Octave 3.8, jcobi/2

Percentage Accurate: 62.3% → 96.9%

Time: 9.9s

Alternatives: 13

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 0.6× speedup

Math

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

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 (fma 4.0 i (fma 2.0 beta 2.0))) alpha)
     (*
      (fma (/ beta (+ (fma 2.0 i beta) 2.0)) (/ beta (fma 2.0 i beta)) 1.0)
      0.5))))

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 * fma(4.0, i, fma(2.0, beta, 2.0))) / alpha;
	} else {
		tmp = fma((beta / (fma(2.0, i, beta) + 2.0)), (beta / fma(2.0, i, beta)), 1.0) * 0.5;
	}
	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 = Float64(Float64(0.5 * fma(4.0, i, fma(2.0, beta, 2.0))) / alpha);
	else
		tmp = Float64(fma(Float64(beta / Float64(fma(2.0, i, beta) + 2.0)), Float64(beta / fma(2.0, i, beta)), 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]}, 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], N[(N[(0.5 * N[(4.0 * i + N[(2.0 * beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta / N[(N[(2.0 * i + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(beta / N[(2.0 * i + beta), $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))) < -0.5

   1. Initial program 5.0%

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

   3. Taylor expanded in alpha around inf

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

      1. *-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. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

   7. Applied rewrites 94.8%

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

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

   1. Initial program 82.1%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 98.6

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

   5. Applied rewrites 98.6%

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

4. Final simplification 97.7%

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

Alternative 2: 95.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.999999995:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{fma}\left(4, i, \mathsf{fma}\left(2, \beta, 2\right)\right)}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha \cdot \alpha}{\left(-2 - \mathsf{fma}\left(2, i, \alpha\right)\right) \cdot \mathsf{fma}\left(2, i, \alpha\right)}, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

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 -0.999999995)
     (/ (* 0.5 (fma 4.0 i (fma 2.0 beta 2.0))) alpha)
     (if (<= t_1 4e-35)
       (fma
        (/ (* alpha alpha) (* (- -2.0 (fma 2.0 i alpha)) (fma 2.0 i alpha)))
        0.5
        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 <= -0.999999995) {
		tmp = (0.5 * fma(4.0, i, fma(2.0, beta, 2.0))) / alpha;
	} else if (t_1 <= 4e-35) {
		tmp = fma(((alpha * alpha) / ((-2.0 - fma(2.0, i, alpha)) * fma(2.0, i, alpha))), 0.5, 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 <= -0.999999995)
		tmp = Float64(Float64(0.5 * fma(4.0, i, fma(2.0, beta, 2.0))) / alpha);
	elseif (t_1 <= 4e-35)
		tmp = fma(Float64(Float64(alpha * alpha) / Float64(Float64(-2.0 - fma(2.0, i, alpha)) * fma(2.0, i, alpha))), 0.5, 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, -0.999999995], N[(N[(0.5 * N[(4.0 * i + N[(2.0 * beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 4e-35], N[(N[(N[(alpha * alpha), $MachinePrecision] / N[(N[(-2.0 - N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision] * N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 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))) < -0.99999999500000003

   1. Initial program 2.9%

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

   3. Taylor expanded in alpha around inf

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

      1. *-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. lower-*.f64 96.2

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

   5. Applied rewrites 96.2%

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

   7. Applied rewrites 96.2%

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

   if -0.99999999500000003 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 4.00000000000000003e-35

   1. Initial program 99.5%

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

   3. Taylor expanded in beta around 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-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 98.2%

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

   if 4.00000000000000003e-35 < (/.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 54.1%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 95.4

