jcobi/2

Percentage Accurate: 63.4% → 97.4%

Time: 10.0s

Alternatives: 12

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: 63.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.998:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, 2, \mathsf{fma}\left(i, 4, 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, \frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.998)
     (/ (/ (fma beta 2.0 (fma i 4.0 2.0)) alpha) 2.0)
     (/
      (fma
       (/ (- beta alpha) (fma 2.0 i (+ alpha beta)))
       (/ (+ alpha beta) (+ alpha (+ beta (fma 2.0 i 2.0))))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.998) {
		tmp = (fma(beta, 2.0, fma(i, 4.0, 2.0)) / alpha) / 2.0;
	} else {
		tmp = fma(((beta - alpha) / fma(2.0, i, (alpha + beta))), ((alpha + beta) / (alpha + (beta + fma(2.0, i, 2.0)))), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.998)
		tmp = Float64(Float64(fma(beta, 2.0, fma(i, 4.0, 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(fma(Float64(Float64(beta - alpha) / fma(2.0, i, Float64(alpha + beta))), Float64(Float64(alpha + beta) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.998], N[(N[(N[(beta * 2.0 + N[(i * 4.0 + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.998

   1. Initial program 2.9%

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

   3. Taylor expanded in 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 87.3

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

   5. Simplified 87.3%

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

   if -0.998 < (/.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 80.1%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.998:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, 2, \mathsf{fma}\left(i, 4, 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, \frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, 2, \mathsf{fma}\left(i, 4, 2\right)\right)}{\alpha}}{2}\\ \mathbf{elif}\;t\_1 \leq 0.9999995:\\ \;\;\;\;\mathsf{fma}\left(\beta, \frac{\beta}{\mathsf{fma}\left(2, i, \beta\right) \cdot \left(2 + \mathsf{fma}\left(2, i, \beta\right)\right)}, 1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 -0.5)
     (/ (/ (fma beta 2.0 (fma i 4.0 2.0)) alpha) 2.0)
     (if (<= t_1 0.9999995)
       (*
        (fma beta (/ beta (* (fma 2.0 i beta) (+ 2.0 (fma 2.0 i beta)))) 1.0)
        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 <= -0.5) {
		tmp = (fma(beta, 2.0, fma(i, 4.0, 2.0)) / alpha) / 2.0;
	} else if (t_1 <= 0.9999995) {
		tmp = fma(beta, (beta / (fma(2.0, i, beta) * (2.0 + fma(2.0, i, beta)))), 1.0) * 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 <= -0.5)
		tmp = Float64(Float64(fma(beta, 2.0, fma(i, 4.0, 2.0)) / alpha) / 2.0);
	elseif (t_1 <= 0.9999995)
		tmp = Float64(fma(beta, Float64(beta / Float64(fma(2.0, i, beta) * Float64(2.0 + fma(2.0, i, beta)))), 1.0) * 0.5);
	else
		tmp = 1.0;
	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[(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, -0.5], N[(N[(N[(beta * 2.0 + N[(i * 4.0 + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$1, 0.9999995], N[(N[(beta * N[(beta / N[(N[(2.0 * i + beta), $MachinePrecision] * N[(2.0 + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision], 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))) < -0.5

   1. Initial program 4.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 86.4

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

   5. Simplified 86.4%

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

   if -0.5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 0.999999500000000041

   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. Step-by-step derivation

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 99.8

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

   7. Simplified 99.8%

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

   8. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 99.8

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

   10. Simplified 99.8%

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

       1. lift-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-fma.f64 N/A

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

       7. div-inv N/A

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

       8. metadata-eval N/A

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

       9. lower-*.f64 99.8

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

   12. Applied egg-rr 99.8%

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

   if 0.999999500000000041 < (/.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 34.3%

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

   3. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 90.8

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

   5. Simplified 90.8%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 90.8

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

   8. Simplified 90.8%

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

   9. Taylor expanded in beta around inf

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

   11. Simplified 90.8%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, 2, \mathsf{fma}\left(i, 4, 2\right)\right)}{\alpha}}{2}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 0.9999995:\\ \;\;\;\;\mathsf{fma}\left(\beta, \frac{\beta}{\mathsf{fma}\left(2, i, \beta\right) \cdot \left(2 + \mathsf{fma}\left(2, i, \beta\right)\right)}, 1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, 2, \mathsf{fma}\left(i, 4, 2\right)\right)}{\alpha}}{2}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \end{array} \end{array} \]

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 -0.5)
     (/ (/ (fma beta 2.0 (fma i 4.0 2.0)) alpha) 2.0)
     (if (<= t_1 2e-11)
       0.5
       (/ (+ 1.0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))) 2.0)))))

