Octave 3.8, jcobi/2

Percentage Accurate: 62.7% → 97.6%

Time: 11.0s

Alternatives: 16

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: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2} \leq -0.999:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta + \alpha}{t\_1}, \frac{\beta - \alpha}{t\_1 + 2}, 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha))) (t_1 (fma i 2.0 (+ beta alpha))))
   (if (<= (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0)) -0.999)
     (* 0.5 (/ (+ (fma 4.0 i (* 2.0 beta)) 2.0) alpha))
     (* (fma (/ (+ beta alpha) t_1) (/ (- beta alpha) (+ t_1 2.0)) 1.0) 0.5))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = fma(i, 2.0, (beta + alpha));
	double tmp;
	if (((((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0)) <= -0.999) {
		tmp = 0.5 * ((fma(4.0, i, (2.0 * beta)) + 2.0) / alpha);
	} else {
		tmp = fma(((beta + alpha) / t_1), ((beta - alpha) / (t_1 + 2.0)), 1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = fma(i, 2.0, Float64(beta + alpha))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0)) <= -0.999)
		tmp = Float64(0.5 * Float64(Float64(fma(4.0, i, Float64(2.0 * beta)) + 2.0) / alpha));
	else
		tmp = Float64(fma(Float64(Float64(beta + alpha) / t_1), Float64(Float64(beta - alpha) / Float64(t_1 + 2.0)), 1.0) * 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision], -0.999], N[(0.5 * N[(N[(N[(4.0 * i + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(beta + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $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))) < -0.998999999999999999

   1. Initial program 2.4%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 94.0

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

   5. Applied rewrites 94.0%

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

   if -0.998999999999999999 < (/.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 77.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.999:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta + \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 2}, 1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.999:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha}{\mathsf{fma}\left(i, 2, \alpha\right)}, \frac{\alpha}{-2 - \mathsf{fma}\left(i, 2, \alpha\right)}, 1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\frac{2 + \left(\beta + \alpha\right)}{\beta - \alpha}} + 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.999)
     (* 0.5 (/ (+ (fma 4.0 i (* 2.0 beta)) 2.0) alpha))
     (if (<= t_1 2e-25)
       (*
        (fma
         (/ alpha (fma i 2.0 alpha))
         (/ alpha (- -2.0 (fma i 2.0 alpha)))
         1.0)
        0.5)
       (* (+ (/ 1.0 (/ (+ 2.0 (+ beta alpha)) (- beta alpha))) 1.0) 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.999) {
		tmp = 0.5 * ((fma(4.0, i, (2.0 * beta)) + 2.0) / alpha);
	} else if (t_1 <= 2e-25) {
		tmp = fma((alpha / fma(i, 2.0, alpha)), (alpha / (-2.0 - fma(i, 2.0, alpha))), 1.0) * 0.5;
	} else {
		tmp = ((1.0 / ((2.0 + (beta + alpha)) / (beta - alpha))) + 1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.999)
		tmp = Float64(0.5 * Float64(Float64(fma(4.0, i, Float64(2.0 * beta)) + 2.0) / alpha));
	elseif (t_1 <= 2e-25)
		tmp = Float64(fma(Float64(alpha / fma(i, 2.0, alpha)), Float64(alpha / Float64(-2.0 - fma(i, 2.0, alpha))), 1.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(2.0 + Float64(beta + alpha)) / Float64(beta - alpha))) + 1.0) * 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.999], N[(0.5 * N[(N[(N[(4.0 * i + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-25], N[(N[(N[(alpha / N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision] * N[(alpha / N[(-2.0 - N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(1.0 / N[(N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 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))) < -0.998999999999999999

   1. Initial program 2.4%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 94.0

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

   5. Applied rewrites 94.0%

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

   if -0.998999999999999999 < (/.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))) < 2.00000000000000008e-25

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 99.2

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

   5. Applied rewrites 99.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 99.2

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

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\alpha}{\mathsf{fma}\left(-1, \mathsf{fma}\left(2, i, \alpha\right), -2\right)}, \frac{\alpha}{\mathsf{fma}\left(2, i, \alpha\right)}, 1\right) \cdot 0.5} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.2%

