Octave 3.8, jcobi/2

Percentage Accurate: 63.4% → 97.4%

Time: 12.8s

Alternatives: 13

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.998:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{i}{\alpha}, \frac{0.5 \cdot \mathsf{fma}\left(2, \beta, 2\right)}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, \frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

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 2.0 (/ i alpha) (/ (* 0.5 (fma 2.0 beta 2.0)) alpha))
     (/
      (fma
       (/ (- beta alpha) (+ alpha (fma 2.0 i 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(2.0, (i / alpha), ((0.5 * fma(2.0, beta, 2.0)) / alpha));
	} else {
		tmp = fma(((beta - alpha) / (alpha + fma(2.0, i, 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 = fma(2.0, Float64(i / alpha), Float64(Float64(0.5 * fma(2.0, beta, 2.0)) / alpha));
	else
		tmp = Float64(fma(Float64(Float64(beta - alpha) / Float64(alpha + fma(2.0, i, 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[(2.0 * N[(i / alpha), $MachinePrecision] + N[(N[(0.5 * N[(2.0 * beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(2.0 * i + 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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. mul0-lft N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 87.3%

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

   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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.998:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{i}{\alpha}, \frac{0.5 \cdot \mathsf{fma}\left(2, \beta, 2\right)}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, \frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{i}{\alpha}, \frac{0.5 \cdot \mathsf{fma}\left(2, \beta, 2\right)}{\alpha}\right)\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta \cdot \beta}{\mathsf{fma}\left(2, i, \beta\right) \cdot \left(2 + \mathsf{fma}\left(2, i, \beta\right)\right)}, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\\ \end{array} \end{array} \]

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 (/ i alpha) (/ (* 0.5 (fma 2.0 beta 2.0)) alpha))
     (if (<= t_1 0.05)
       (fma
        (/ (* beta beta) (* (fma 2.0 i beta) (+ 2.0 (fma 2.0 i beta))))
        0.5
        0.5)
       (+ 0.5 (* 0.5 (/ (- beta alpha) (+ beta (+ alpha 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, (i / alpha), ((0.5 * fma(2.0, beta, 2.0)) / alpha));
	} else if (t_1 <= 0.05) {
		tmp = fma(((beta * beta) / (fma(2.0, i, beta) * (2.0 + fma(2.0, i, beta)))), 0.5, 0.5);
	} else {
		tmp = 0.5 + (0.5 * ((beta - alpha) / (beta + (alpha + 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 = fma(2.0, Float64(i / alpha), Float64(Float64(0.5 * fma(2.0, beta, 2.0)) / alpha));
	elseif (t_1 <= 0.05)
		tmp = fma(Float64(Float64(beta * beta) / Float64(fma(2.0, i, beta) * Float64(2.0 + fma(2.0, i, beta)))), 0.5, 0.5);
	else
		tmp = Float64(0.5 + Float64(0.5 * Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 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[(2.0 * N[(i / alpha), $MachinePrecision] + N[(N[(0.5 * N[(2.0 * beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.05], N[(N[(N[(beta * beta), $MachinePrecision] / N[(N[(2.0 * i + beta), $MachinePrecision] * N[(2.0 + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision], N[(0.5 + N[(0.5 * N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. mul0-lft N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 86.4

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 86.4%

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

   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))) < 0.050000000000000003

   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 alpha around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

   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

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 91.1

          \[\leadsto 0.5 + 0.5 \cdot \frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right)} + \beta} \]

   5. Applied rewrites 91.1%

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

4. Final simplification 94.5%

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

Alternative 3: 94.8% accurate, 0.5× speedup

Math

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

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

Derivation

1. Split input into 3 regimes

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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. mul0-lft N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 86.4

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 86.4%

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

   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

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

   5. Applied rewrites 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

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 89.1

          \[\leadsto 0.5 + 0.5 \cdot \frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right)} + \beta} \]

   5. Applied rewrites 89.1%

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

4. Final simplification 93.6%

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

Alternative 4: 94.8% accurate, 0.5× speedup

Math

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

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

Derivation

1. Split input into 3 regimes

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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. mul0-lft N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 86.4

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 86.4%

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

   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

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

   5. Applied rewrites 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

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 89.1

          \[\leadsto 0.5 + 0.5 \cdot \frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right)} + \beta} \]