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

   5. Applied rewrites 95.4%

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

   7. Applied rewrites 95.4%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.999999995:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{fma}\left(4, i, \mathsf{fma}\left(2, \beta, 2\right)\right)}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 4 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha \cdot \alpha}{\left(-2 - \mathsf{fma}\left(2, i, \alpha\right)\right) \cdot \mathsf{fma}\left(2, i, \alpha\right)}, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{fma}\left(4, i, \mathsf{fma}\left(2, \beta, 2\right)\right)}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 0.99999:\\ \;\;\;\;\mathsf{fma}\left(\beta, \frac{\beta}{\left(\mathsf{fma}\left(i, 2, \beta\right) + 2\right) \cdot \mathsf{fma}\left(i, 2, \beta\right)}, 1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 -0.5)
     (/ (* 0.5 (fma 4.0 i (fma 2.0 beta 2.0))) alpha)
     (if (<= t_1 0.99999)
       (*
        (fma beta (/ beta (* (+ (fma i 2.0 beta) 2.0) (fma i 2.0 beta))) 1.0)
        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 = (0.5 * fma(4.0, i, fma(2.0, beta, 2.0))) / alpha;
	} else if (t_1 <= 0.99999) {
		tmp = fma(beta, (beta / ((fma(i, 2.0, beta) + 2.0) * fma(i, 2.0, beta))), 1.0) * 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 <= -0.5)
		tmp = Float64(Float64(0.5 * fma(4.0, i, fma(2.0, beta, 2.0))) / alpha);
	elseif (t_1 <= 0.99999)
		tmp = Float64(fma(beta, Float64(beta / Float64(Float64(fma(i, 2.0, beta) + 2.0) * fma(i, 2.0, beta))), 1.0) * 0.5);
	else
		tmp = 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[(0.5 * N[(4.0 * i + N[(2.0 * beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 0.99999], N[(N[(beta * N[(beta / N[(N[(N[(i * 2.0 + beta), $MachinePrecision] + 2.0), $MachinePrecision] * N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.0%

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

   3. Taylor expanded in alpha around inf

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

      1. *-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. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

   7. Applied rewrites 94.8%

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

   if -0.5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 0.999990000000000046

   1. Initial program 100.0%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 97.8

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

   5. Applied rewrites 97.8%

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

   7. Applied rewrites 97.8%

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

   if 0.999990000000000046 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 47.1%

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

   3. Taylor expanded in 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 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{fma}\left(4, i, \mathsf{fma}\left(2, \beta, 2\right)\right)}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 0.99999:\\ \;\;\;\;\mathsf{fma}\left(\beta, \frac{\beta}{\left(\mathsf{fma}\left(i, 2, \beta\right) + 2\right) \cdot \mathsf{fma}\left(i, 2, \beta\right)}, 1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{fma}\left(4, i, \mathsf{fma}\left(2, \beta, 2\right)\right)}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-35}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

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 (fma 2.0 beta 2.0))) alpha)
     (if (<= t_1 4e-35)
       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, fma(2.0, beta, 2.0))) / alpha;
	} else if (t_1 <= 4e-35) {
		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(Float64(0.5 * fma(4.0, i, fma(2.0, beta, 2.0))) / alpha);
	elseif (t_1 <= 4e-35)
		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[(N[(0.5 * N[(4.0 * i + N[(2.0 * beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 4e-35], 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 5.0%

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

   3. Taylor expanded in alpha around inf

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

      1. *-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. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

   7. Applied rewrites 94.8%

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

   if -0.5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 4.00000000000000003e-35

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 97.3%

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

   if 4.00000000000000003e-35 < (/.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 54.1%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 95.4

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

   5. Applied rewrites 95.4%

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

   7. Applied rewrites 95.4%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{fma}\left(4, i, \mathsf{fma}\left(2, \beta, 2\right)\right)}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 4 \cdot 10^{-35}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-35}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

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)
     (* (/ (fma 4.0 i 2.0) alpha) 0.5)
     (if (<= t_1 4e-35)
       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 = (fma(4.0, i, 2.0) / alpha) * 0.5;
	} else if (t_1 <= 4e-35) {
		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(Float64(fma(4.0, i, 2.0) / alpha) * 0.5);
	elseif (t_1 <= 4e-35)
		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[(N[(N[(4.0 * i + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 4e-35], 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 5.0%

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

   3. Taylor expanded in alpha around inf

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

      1. *-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. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 79.9%

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

   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))) < 4.00000000000000003e-35

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 97.3%

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

   if 4.00000000000000003e-35 < (/.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 54.1%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 95.4

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

   5. Applied rewrites 95.4%

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

   7. Applied rewrites 95.4%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 4 \cdot 10^{-35}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-35}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\beta - \alpha}{2 + \beta} + 1\right) \cdot 0.5\\ \end{array} \end{array} \]

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 -0.5)
     (* (/ (fma 4.0 i 2.0) alpha) 0.5)
     (if (<= t_1 4e-35) 0.5 (* (+ (/ (- beta alpha) (+ 2.0 beta)) 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 <= -0.5) {
		tmp = (fma(4.0, i, 2.0) / alpha) * 0.5;
	} else if (t_1 <= 4e-35) {
		tmp = 0.5;
	} else {
		tmp = (((beta - alpha) / (2.0 + beta)) + 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 <= -0.5)
		tmp = Float64(Float64(fma(4.0, i, 2.0) / alpha) * 0.5);
	elseif (t_1 <= 4e-35)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(2.0 + beta)) + 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, -0.5], N[(N[(N[(4.0 * i + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 4e-35], 0.5, N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + beta), $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))) < -0.5

