Octave 3.8, jcobi/2

Percentage Accurate: 62.7% → 98.2%

Time: 13.0s

Alternatives: 14

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.7% 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: 98.2% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (+ beta 2.0))) (t_1 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ 2.0 t_1)) -0.5)
     (/
      (fma
       -0.5
       (* (+ beta 2.0) (/ t_0 alpha))
       (fma
        0.5
        (fma (+ beta 2.0) (* (+ beta 2.0) (/ t_0 (* alpha alpha))) t_0)
        (* 2.0 i)))
      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 = beta + (beta + 2.0);
	double t_1 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.5) {
		tmp = fma(-0.5, ((beta + 2.0) * (t_0 / alpha)), fma(0.5, fma((beta + 2.0), ((beta + 2.0) * (t_0 / (alpha * alpha))), t_0), (2.0 * i))) / 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(beta + Float64(beta + 2.0))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(2.0 + t_1)) <= -0.5)
		tmp = Float64(fma(-0.5, Float64(Float64(beta + 2.0) * Float64(t_0 / alpha)), fma(0.5, fma(Float64(beta + 2.0), Float64(Float64(beta + 2.0) * Float64(t_0 / Float64(alpha * alpha))), t_0), Float64(2.0 * i))) / 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[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(-0.5 * N[(N[(beta + 2.0), $MachinePrecision] * N[(t$95$0 / alpha), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(t$95$0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $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))) < -0.5

   1. Initial program 3.8%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Simplified 7.9%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-/.f64 36.3

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

   8. Simplified 36.3%

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

   9. Taylor expanded in alpha around inf

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

       1. lower-/.f64 N/A

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

   11. Simplified 89.0%

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

   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. Initial program 80.7%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 97.6%

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

Alternative 2: 96.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\beta + \mathsf{fma}\left(2, i, 1\right)}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 0.99999998:\\ \;\;\;\;\mathsf{fma}\left(\beta \cdot \frac{\beta}{\mathsf{fma}\left(2, i, \beta\right) \cdot \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\left(-2 - \alpha\right) - \alpha}{\beta}, 1\right)\\ \end{array} \end{array} \]

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)
     (/ (+ beta (fma 2.0 i 1.0)) alpha)
     (if (<= t_1 0.99999998)
       (fma
        (* beta (/ beta (* (fma 2.0 i beta) (+ beta (fma 2.0 i 2.0)))))
        0.5
        0.5)
       (fma 0.5 (/ (- (- -2.0 alpha) alpha) beta) 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 = (beta + fma(2.0, i, 1.0)) / alpha;
	} else if (t_1 <= 0.99999998) {
		tmp = fma((beta * (beta / (fma(2.0, i, beta) * (beta + fma(2.0, i, 2.0))))), 0.5, 0.5);
	} else {
		tmp = fma(0.5, (((-2.0 - alpha) - alpha) / beta), 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(beta + fma(2.0, i, 1.0)) / alpha);
	elseif (t_1 <= 0.99999998)
		tmp = fma(Float64(beta * Float64(beta / Float64(fma(2.0, i, beta) * Float64(beta + fma(2.0, i, 2.0))))), 0.5, 0.5);
	else
		tmp = fma(0.5, Float64(Float64(Float64(-2.0 - alpha) - alpha) / beta), 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[(beta + N[(2.0 * i + 1.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 0.99999998], N[(N[(beta * N[(beta / N[(N[(2.0 * i + beta), $MachinePrecision] * N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision], N[(0.5 * N[(N[(N[(-2.0 - alpha), $MachinePrecision] - alpha), $MachinePrecision] / beta), $MachinePrecision] + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 3.8%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Simplified 7.9%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 88.1

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

   8. Simplified 88.1%

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

   9. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

       2. associate-+l+ N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 88.1

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

   11. Simplified 88.1%

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

   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))) < 0.999999980000000011

   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 99.4

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

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\beta \cdot \beta}{\mathsf{fma}\left(2, i, \beta\right) \cdot \left(2 + \mathsf{fma}\left(2, i, \beta\right)\right)}, 0.5, 0.5\right)} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 99.4