   5. Applied rewrites 89.1%

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

4. Final simplification 93.6%

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

Alternative 5: 94.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(i, 4, 2\right), \beta\right)}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -0.5)
     (/ (fma 0.5 (fma i 4.0 2.0) beta) alpha)
     (if (<= t_1 0.05) 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(0.5, fma(i, 4.0, 2.0), beta) / alpha;
	} else if (t_1 <= 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))
	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(fma(0.5, fma(i, 4.0, 2.0), beta) / alpha);
	elseif (t_1 <= 0.05)
		tmp = 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[(0.5 * N[(i * 4.0 + 2.0), $MachinePrecision] + beta), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 0.05], 0.5, 1.0]]]]

Derivation

1. Split input into 3 regimes

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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. mul0-lft N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 86.4

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 86.4%

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

   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))) < 0.050000000000000003

   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

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

   5. Applied rewrites 97.7%

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

   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 beta around inf

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

   5. Applied rewrites 89.1%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(i, 4, 2\right), \beta\right)}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 0.05:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.998:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{fma}\left(i, 4, 2\right)}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -0.998)
     (/ (* 0.5 (fma i 4.0 2.0)) alpha)
     (if (<= t_1 0.05) 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 <= -0.998) {
		tmp = (0.5 * fma(i, 4.0, 2.0)) / alpha;
	} else if (t_1 <= 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))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0))
	tmp = 0.0
	if (t_1 <= -0.998)
		tmp = Float64(Float64(0.5 * fma(i, 4.0, 2.0)) / alpha);
	elseif (t_1 <= 0.05)
		tmp = 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, -0.998], N[(N[(0.5 * N[(i * 4.0 + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 0.05], 0.5, 1.0]]]]

Derivation

1. Split input into 3 regimes

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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. mul0-lft N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 70.6%

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

   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))) < 0.050000000000000003

   1. Initial program 99.9%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 97.1%

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

   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 beta around inf

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

   5. Applied rewrites 89.1%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.998:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{fma}\left(i, 4, 2\right)}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 0.05:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{fma}\left(2, \beta, 2\right)}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -0.5)
     (/ (* 0.5 (fma 2.0 beta 2.0)) alpha)
     (if (<= t_1 0.05) 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 = (0.5 * fma(2.0, beta, 2.0)) / alpha;
	} else if (t_1 <= 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))
	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(0.5 * fma(2.0, beta, 2.0)) / alpha);
	elseif (t_1 <= 0.05)
		tmp = 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[(0.5 * N[(2.0 * beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 0.05], 0.5, 1.0]]]]

Derivation

1. Split input into 3 regimes

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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. mul0-lft N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 86.4

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 57.3%

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

   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))) < 0.050000000000000003

   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

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

   5. Applied rewrites 97.7%

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

   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 beta around inf

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

   5. Applied rewrites 89.1%

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

4. Final simplification 85.9%

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

Alternative 8: 87.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{0.5}{\alpha} \cdot \mathsf{fma}\left(\beta, 2, 2\right)\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -0.5)
     (* (/ 0.5 alpha) (fma beta 2.0 2.0))
     (if (<= t_1 0.05) 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 = (0.5 / alpha) * fma(beta, 2.0, 2.0);
	} else if (t_1 <= 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))
	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(0.5 / alpha) * fma(beta, 2.0, 2.0));
	elseif (t_1 <= 0.05)
		tmp = 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[(0.5 / alpha), $MachinePrecision] * N[(beta * 2.0 + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.05], 0.5, 1.0]]]]

Derivation

1. Split input into 3 regimes

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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. mul0-lft N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 86.4

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 57.3%

      \[\leadsto \frac{\mathsf{fma}\left(2, \beta, 2\right) \cdot 0.5}{\alpha} \]
   9. Step-by-step derivation