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

   if 2.00000000000000008e-25 < (/.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 24.1%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\left(1 + \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.0%

      \[\leadsto \left(1 + \frac{1}{\frac{2 + \left(\alpha + \beta\right)}{\beta - \alpha}}\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.999:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha}{\mathsf{fma}\left(i, 2, \alpha\right)}, \frac{\alpha}{-2 - \mathsf{fma}\left(i, 2, \alpha\right)}, 1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\frac{2 + \left(\beta + \alpha\right)}{\beta - \alpha}} + 1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.999:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\alpha \cdot \alpha}{\left(-2 - \mathsf{fma}\left(i, 2, \alpha\right)\right) \cdot \mathsf{fma}\left(i, 2, \alpha\right)}, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\frac{2 + \left(\beta + \alpha\right)}{\beta - \alpha}} + 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.999)
     (* 0.5 (/ (+ (fma 4.0 i (* 2.0 beta)) 2.0) alpha))
     (if (<= t_1 2e-25)
       (fma
        0.5
        (/ (* alpha alpha) (* (- -2.0 (fma i 2.0 alpha)) (fma i 2.0 alpha)))
        0.5)
       (* (+ (/ 1.0 (/ (+ 2.0 (+ beta alpha)) (- beta alpha))) 1.0) 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.999) {
		tmp = 0.5 * ((fma(4.0, i, (2.0 * beta)) + 2.0) / alpha);
	} else if (t_1 <= 2e-25) {
		tmp = fma(0.5, ((alpha * alpha) / ((-2.0 - fma(i, 2.0, alpha)) * fma(i, 2.0, alpha))), 0.5);
	} else {
		tmp = ((1.0 / ((2.0 + (beta + alpha)) / (beta - alpha))) + 1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.999)
		tmp = Float64(0.5 * Float64(Float64(fma(4.0, i, Float64(2.0 * beta)) + 2.0) / alpha));
	elseif (t_1 <= 2e-25)
		tmp = fma(0.5, Float64(Float64(alpha * alpha) / Float64(Float64(-2.0 - fma(i, 2.0, alpha)) * fma(i, 2.0, alpha))), 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(2.0 + Float64(beta + alpha)) / Float64(beta - alpha))) + 1.0) * 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.999], N[(0.5 * N[(N[(N[(4.0 * i + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-25], N[(0.5 * N[(N[(alpha * alpha), $MachinePrecision] / N[(N[(-2.0 - N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision] * N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(1.0 / N[(N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 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))) < -0.998999999999999999

   1. Initial program 2.4%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 94.0

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

   5. Applied rewrites 94.0%

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

   if -0.998999999999999999 < (/.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))) < 2.00000000000000008e-25

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   if 2.00000000000000008e-25 < (/.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 24.1%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\left(1 + \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.0%

      \[\leadsto \left(1 + \frac{1}{\frac{2 + \left(\alpha + \beta\right)}{\beta - \alpha}}\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.999:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\alpha \cdot \alpha}{\left(-2 - \mathsf{fma}\left(i, 2, \alpha\right)\right) \cdot \mathsf{fma}\left(i, 2, \alpha\right)}, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\frac{2 + \left(\beta + \alpha\right)}{\beta - \alpha}} + 1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := t\_0 + 2\\ t_2 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_1}\\ \mathbf{if}\;t\_2 \leq -0.999:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2}{\alpha}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{-\alpha}{t\_1} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\frac{2 + \left(\beta + \alpha\right)}{\beta - \alpha}} + 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (+ t_0 2.0))
        (t_2 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) t_1)))
   (if (<= t_2 -0.999)
     (* 0.5 (/ (+ (fma 4.0 i (* 2.0 beta)) 2.0) alpha))
     (if (<= t_2 2e-25)
       (/ (+ (/ (- alpha) t_1) 1.0) 2.0)
       (* (+ (/ 1.0 (/ (+ 2.0 (+ beta alpha)) (- beta alpha))) 1.0) 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = t_0 + 2.0;
	double t_2 = (((beta - alpha) * (beta + alpha)) / t_0) / t_1;
	double tmp;
	if (t_2 <= -0.999) {
		tmp = 0.5 * ((fma(4.0, i, (2.0 * beta)) + 2.0) / alpha);
	} else if (t_2 <= 2e-25) {
		tmp = ((-alpha / t_1) + 1.0) / 2.0;
	} else {
		tmp = ((1.0 / ((2.0 + (beta + alpha)) / (beta - alpha))) + 1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(t_0 + 2.0)
	t_2 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / t_1)
	tmp = 0.0
	if (t_2 <= -0.999)
		tmp = Float64(0.5 * Float64(Float64(fma(4.0, i, Float64(2.0 * beta)) + 2.0) / alpha));
	elseif (t_2 <= 2e-25)
		tmp = Float64(Float64(Float64(Float64(-alpha) / t_1) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(2.0 + Float64(beta + alpha)) / Float64(beta - alpha))) + 1.0) * 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -0.999], N[(0.5 * N[(N[(N[(4.0 * i + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-25], N[(N[(N[((-alpha) / t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 / N[(N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 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))) < -0.998999999999999999