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

      9. lift-+.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 99.4

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

   7. Applied egg-rr 99.4%

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

   if 0.999999980000000011 < (/.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 44.6%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. 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. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 93.5

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

   5. Simplified 93.5%

      \[\leadsto \color{blue}{0.5 + 0.5 \cdot \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. distribute-lft1-in N/A

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 93.6

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

   7. Applied egg-rr 93.6%

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

   8. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. metadata-eval 93.6

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

   10. Simplified 93.6%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{\beta + \mathsf{fma}\left(2, i, 1\right)}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 0.99999998:\\ \;\;\;\;\mathsf{fma}\left(\beta \cdot \frac{\beta}{\mathsf{fma}\left(2, i, \beta\right) \cdot \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\left(-2 - \alpha\right) - \alpha}{\beta}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\beta + \mathsf{fma}\left(2, i, 1\right)}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 10^{-40}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -0.5)
     (/ (+ beta (fma 2.0 i 1.0)) alpha)
     (if (<= t_1 1e-40)
       0.5
       (+ 0.5 (* 0.5 (/ (- beta alpha) (+ (+ alpha beta) 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 = (beta + fma(2.0, i, 1.0)) / alpha;
	} else if (t_1 <= 1e-40) {
		tmp = 0.5;
	} else {
		tmp = 0.5 + (0.5 * ((beta - alpha) / ((alpha + beta) + 2.0)));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0))
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = Float64(Float64(beta + fma(2.0, i, 1.0)) / alpha);
	elseif (t_1 <= 1e-40)
		tmp = 0.5;
	else
		tmp = Float64(0.5 + Float64(0.5 * Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0))));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, 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[(beta + N[(2.0 * i + 1.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 1e-40], 0.5, N[(0.5 + N[(0.5 * N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $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 3.8%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Simplified 7.9%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 88.1

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

   8. Simplified 88.1%

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

   9. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

       2. associate-+l+ N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 88.1

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

   11. Simplified 88.1%

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

   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))) < 9.9999999999999993e-41

   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. Simplified 100.0%

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

   if 9.9999999999999993e-41 < (/.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 53.2%

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

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(1 + \frac{\beta}{2 + \left(\alpha + \beta\right)}\right) - \frac{\alpha}{2 + \left(\alpha + \beta\right)}\right)} \]
   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. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 93.5

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

   5. Simplified 93.5%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{\beta + \mathsf{fma}\left(2, i, 1\right)}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 10^{-40}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\beta + \mathsf{fma}\left(2, i, 1\right)}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 10^{-40}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\beta + 2}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -0.5)
     (/ (+ beta (fma 2.0 i 1.0)) alpha)
     (if (<= t_1 1e-40) 0.5 (fma (/ beta (+ beta 2.0)) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -0.5) {
		tmp = (beta + fma(2.0, i, 1.0)) / alpha;
	} else if (t_1 <= 1e-40) {
		tmp = 0.5;
	} else {
		tmp = fma((beta / (beta + 2.0)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0))
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = Float64(Float64(beta + fma(2.0, i, 1.0)) / alpha);
	elseif (t_1 <= 1e-40)
		tmp = 0.5;
	else
		tmp = fma(Float64(beta / Float64(beta + 2.0)), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.5], N[(N[(beta + N[(2.0 * i + 1.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 1e-40], 0.5, N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 3.8%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Simplified 7.9%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 88.1

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

   8. Simplified 88.1%

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

   9. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

       2. associate-+l+ N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 88.1

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

   11. Simplified 88.1%

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

   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))) < 9.9999999999999993e-41

   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. Simplified 100.0%

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

   if 9.9999999999999993e-41 < (/.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 53.2%