   1. Initial program 5.0%

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

   3. Taylor expanded in alpha around inf

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

      1. *-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. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 79.9%

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

   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))) < 4.00000000000000003e-35

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 97.3%

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

   if 4.00000000000000003e-35 < (/.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 54.1%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 95.4

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

   5. Applied rewrites 95.4%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 94.3%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 4 \cdot 10^{-35}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\beta - \alpha}{2 + \beta} + 1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-35}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\beta}{2 + \beta} + 1\right) \cdot 0.5\\ \end{array} \end{array} \]

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 -0.5)
     (* (/ (fma 4.0 i 2.0) alpha) 0.5)
     (if (<= t_1 4e-35) 0.5 (* (+ (/ beta (+ 2.0 beta)) 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 <= -0.5) {
		tmp = (fma(4.0, i, 2.0) / alpha) * 0.5;
	} else if (t_1 <= 4e-35) {
		tmp = 0.5;
	} else {
		tmp = ((beta / (2.0 + beta)) + 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 <= -0.5)
		tmp = Float64(Float64(fma(4.0, i, 2.0) / alpha) * 0.5);
	elseif (t_1 <= 4e-35)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(beta / Float64(2.0 + beta)) + 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, -0.5], N[(N[(N[(4.0 * i + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 4e-35], 0.5, N[(N[(N[(beta / N[(2.0 + beta), $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))) < -0.5

   1. Initial program 5.0%

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

   3. Taylor expanded in alpha around inf

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

      1. *-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. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 79.9%

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

   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))) < 4.00000000000000003e-35

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 97.3%

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

   if 4.00000000000000003e-35 < (/.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 54.1%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 94.3%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 4 \cdot 10^{-35}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\beta}{2 + \beta} + 1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \beta, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-35}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\beta}{2 + \beta} + 1\right) \cdot 0.5\\ \end{array} \end{array} \]

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 -0.5)
     (* (/ (fma 2.0 beta 2.0) alpha) 0.5)
     (if (<= t_1 4e-35) 0.5 (* (+ (/ beta (+ 2.0 beta)) 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 <= -0.5) {
		tmp = (fma(2.0, beta, 2.0) / alpha) * 0.5;
	} else if (t_1 <= 4e-35) {
		tmp = 0.5;
	} else {
		tmp = ((beta / (2.0 + beta)) + 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 <= -0.5)
		tmp = Float64(Float64(fma(2.0, beta, 2.0) / alpha) * 0.5);
	elseif (t_1 <= 4e-35)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(beta / Float64(2.0 + beta)) + 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, -0.5], N[(N[(N[(2.0 * beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 4e-35], 0.5, N[(N[(N[(beta / N[(2.0 + beta), $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))) < -0.5

   1. Initial program 5.0%

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

   3. Taylor expanded in alpha around inf

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

      1. *-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. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 71.6%

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

   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))) < 4.00000000000000003e-35

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 97.3%

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

   if 4.00000000000000003e-35 < (/.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 54.1%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 94.3%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \beta, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 4 \cdot 10^{-35}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\beta}{2 + \beta} + 1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.999999:\\ \;\;\;\;\frac{i}{\alpha} \cdot 2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-35}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\beta}{2 + \beta} + 1\right) \cdot 0.5\\ \end{array} \end{array} \]