   1. Initial program 2.4%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 94.0

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

   5. Applied rewrites 94.0%

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

   if -0.998999999999999999 < (/.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))) < 2.00000000000000008e-25

   1. Initial program 99.9%

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

   3. Taylor expanded in alpha around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 2.00000000000000008e-25 < (/.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 24.1%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\left(1 + \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.0%

      \[\leadsto \left(1 + \frac{1}{\frac{2 + \left(\alpha + \beta\right)}{\beta - \alpha}}\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.999:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{-\alpha}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\frac{2 + \left(\beta + \alpha\right)}{\beta - \alpha}} + 1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.999:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha}{\mathsf{fma}\left(-1, \mathsf{fma}\left(2, i, \alpha\right), -2\right)}, 1, 1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\frac{2 + \left(\beta + \alpha\right)}{\beta - \alpha}} + 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.999)
     (* 0.5 (/ (+ (fma 4.0 i (* 2.0 beta)) 2.0) alpha))
     (if (<= t_1 2e-25)
       (* (fma (/ alpha (fma -1.0 (fma 2.0 i alpha) -2.0)) 1.0 1.0) 0.5)
       (* (+ (/ 1.0 (/ (+ 2.0 (+ beta alpha)) (- beta alpha))) 1.0) 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.999) {
		tmp = 0.5 * ((fma(4.0, i, (2.0 * beta)) + 2.0) / alpha);
	} else if (t_1 <= 2e-25) {
		tmp = fma((alpha / fma(-1.0, fma(2.0, i, alpha), -2.0)), 1.0, 1.0) * 0.5;
	} else {
		tmp = ((1.0 / ((2.0 + (beta + alpha)) / (beta - alpha))) + 1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.999)
		tmp = Float64(0.5 * Float64(Float64(fma(4.0, i, Float64(2.0 * beta)) + 2.0) / alpha));
	elseif (t_1 <= 2e-25)
		tmp = Float64(fma(Float64(alpha / fma(-1.0, fma(2.0, i, alpha), -2.0)), 1.0, 1.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(2.0 + Float64(beta + alpha)) / Float64(beta - alpha))) + 1.0) * 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.999], N[(0.5 * N[(N[(N[(4.0 * i + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-25], N[(N[(N[(alpha / N[(-1.0 * N[(2.0 * i + alpha), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] * 1.0 + 1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(1.0 / N[(N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 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))) < -0.998999999999999999

   1. Initial program 2.4%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 94.0

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

   5. Applied rewrites 94.0%

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

   if -0.998999999999999999 < (/.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))) < 2.00000000000000008e-25