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

   3. Taylor expanded in alpha around 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 50.5

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

   5. Simplified 50.5%

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

   6. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 91.8

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

   8. Simplified 91.8%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{\beta + \mathsf{fma}\left(2, i, 1\right)}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 10^{-40}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\beta + 2}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\beta + \mathsf{fma}\left(2, i, 1\right)}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + \mathsf{fma}\left(i, -2, -1\right)}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -0.5)
     (/ (+ beta (fma 2.0 i 1.0)) alpha)
     (if (<= t_1 0.1) 0.5 (/ (+ beta (fma i -2.0 -1.0)) beta)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -0.5) {
		tmp = (beta + fma(2.0, i, 1.0)) / alpha;
	} else if (t_1 <= 0.1) {
		tmp = 0.5;
	} else {
		tmp = (beta + fma(i, -2.0, -1.0)) / beta;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0))
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = Float64(Float64(beta + fma(2.0, i, 1.0)) / alpha);
	elseif (t_1 <= 0.1)
		tmp = 0.5;
	else
		tmp = Float64(Float64(beta + fma(i, -2.0, -1.0)) / beta);
	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[(beta + N[(2.0 * i + 1.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 0.1], 0.5, N[(N[(beta + N[(i * -2.0 + -1.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 3.8%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Simplified 7.9%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 88.1

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

   8. Simplified 88.1%

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

   9. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

       2. associate-+l+ N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 88.1

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

   11. Simplified 88.1%

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

   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))) < 0.10000000000000001

   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. Simplified 99.4%

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

   if 0.10000000000000001 < (/.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 46.8%

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

   3. Taylor expanded in alpha around 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 44.7

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

   5. Simplified 44.7%

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

   6. Taylor expanded in beta around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{2 + 4 \cdot i}{\beta} + 1} \]

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 91.1

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

   8. Simplified 91.1%

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

   9. Taylor expanded in beta around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt-in N/A

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

       5. metadata-eval N/A

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

       6. *-commutative N/A

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

       7. associate-*l* N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 91.1

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

   11. Simplified 91.1%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{\beta + \mathsf{fma}\left(2, i, 1\right)}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 0.1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + \mathsf{fma}\left(i, -2, -1\right)}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\beta + \mathsf{fma}\left(2, i, 1\right)}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -0.5)
     (/ (+ beta (fma 2.0 i 1.0)) alpha)
     (if (<= t_1 0.1) 0.5 (+ 1.0 (/ -1.0 beta))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -0.5) {
		tmp = (beta + fma(2.0, i, 1.0)) / alpha;
	} else if (t_1 <= 0.1) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0))
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = Float64(Float64(beta + fma(2.0, i, 1.0)) / alpha);
	elseif (t_1 <= 0.1)
		tmp = 0.5;
	else
		tmp = Float64(1.0 + Float64(-1.0 / beta));
	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[(beta + N[(2.0 * i + 1.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 0.1], 0.5, N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 3.8%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Simplified 7.9%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 88.1

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

   8. Simplified 88.1%

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

   9. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

       2. associate-+l+ N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 88.1

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

   11. Simplified 88.1%

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

   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))) < 0.10000000000000001

   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. Simplified 99.4%

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

   if 0.10000000000000001 < (/.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 46.8%

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

   3. Taylor expanded in alpha around 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 44.7

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

   5. Simplified 44.7%

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

   6. Taylor expanded in beta around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{2 + 4 \cdot i}{\beta} + 1} \]

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 91.1

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

   8. Simplified 91.1%

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

   9. Taylor expanded in i around 0

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

       1. sub-neg N/A

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

       2. distribute-neg-frac N/A

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

       3. metadata-eval N/A

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

       4. lower-+.f64 N/A

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

       5. lower-/.f64 90.7

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

   11. Simplified 90.7%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{\beta + \mathsf{fma}\left(2, i, 1\right)}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 0.1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, i, 1\right)}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -0.5)
     (/ (fma 2.0 i 1.0) alpha)
     (if (<= t_1 0.1) 0.5 (+ 1.0 (/ -1.0 beta))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -0.5) {
		tmp = fma(2.0, i, 1.0) / alpha;
	} else if (t_1 <= 0.1) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0))
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = Float64(fma(2.0, i, 1.0) / alpha);
	elseif (t_1 <= 0.1)
		tmp = 0.5;
	else
		tmp = Float64(1.0 + Float64(-1.0 / beta));
	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[(2.0 * i + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 0.1], 0.5, N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 3.8%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Simplified 7.9%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 88.1