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 -0.999999)
     (* (/ i alpha) 2.0)
     (if (<= t_1 4e-35) 0.5 (* (+ (/ beta (+ 2.0 beta)) 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 <= -0.999999) {
		tmp = (i / alpha) * 2.0;
	} else if (t_1 <= 4e-35) {
		tmp = 0.5;
	} else {
		tmp = ((beta / (2.0 + beta)) + 1.0) * 0.5;
	}
	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 <= (-0.999999d0)) then
        tmp = (i / alpha) * 2.0d0
    else if (t_1 <= 4d-35) then
        tmp = 0.5d0
    else
        tmp = ((beta / (2.0d0 + beta)) + 1.0d0) * 0.5d0
    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 <= -0.999999) {
		tmp = (i / alpha) * 2.0;
	} else if (t_1 <= 4e-35) {
		tmp = 0.5;
	} else {
		tmp = ((beta / (2.0 + beta)) + 1.0) * 0.5;
	}
	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 <= -0.999999:
		tmp = (i / alpha) * 2.0
	elif t_1 <= 4e-35:
		tmp = 0.5
	else:
		tmp = ((beta / (2.0 + beta)) + 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 <= -0.999999)
		tmp = Float64(Float64(i / alpha) * 2.0);
	elseif (t_1 <= 4e-35)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(beta / Float64(2.0 + beta)) + 1.0) * 0.5);
	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 <= -0.999999)
		tmp = (i / alpha) * 2.0;
	elseif (t_1 <= 4e-35)
		tmp = 0.5;
	else
		tmp = ((beta / (2.0 + beta)) + 1.0) * 0.5;
	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, -0.999999], N[(N[(i / alpha), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 4e-35], 0.5, N[(N[(N[(beta / N[(2.0 + beta), $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))) < -0.999998999999999971

   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 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. lower-*.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 29.0%

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

   if -0.999998999999999971 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 4.00000000000000003e-35

   1. Initial program 99.8%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 96.6%

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

   if 4.00000000000000003e-35 < (/.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 54.1%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 94.3%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.999999:\\ \;\;\;\;\frac{i}{\alpha} \cdot 2\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 4 \cdot 10^{-35}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\beta}{2 + \beta} + 1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.999999:\\ \;\;\;\;\frac{i}{\alpha} \cdot 2\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -0.999999) (* (/ i alpha) 2.0) (if (<= t_1 0.5) 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 <= -0.999999) {
		tmp = (i / alpha) * 2.0;
	} else if (t_1 <= 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) :: 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 <= (-0.999999d0)) then
        tmp = (i / alpha) * 2.0d0
    else if (t_1 <= 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 t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -0.999999) {
		tmp = (i / alpha) * 2.0;
	} else if (t_1 <= 0.5) {
		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 <= -0.999999:
		tmp = (i / alpha) * 2.0
	elif t_1 <= 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))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -0.999999)
		tmp = Float64(Float64(i / alpha) * 2.0);
	elseif (t_1 <= 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);
	t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	tmp = 0.0;
	if (t_1 <= -0.999999)
		tmp = (i / alpha) * 2.0;
	elseif (t_1 <= 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]}, 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, -0.999999], N[(N[(i / alpha), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 0.5], 0.5, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999998999999999971

   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 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. lower-*.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 29.0%

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

   if -0.999998999999999971 < (/.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 99.8%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 94.7%

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

   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 48.7%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 94.8%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.999999:\\ \;\;\;\;\frac{i}{\alpha} \cdot 2\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.1% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha)))
        (t_1 (/ (/ (* (- beta alpha) (+ beta alpha)) t_0) (+ t_0 2.0))))
   (if (<= t_1 -1.0) (/ beta alpha) (if (<= t_1 0.5) 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 <= -1.0) {
		tmp = beta / alpha;
	} else if (t_1 <= 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) :: 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 <= (-1.0d0)) then
        tmp = beta / alpha
    else if (t_1 <= 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 t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = beta / alpha;
	} else if (t_1 <= 0.5) {
		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 <= -1.0:
		tmp = beta / alpha
	elif t_1 <= 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))
	t_1 = Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_0) / Float64(t_0 + 2.0))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(beta / alpha);
	elseif (t_1 <= 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);
	t_1 = (((beta - alpha) * (beta + alpha)) / t_0) / (t_0 + 2.0);
	tmp = 0.0;
	if (t_1 <= -1.0)
		tmp = beta / alpha;
	elseif (t_1 <= 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]}, 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, -1.0], N[(beta / alpha), $MachinePrecision], If[LessEqual[t$95$1, 0.5], 0.5, 1.0]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 2.3%

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

   3. Taylor expanded in 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. lower-*.f64 96.1

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

   5. Applied rewrites 96.1%

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

   6. Taylor expanded in beta around inf

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

   8. Applied rewrites 20.0%

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

   if -1 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 0.5

   1. Initial program 98.3%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 92.8%

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

   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 48.7%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 94.8%

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

4. Final simplification 77.1%

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

Alternative 12: 76.0% 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 69.2%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 67.4%

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

   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 48.7%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 94.8%

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

4. Final simplification 74.7%

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

Alternative 13: 60.7% 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 63.7%

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

3. Taylor expanded in i around inf

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

5. Applied rewrites 55.1%

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

Reproduce

herbie shell --seed 2024331 
(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))