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 99.2

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

   5. Applied rewrites 99.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 99.2

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 98.4%

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

   if 2.00000000000000008e-25 < (/.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 24.1%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\left(1 + \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.0%

      \[\leadsto \left(1 + \frac{1}{\frac{2 + \left(\alpha + \beta\right)}{\beta - \alpha}}\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.999:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha}{\mathsf{fma}\left(-1, \mathsf{fma}\left(2, i, \alpha\right), -2\right)}, 1, 1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\frac{2 + \left(\beta + \alpha\right)}{\beta - \alpha}} + 1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.999:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha}{\mathsf{fma}\left(-1, \mathsf{fma}\left(2, i, \alpha\right), -2\right)}, 1, 1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.999)
     (* 0.5 (/ (+ (fma 4.0 i (* 2.0 beta)) 2.0) alpha))
     (if (<= t_1 2e-25)
       (* (fma (/ alpha (fma -1.0 (fma 2.0 i alpha) -2.0)) 1.0 1.0) 0.5)
       (fma (/ (- beta alpha) (+ 2.0 (+ beta alpha))) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.999) {
		tmp = 0.5 * ((fma(4.0, i, (2.0 * beta)) + 2.0) / alpha);
	} else if (t_1 <= 2e-25) {
		tmp = fma((alpha / fma(-1.0, fma(2.0, i, alpha), -2.0)), 1.0, 1.0) * 0.5;
	} else {
		tmp = fma(((beta - alpha) / (2.0 + (beta + alpha))), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.999)
		tmp = Float64(0.5 * Float64(Float64(fma(4.0, i, Float64(2.0 * beta)) + 2.0) / alpha));
	elseif (t_1 <= 2e-25)
		tmp = Float64(fma(Float64(alpha / fma(-1.0, fma(2.0, i, alpha), -2.0)), 1.0, 1.0) * 0.5);
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha))), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.999], N[(0.5 * N[(N[(N[(4.0 * i + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-25], N[(N[(N[(alpha / N[(-1.0 * N[(2.0 * i + alpha), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] * 1.0 + 1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $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))) < -0.998999999999999999

   1. Initial program 2.4%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 94.0

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

   5. Applied rewrites 94.0%

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

   if -0.998999999999999999 < (/.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))) < 2.00000000000000008e-25

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 99.2

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

   5. Applied rewrites 99.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 99.2

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 98.4%

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

   if 2.00000000000000008e-25 < (/.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 24.1%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\left(1 + \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.0%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.999:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha}{\mathsf{fma}\left(-1, \mathsf{fma}\left(2, i, \alpha\right), -2\right)}, 1, 1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-25}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.5)
     (* 0.5 (/ (+ (fma 4.0 i (* 2.0 beta)) 2.0) alpha))
     (if (<= t_1 2e-25)
       0.5
       (fma (/ (- beta alpha) (+ 2.0 (+ beta alpha))) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.5) {
		tmp = 0.5 * ((fma(4.0, i, (2.0 * beta)) + 2.0) / alpha);
	} else if (t_1 <= 2e-25) {
		tmp = 0.5;
	} else {
		tmp = fma(((beta - alpha) / (2.0 + (beta + alpha))), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = Float64(0.5 * Float64(Float64(fma(4.0, i, Float64(2.0 * beta)) + 2.0) / alpha));
	elseif (t_1 <= 2e-25)
		tmp = 0.5;
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha))), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.5], N[(0.5 * N[(N[(N[(4.0 * i + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-25], 0.5, N[(N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $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 5.5%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. neg-sub0 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 91.7

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

   5. Applied rewrites 91.7%

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

   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))) < 2.00000000000000008e-25

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 99.0%

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

   if 2.00000000000000008e-25 < (/.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 24.1%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\left(1 + \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.0%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.5:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 2 \cdot 10^{-25}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, 4, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-25}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.8)
     (* (/ (fma i 4.0 2.0) alpha) 0.5)
     (if (<= t_1 2e-25)
       0.5
       (fma (/ (- beta alpha) (+ 2.0 (+ beta alpha))) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.8) {
		tmp = (fma(i, 4.0, 2.0) / alpha) * 0.5;
	} else if (t_1 <= 2e-25) {
		tmp = 0.5;
	} else {
		tmp = fma(((beta - alpha) / (2.0 + (beta + alpha))), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.8)
		tmp = Float64(Float64(fma(i, 4.0, 2.0) / alpha) * 0.5);
	elseif (t_1 <= 2e-25)
		tmp = 0.5;
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha))), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.8], N[(N[(N[(i * 4.0 + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 2e-25], 0.5, N[(N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $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))) < -0.80000000000000004