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

   8. Simplified 88.1%

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

   9. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 74.8

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

   11. Simplified 74.8%

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

   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))) < 0.10000000000000001

   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. Simplified 99.4%

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

   if 0.10000000000000001 < (/.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 46.8%

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

   3. Taylor expanded in alpha around 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 44.7

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

   5. Simplified 44.7%

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

   6. Taylor expanded in beta around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{2 + 4 \cdot i}{\beta} + 1} \]

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 91.1

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

   8. Simplified 91.1%

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

   9. Taylor expanded in i around 0

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

       1. sub-neg N/A

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

       2. distribute-neg-frac N/A

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

       3. metadata-eval N/A

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

       4. lower-+.f64 N/A

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

       5. lower-/.f64 90.7

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

   11. Simplified 90.7%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, i, 1\right)}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 0.1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -0.5)
     (/ (+ beta 1.0) alpha)
     (if (<= t_1 0.1) 0.5 (+ 1.0 (/ -1.0 beta))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -0.5) {
		tmp = (beta + 1.0) / alpha;
	} else if (t_1 <= 0.1) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)
    if (t_1 <= (-0.5d0)) then
        tmp = (beta + 1.0d0) / alpha
    else if (t_1 <= 0.1d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 + ((-1.0d0) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -0.5) {
		tmp = (beta + 1.0) / alpha;
	} else if (t_1 <= 0.1) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)
	tmp = 0
	if t_1 <= -0.5:
		tmp = (beta + 1.0) / alpha
	elif t_1 <= 0.1:
		tmp = 0.5
	else:
		tmp = 1.0 + (-1.0 / beta)
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0))
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	elseif (t_1 <= 0.1)
		tmp = 0.5;
	else
		tmp = Float64(1.0 + Float64(-1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	tmp = 0.0;
	if (t_1 <= -0.5)
		tmp = (beta + 1.0) / alpha;
	elseif (t_1 <= 0.1)
		tmp = 0.5;
	else
		tmp = 1.0 + (-1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.5], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 0.1], 0.5, N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 3.8%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Simplified 7.9%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 88.1

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

   8. Simplified 88.1%

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

   9. Taylor expanded in i around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. metadata-eval N/A

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

       5. *-lft-identity N/A

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

       6. metadata-eval N/A

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

       7. lower-+.f64 56.9

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

   11. Simplified 56.9%

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

   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))) < 0.10000000000000001

   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. Simplified 99.4%

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

   if 0.10000000000000001 < (/.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 46.8%

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

   3. Taylor expanded in alpha around 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 44.7

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

   5. Simplified 44.7%

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

   6. Taylor expanded in beta around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{2 + 4 \cdot i}{\beta} + 1} \]

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 91.1

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

   8. Simplified 91.1%

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

   9. Taylor expanded in i around 0

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

       1. sub-neg N/A

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

       2. distribute-neg-frac N/A

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

       3. metadata-eval N/A

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

       4. lower-+.f64 N/A

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

       5. lower-/.f64 90.7

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

   11. Simplified 90.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 0.1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.2% accurate, 0.6× speedup

Math

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

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))
        -0.9999995)
     (/ (+ beta (fma 2.0 i 1.0)) 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)) <= -0.9999995) {
		tmp = (beta + fma(2.0, i, 1.0)) / 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)) <= -0.9999995)
		tmp = Float64(Float64(beta + fma(2.0, i, 1.0)) / 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], -0.9999995], N[(N[(beta + N[(2.0 * i + 1.0), $MachinePrecision]), $MachinePrecision] / alpha), $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))) < -0.999999500000000041