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 <= -0.5) {
		tmp = (fma(beta, 2.0, fma(i, 4.0, 2.0)) / alpha) / 2.0;
	} else if (t_1 <= 2e-11) {
		tmp = 0.5;
	} else {
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	}
	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 <= -0.5)
		tmp = Float64(Float64(fma(beta, 2.0, fma(i, 4.0, 2.0)) / alpha) / 2.0);
	elseif (t_1 <= 2e-11)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(Float64(alpha + 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]}, 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, -0.5], N[(N[(N[(beta * 2.0 + N[(i * 4.0 + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$1, 2e-11], 0.5, N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $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))) < -0.5

   1. Initial program 4.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 86.4

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

   5. Simplified 86.4%

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

   if -0.5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 1.99999999999999988e-11

   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 inf

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

   5. Simplified 99.3%

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

      1. metadata-eval 99.3

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

   7. Applied egg-rr 99.3%

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

   if 1.99999999999999988e-11 < (/.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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 89.1

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

   5. Simplified 89.1%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, 2, \mathsf{fma}\left(i, 4, 2\right)\right)}{\alpha}}{2}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(2, \beta, 2\right)}{\alpha}}{2}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \end{array} \end{array} \]

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 -0.5)
     (/ (/ (fma 2.0 beta 2.0) alpha) 2.0)
     (if (<= t_1 2e-11)
       0.5
       (/ (+ 1.0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))) 2.0)))))

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 <= -0.5) {
		tmp = (fma(2.0, beta, 2.0) / alpha) / 2.0;
	} else if (t_1 <= 2e-11) {
		tmp = 0.5;
	} else {
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	}
	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 <= -0.5)
		tmp = Float64(Float64(fma(2.0, beta, 2.0) / alpha) / 2.0);
	elseif (t_1 <= 2e-11)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(Float64(alpha + 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]}, 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, -0.5], N[(N[(N[(2.0 * beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$1, 2e-11], 0.5, N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $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))) < -0.5

   1. Initial program 4.2%

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

   3. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 6.5

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

   5. Simplified 6.5%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 57.3

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

   8. Simplified 57.3%

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

   if -0.5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 1.99999999999999988e-11

   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 inf

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

   5. Simplified 99.3%

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

      1. metadata-eval 99.3

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

   7. Applied egg-rr 99.3%

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

   if 1.99999999999999988e-11 < (/.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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 89.1

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

   5. Simplified 89.1%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(2, \beta, 2\right)}{\alpha}}{2}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(2, \beta, 2\right)}{\alpha}}{2}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\beta + 2}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

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 -0.5)
     (/ (/ (fma 2.0 beta 2.0) alpha) 2.0)
     (if (<= t_1 2e-11) 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 <= -0.5) {
		tmp = (fma(2.0, beta, 2.0) / alpha) / 2.0;
	} else if (t_1 <= 2e-11) {
		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 <= -0.5)
		tmp = Float64(Float64(fma(2.0, beta, 2.0) / alpha) / 2.0);
	elseif (t_1 <= 2e-11)
		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, -0.5], N[(N[(N[(2.0 * beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$1, 2e-11], 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))) < -0.5

   1. Initial program 4.2%

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

   3. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 6.5

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

   5. Simplified 6.5%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 57.3

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

   8. Simplified 57.3%

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

   if -0.5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 1.99999999999999988e-11

   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 inf

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

   5. Simplified 99.3%

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

      1. metadata-eval 99.3

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

   7. Applied egg-rr 99.3%

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

   if 1.99999999999999988e-11 < (/.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. Step-by-step derivation

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 100.0

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

   7. Simplified 100.0%

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

   8. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 100.0

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

   10. Simplified 100.0%

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

   11. Taylor expanded in i around 0

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(1 + \frac{\beta}{2 + \beta}\right)} \]
   12. 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 89.1

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

   13. Simplified 89.1%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(2, \beta, 2\right)}{\alpha}}{2}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\beta + 2}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.998:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, 2, \mathsf{fma}\left(i, 4, 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, \frac{\beta}{2 + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.998)
     (/ (/ (fma beta 2.0 (fma i 4.0 2.0)) alpha) 2.0)
     (/
      (fma
       (/ (- beta alpha) (fma 2.0 i (+ alpha beta)))
       (/ beta (+ 2.0 (fma 2.0 i beta)))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.998) {
		tmp = (fma(beta, 2.0, fma(i, 4.0, 2.0)) / alpha) / 2.0;
	} else {
		tmp = fma(((beta - alpha) / fma(2.0, i, (alpha + beta))), (beta / (2.0 + fma(2.0, i, beta))), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.998)
		tmp = Float64(Float64(fma(beta, 2.0, fma(i, 4.0, 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(fma(Float64(Float64(beta - alpha) / fma(2.0, i, Float64(alpha + beta))), Float64(beta / Float64(2.0 + fma(2.0, i, beta))), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.998], N[(N[(N[(beta * 2.0 + N[(i * 4.0 + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta / N[(2.0 + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.998