   1. Initial program 3.9%

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

   3. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 12.9

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

   5. Applied rewrites 12.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 12.9

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

   7. Applied rewrites 12.9%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 80.7%

       \[\leadsto \frac{2 - -4 \cdot i}{\color{blue}{\alpha}} \cdot 0.5 \]
   11. Step-by-step derivation

   12. Applied rewrites 80.7%

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

   if -0.80000000000000004 < (/.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))) < 2.00000000000000008e-25

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 98.4%

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

   if 2.00000000000000008e-25 < (/.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 24.1%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\left(1 + \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.0%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, 4, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 2 \cdot 10^{-25}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, 4, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-25}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\beta - \alpha, \frac{0.5}{\left(2 + \beta\right) + \alpha}, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.8)
     (* (/ (fma i 4.0 2.0) alpha) 0.5)
     (if (<= t_1 2e-25)
       0.5
       (fma (- beta alpha) (/ 0.5 (+ (+ 2.0 beta) alpha)) 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.8) {
		tmp = (fma(i, 4.0, 2.0) / alpha) * 0.5;
	} else if (t_1 <= 2e-25) {
		tmp = 0.5;
	} else {
		tmp = fma((beta - alpha), (0.5 / ((2.0 + beta) + alpha)), 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.8)
		tmp = Float64(Float64(fma(i, 4.0, 2.0) / alpha) * 0.5);
	elseif (t_1 <= 2e-25)
		tmp = 0.5;
	else
		tmp = fma(Float64(beta - alpha), Float64(0.5 / Float64(Float64(2.0 + beta) + alpha)), 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.8], N[(N[(N[(i * 4.0 + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 2e-25], 0.5, N[(N[(beta - alpha), $MachinePrecision] * N[(0.5 / N[(N[(2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision] + 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))) < -0.80000000000000004

   1. Initial program 3.9%

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

   3. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 12.9

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

   5. Applied rewrites 12.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 12.9

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

   7. Applied rewrites 12.9%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 80.7%

       \[\leadsto \frac{2 - -4 \cdot i}{\color{blue}{\alpha}} \cdot 0.5 \]
   11. Step-by-step derivation

   12. Applied rewrites 80.7%

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

   if -0.80000000000000004 < (/.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))) < 2.00000000000000008e-25

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 98.4%

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

   if 2.00000000000000008e-25 < (/.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 24.1%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\left(1 + \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.0%

      \[\leadsto \left(1 + \frac{1}{\frac{2 + \left(\alpha + \beta\right)}{\beta - \alpha}}\right) \cdot 0.5 \]
   8. Step-by-step derivation

   9. Applied rewrites 98.0%

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

   11. Applied rewrites 97.9%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, 4, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 2 \cdot 10^{-25}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\beta - \alpha, \frac{0.5}{\left(2 + \beta\right) + \alpha}, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, 4, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-19}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, \alpha, 2\right)}{\beta}, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.8)
     (* (/ (fma i 4.0 2.0) alpha) 0.5)
     (if (<= t_1 5e-19) 0.5 (fma (/ (fma 2.0 alpha 2.0) beta) -0.5 1.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.8) {
		tmp = (fma(i, 4.0, 2.0) / alpha) * 0.5;
	} else if (t_1 <= 5e-19) {
		tmp = 0.5;
	} else {
		tmp = fma((fma(2.0, alpha, 2.0) / beta), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.8)
		tmp = Float64(Float64(fma(i, 4.0, 2.0) / alpha) * 0.5);
	elseif (t_1 <= 5e-19)
		tmp = 0.5;
	else
		tmp = fma(Float64(fma(2.0, alpha, 2.0) / beta), -0.5, 1.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.8], N[(N[(N[(i * 4.0 + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 5e-19], 0.5, N[(N[(N[(2.0 * alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision] * -0.5 + 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))) < -0.80000000000000004

   1. Initial program 3.9%

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

   3. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 12.9

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

   5. Applied rewrites 12.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 12.9

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

   7. Applied rewrites 12.9%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 80.7%

       \[\leadsto \frac{2 - -4 \cdot i}{\color{blue}{\alpha}} \cdot 0.5 \]
   11. Step-by-step derivation