   1. Initial program 2.6%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Simplified 6.7%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 88.8

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

   8. Simplified 88.8%

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

   9. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

       2. associate-+l+ N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 88.8

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

   11. Simplified 88.8%

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

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

   1. Initial program 80.6%

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 97.5%

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

Alternative 10: 97.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 3.8%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Simplified 7.9%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 88.1

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

   8. Simplified 88.1%

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

   9. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

       2. associate-+l+ N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 88.1

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

   11. Simplified 88.1%

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

   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. Initial program 80.7%

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

   3. Taylor expanded in 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 79.5

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

   5. Simplified 79.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\beta \cdot \beta}{\mathsf{fma}\left(2, i, \beta\right) \cdot \left(2 + \mathsf{fma}\left(2, i, \beta\right)\right)}, 0.5, 0.5\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   7. Applied egg-rr 99.3%

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

4. Final simplification 96.9%

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

Alternative 11: 76.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq 0.1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) 0.1)
     0.5
     (+ 1.0 (/ -1.0 beta)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= 0.1) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)) <= 0.1d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 + ((-1.0d0) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= 0.1) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= 0.1:
		tmp = 0.5
	else:
		tmp = 1.0 + (-1.0 / beta)
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= 0.1)
		tmp = 0.5;
	else
		tmp = Float64(1.0 + Float64(-1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= 0.1)
		tmp = 0.5;
	else
		tmp = 1.0 + (-1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], 0.1], 0.5, N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 0.10000000000000001

   1. Initial program 71.1%

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

   3. Taylor expanded in i around inf

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

   5. Simplified 73.9%

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

   if 0.10000000000000001 < (/.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 46.8%

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

   3. Taylor expanded in alpha around 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 44.7

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

   5. Simplified 44.7%

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

   6. Taylor expanded in beta around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{2 + 4 \cdot i}{\beta} + 1} \]

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 91.1

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

   8. Simplified 91.1%

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

   9. Taylor expanded in i around 0

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

       1. sub-neg N/A

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

       2. distribute-neg-frac N/A

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

       3. metadata-eval N/A

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

       4. lower-+.f64 N/A

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

       5. lower-/.f64 90.7

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

   11. Simplified 90.7%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 0.1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq 0.1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 0.1)
     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)) <= 0.1) {
		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)) <= 0.1d0) 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)) <= 0.1) {
		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)) <= 0.1:
		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)) <= 0.1)
		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)) <= 0.1)
		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], 0.1], 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))) < 0.10000000000000001

   1. Initial program 71.1%

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

   3. Taylor expanded in i around inf

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

   5. Simplified 73.9%

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

   if 0.10000000000000001 < (/.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 46.8%

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

   3. Taylor expanded in beta around inf

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

   5. Simplified 90.7%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 0.1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.5% 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 64.2%

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

3. Taylor expanded in i around inf

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

5. Simplified 59.8%

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

Alternative 14: 3.5% accurate, 73.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 0.0)

C

double code(double alpha, double beta, double i) {
	return 0.0;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.0d0
end function

Java

public static double code(double alpha, double beta, double i) {
	return 0.0;
}

Python

def code(alpha, beta, i):
	return 0.0

Julia

function code(alpha, beta, i)
	return 0.0
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 0.0;
end

Wolfram

code[alpha_, beta_, i_] := 0.0

Derivation

1. Initial program 64.2%

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

3. Taylor expanded in i around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(1 + \frac{\beta}{2 + \left(\alpha + \beta\right)}\right) - \frac{\alpha}{2 + \left(\alpha + \beta\right)}\right)} \]
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. lower-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 70.2

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

5. Simplified 70.2%

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

6. Taylor expanded in alpha around inf

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

8. Simplified 3.5%

   \[\leadsto 0.5 + \color{blue}{-0.5} \]
9. Step-by-step derivation

   1. metadata-eval 3.5

      \[\leadsto \color{blue}{0} \]

10. Applied egg-rr 3.5%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024208 
(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))