   10. Applied rewrites 57.2%

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

   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))) < 0.050000000000000003

   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

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

   5. Applied rewrites 97.7%

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

   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 beta around inf

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

   5. Applied rewrites 89.1%

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

4. Final simplification 85.9%

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

Alternative 9: 80.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{2 \cdot i}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -1.0) (/ (* 2.0 i) alpha) (if (<= t_1 0.05) 0.5 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = (2.0 * i) / alpha;
	} else if (t_1 <= 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) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)
    if (t_1 <= (-1.0d0)) then
        tmp = (2.0d0 * i) / alpha
    else if (t_1 <= 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 t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = (2.0 * i) / alpha;
	} else if (t_1 <= 0.05) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)
	tmp = 0
	if t_1 <= -1.0:
		tmp = (2.0 * i) / alpha
	elif t_1 <= 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))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(Float64(2.0 * i) / alpha);
	elseif (t_1 <= 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);
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	tmp = 0.0;
	if (t_1 <= -1.0)
		tmp = (2.0 * i) / alpha;
	elseif (t_1 <= 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]}, Block[{t$95$1 = N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[(2.0 * i), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 0.05], 0.5, 1.0]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 2.0%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. mul0-lft N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 87.0

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

   5. Applied rewrites 87.0%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 35.1%

      \[\leadsto \frac{2 \cdot i}{\alpha} \]

   if -1 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 0.050000000000000003

   1. Initial program 98.1%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 95.1%

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

   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 beta around inf

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

   5. Applied rewrites 89.1%

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

4. Final simplification 80.1%

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

Alternative 10: 96.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.998:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{i}{\alpha}, \frac{0.5 \cdot \mathsf{fma}\left(2, \beta, 2\right)}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, \frac{\beta}{2 + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

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)) -0.998)
     (fma 2.0 (/ i alpha) (/ (* 0.5 (fma 2.0 beta 2.0)) alpha))
     (/
      (fma
       (/ (- beta alpha) (+ alpha (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)) <= -0.998) {
		tmp = fma(2.0, (i / alpha), ((0.5 * fma(2.0, beta, 2.0)) / alpha));
	} else {
		tmp = fma(((beta - alpha) / (alpha + 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)) <= -0.998)
		tmp = fma(2.0, Float64(i / alpha), Float64(Float64(0.5 * fma(2.0, beta, 2.0)) / alpha));
	else
		tmp = Float64(fma(Float64(Float64(beta - alpha) / Float64(alpha + 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], -0.998], N[(2.0 * N[(i / alpha), $MachinePrecision] + N[(N[(0.5 * N[(2.0 * beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(2.0 * i + 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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. mul0-lft N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 87.3%

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

   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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 99.8

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

   7. Applied rewrites 99.8%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.998:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{i}{\alpha}, \frac{0.5 \cdot \mathsf{fma}\left(2, \beta, 2\right)}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, \frac{\beta}{2 + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.6% accurate, 0.6× speedup

Math

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

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)) -0.5)
     (fma 2.0 (/ i alpha) (/ (* 0.5 (fma 2.0 beta 2.0)) alpha))
     (*
      0.5
      (fma (/ beta (fma 2.0 i beta)) (/ beta (+ beta (fma 2.0 i 2.0))) 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)) <= -0.5) {
		tmp = fma(2.0, (i / alpha), ((0.5 * fma(2.0, beta, 2.0)) / alpha));
	} else {
		tmp = 0.5 * fma((beta / fma(2.0, i, beta)), (beta / (beta + fma(2.0, i, 2.0))), 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)) <= -0.5)
		tmp = fma(2.0, Float64(i / alpha), Float64(Float64(0.5 * fma(2.0, beta, 2.0)) / alpha));
	else
		tmp = Float64(0.5 * fma(Float64(beta / fma(2.0, i, beta)), Float64(beta / Float64(beta + fma(2.0, i, 2.0))), 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]}, 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[(2.0 * N[(i / alpha), $MachinePrecision] + N[(N[(0.5 * N[(2.0 * beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(beta / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] * N[(beta / N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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))) < -0.5

   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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. mul0-lft N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 86.4

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 86.4%

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

   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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 99.9

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

   7. Applied rewrites 99.9%

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

   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. Applied rewrites 99.9%

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

       1. lift-/.f64 N/A

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

       2. div-inv N/A

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

       3. metadata-eval N/A

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

       4. lower-*.f64 99.9

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

   12. Applied rewrites 99.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\beta}{\mathsf{fma}\left(2, i, \beta\right)}, \frac{\beta}{\beta + \mathsf{fma}\left(2, i, 2\right)}, 1\right) \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

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

Alternative 12: 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

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

   5. Applied rewrites 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 beta around inf

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

   5. Applied rewrites 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 13: 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

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

5. Applied rewrites 61.0%

   \[\leadsto \color{blue}{0.5} \]
6. 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))