   12. Applied rewrites 80.7%

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

   if -0.80000000000000004 < (/.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))) < 5.0000000000000004e-19

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 98.4%

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

   if 5.0000000000000004e-19 < (/.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 21.5%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in beta around inf

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

   8. Applied rewrites 96.7%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, 4, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 5 \cdot 10^{-19}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, \alpha, 2\right)}{\beta}, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, 4, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-19}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2 \cdot \alpha}{\beta}, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.8)
     (* (/ (fma i 4.0 2.0) alpha) 0.5)
     (if (<= t_1 5e-19) 0.5 (fma (/ (* 2.0 alpha) beta) -0.5 1.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.8) {
		tmp = (fma(i, 4.0, 2.0) / alpha) * 0.5;
	} else if (t_1 <= 5e-19) {
		tmp = 0.5;
	} else {
		tmp = fma(((2.0 * alpha) / beta), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.8)
		tmp = Float64(Float64(fma(i, 4.0, 2.0) / alpha) * 0.5);
	elseif (t_1 <= 5e-19)
		tmp = 0.5;
	else
		tmp = fma(Float64(Float64(2.0 * alpha) / beta), -0.5, 1.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.8], N[(N[(N[(i * 4.0 + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 5e-19], 0.5, N[(N[(N[(2.0 * alpha), $MachinePrecision] / beta), $MachinePrecision] * -0.5 + 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))) < -0.80000000000000004

   1. Initial program 3.9%

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

   3. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 12.9

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

   5. Applied rewrites 12.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 12.9

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

   7. Applied rewrites 12.9%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 80.7%

       \[\leadsto \frac{2 - -4 \cdot i}{\color{blue}{\alpha}} \cdot 0.5 \]
   11. Step-by-step derivation

   12. Applied rewrites 80.7%

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

   if -0.80000000000000004 < (/.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))) < 5.0000000000000004e-19

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 98.4%

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

   if 5.0000000000000004e-19 < (/.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 21.5%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in beta around inf

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

   8. Applied rewrites 96.7%

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

   9. Taylor expanded in alpha around inf

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

   11. Applied rewrites 96.7%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, 4, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 5 \cdot 10^{-19}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2 \cdot \alpha}{\beta}, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 87.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, 2, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-19}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2 \cdot \alpha}{\beta}, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.5)
     (* (/ (fma beta 2.0 2.0) alpha) 0.5)
     (if (<= t_1 5e-19) 0.5 (fma (/ (* 2.0 alpha) beta) -0.5 1.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.5) {
		tmp = (fma(beta, 2.0, 2.0) / alpha) * 0.5;
	} else if (t_1 <= 5e-19) {
		tmp = 0.5;
	} else {
		tmp = fma(((2.0 * alpha) / beta), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = Float64(Float64(fma(beta, 2.0, 2.0) / alpha) * 0.5);
	elseif (t_1 <= 5e-19)
		tmp = 0.5;
	else
		tmp = fma(Float64(Float64(2.0 * alpha) / beta), -0.5, 1.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.5], N[(N[(N[(beta * 2.0 + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 5e-19], 0.5, N[(N[(N[(2.0 * alpha), $MachinePrecision] / beta), $MachinePrecision] * -0.5 + 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 5.5%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 7.3

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

   5. Applied rewrites 7.3%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 65.8%

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

   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))) < 5.0000000000000004e-19

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 99.0%

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

   if 5.0000000000000004e-19 < (/.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 21.5%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in beta around inf