   1. Initial program 2.9%

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

   3. Taylor expanded in 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 87.3

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

   5. Simplified 87.3%

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

   if -0.998 < (/.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 80.1%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 99.8

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

   7. Simplified 99.8%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.998:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, 2, \mathsf{fma}\left(i, 4, 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, \frac{\beta}{2 + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 86.4

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

   5. Simplified 86.4%

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

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

   1. Initial program 80.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 99.9

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

   7. Simplified 99.9%

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

   8. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 99.9

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

   10. Simplified 99.9%

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

4. Final simplification 96.6%

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

Alternative 8: 96.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t\_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_1} \leq -0.998:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, 2, \mathsf{fma}\left(i, 4, 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{t\_1}}{2}\\ \end{array} \end{array} \]

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) -0.998)
     (/ (/ (fma beta 2.0 (fma i 4.0 2.0)) alpha) 2.0)
     (/ (+ 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) <= -0.998) {
		tmp = (fma(beta, 2.0, fma(i, 4.0, 2.0)) / alpha) / 2.0;
	} 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) <= -0.998)
		tmp = Float64(Float64(fma(beta, 2.0, fma(i, 4.0, 2.0)) / alpha) / 2.0);
	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], -0.998], N[(N[(N[(beta * 2.0 + N[(i * 4.0 + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $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))) < -0.998

   1. Initial program 2.9%

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

   3. Taylor expanded in 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 87.3

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

   5. Simplified 87.3%

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

   if -0.998 < (/.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 80.1%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 97.7

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

   5. Simplified 97.7%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.998:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, 2, \mathsf{fma}\left(i, 4, 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\beta + 2}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) 2e-11)
     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 tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= 2e-11) {
		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))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= 2e-11)
		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]}, 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], 2e-11], 0.5, N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $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))) < 1.99999999999999988e-11

   1. Initial program 68.9%

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

   3. Taylor expanded in i around inf

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

   5. Simplified 72.6%

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

      1. metadata-eval 72.6

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

   7. Applied egg-rr 72.6%

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

   if 1.99999999999999988e-11 < (/.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. Step-by-step derivation

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 100.0

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

   7. Simplified 100.0%

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

   8. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 100.0

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

   10. Simplified 100.0%

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

   11. Taylor expanded in i around 0

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(1 + \frac{\beta}{2 + \beta}\right)} \]
   12. 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 89.1

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

   13. Simplified 89.1%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\beta + 2}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq 0.05:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) 0.05)
     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 tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= 0.05) {
		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) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)) <= 0.05d0) 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 tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= 0.05) {
		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)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= 0.05:
		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))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= 0.05)
		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);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= 0.05)
		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]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], 0.05], 0.5, N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 0.050000000000000003

   1. Initial program 69.5%

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

   3. Taylor expanded in i around inf

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

   5. Simplified 72.1%

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

      1. metadata-eval 72.1

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

   7. Applied egg-rr 72.1%

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

   if 0.050000000000000003 < (/.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 36.5%

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

   3. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 91.1

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

   5. Simplified 91.1%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 91.1

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

   8. Simplified 91.1%

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

   9. Taylor expanded in beta around inf

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

       1. sub-neg N/A

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

       2. distribute-neg-frac N/A

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

       3. metadata-eval N/A

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

       4. lower-+.f64 N/A

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

       5. lower-/.f64 89.9

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

   11. Simplified 89.9%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 0.05:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq 0.05:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 0.05)
     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)) <= 0.05) {
		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)) <= 0.05d0) 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)) <= 0.05) {
		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)) <= 0.05:
		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)) <= 0.05)
		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)) <= 0.05)
		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], 0.05], 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))) < 0.050000000000000003

   1. Initial program 69.5%

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

   3. Taylor expanded in i around inf

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

   5. Simplified 72.1%

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

      1. metadata-eval 72.1

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

   7. Applied egg-rr 72.1%

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

   if 0.050000000000000003 < (/.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 36.5%

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

   3. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 91.1

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

   5. Simplified 91.1%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 91.1

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

   8. Simplified 91.1%

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

   9. Taylor expanded in beta around inf

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

   11. Simplified 89.1%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 0.05:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.1% 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 61.7%

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

3. Taylor expanded in i around inf

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

5. Simplified 61.0%

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

   1. metadata-eval 61.0

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

7. Applied egg-rr 61.0%

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

Reproduce

herbie shell --seed 2024219 
(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))