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

   8. Applied rewrites 96.7%

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

   9. Taylor expanded in alpha around inf

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

   11. Applied rewrites 96.7%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, 2, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 5 \cdot 10^{-19}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2 \cdot \alpha}{\beta}, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.8:\\ \;\;\;\;\frac{2}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-19}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2 \cdot \alpha}{\beta}, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.8)
     (* (/ 2.0 alpha) 0.5)
     (if (<= t_1 5e-19) 0.5 (fma (/ (* 2.0 alpha) beta) -0.5 1.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.8) {
		tmp = (2.0 / alpha) * 0.5;
	} else if (t_1 <= 5e-19) {
		tmp = 0.5;
	} else {
		tmp = fma(((2.0 * alpha) / beta), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.8)
		tmp = Float64(Float64(2.0 / alpha) * 0.5);
	elseif (t_1 <= 5e-19)
		tmp = 0.5;
	else
		tmp = fma(Float64(Float64(2.0 * alpha) / beta), -0.5, 1.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.8], N[(N[(2.0 / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 5e-19], 0.5, N[(N[(N[(2.0 * alpha), $MachinePrecision] / beta), $MachinePrecision] * -0.5 + 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))) < -0.80000000000000004

   1. Initial program 3.9%

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

   3. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 12.9

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

   5. Applied rewrites 12.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 12.9

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

   7. Applied rewrites 12.9%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 80.7%

       \[\leadsto \frac{2 - -4 \cdot i}{\color{blue}{\alpha}} \cdot 0.5 \]

   11. Taylor expanded in i around 0

       \[\leadsto \frac{2}{\alpha} \cdot \frac{1}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 54.2%

       \[\leadsto \frac{2}{\alpha} \cdot 0.5 \]

   if -0.80000000000000004 < (/.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))) < 5.0000000000000004e-19

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 98.4%

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

   if 5.0000000000000004e-19 < (/.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 21.5%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in beta around inf

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

   8. Applied rewrites 96.7%

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

   9. Taylor expanded in alpha around inf

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

   11. Applied rewrites 96.7%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.8:\\ \;\;\;\;\frac{2}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 5 \cdot 10^{-19}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2 \cdot \alpha}{\beta}, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 84.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.8:\\ \;\;\;\;\frac{2}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-19}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.8) (* (/ 2.0 alpha) 0.5) (if (<= t_1 5e-19) 0.5 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.8) {
		tmp = (2.0 / alpha) * 0.5;
	} else if (t_1 <= 5e-19) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (i * 2.0d0) + (beta + alpha)
    t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0d0)
    if (t_1 <= (-0.8d0)) then
        tmp = (2.0d0 / alpha) * 0.5d0
    else if (t_1 <= 5d-19) 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 = (i * 2.0) + (beta + alpha);
	double t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.8) {
		tmp = (2.0 / alpha) * 0.5;
	} else if (t_1 <= 5e-19) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (i * 2.0) + (beta + alpha)
	t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0)
	tmp = 0
	if t_1 <= -0.8:
		tmp = (2.0 / alpha) * 0.5
	elif t_1 <= 5e-19:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.8)
		tmp = Float64(Float64(2.0 / alpha) * 0.5);
	elseif (t_1 <= 5e-19)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (i * 2.0) + (beta + alpha);
	t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	tmp = 0.0;
	if (t_1 <= -0.8)
		tmp = (2.0 / alpha) * 0.5;
	elseif (t_1 <= 5e-19)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.8], N[(N[(2.0 / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 5e-19], 0.5, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.80000000000000004

   1. Initial program 3.9%

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

   3. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 12.9

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

   5. Applied rewrites 12.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 12.9

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

   7. Applied rewrites 12.9%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 80.7%

       \[\leadsto \frac{2 - -4 \cdot i}{\color{blue}{\alpha}} \cdot 0.5 \]

   11. Taylor expanded in i around 0

       \[\leadsto \frac{2}{\alpha} \cdot \frac{1}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 54.2%

       \[\leadsto \frac{2}{\alpha} \cdot 0.5 \]

   if -0.80000000000000004 < (/.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))) < 5.0000000000000004e-19

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 98.4%

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

   if 5.0000000000000004e-19 < (/.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 21.5%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 96.6%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.8:\\ \;\;\;\;\frac{2}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 5 \cdot 10^{-19}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 77.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.3%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 73.2%

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

   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 21.5%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 96.6%

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

4. Final simplification 78.5%

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

Alternative 16: 61.8% 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 60.8%

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

3. Taylor expanded in i around inf

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

5. Applied rewrites 61.2%

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

Reproduce

herbie shell --seed 2024270 
(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))