Octave 3.8, jcobi/2

Percentage Accurate: 63.1% → 97.7%

Time: 14.6s

Alternatives: 14

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 63.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.999995:\\ \;\;\;\;\frac{\left(2 + \mathsf{fma}\left(\beta, 2, i \cdot 4\right)\right) \cdot 0.5}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, \frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.999995)
     (/ (* (+ 2.0 (fma beta 2.0 (* i 4.0))) 0.5) alpha)
     (/
      (fma
       (/ (- beta alpha) (+ alpha (fma 2.0 i beta)))
       (/ (+ alpha beta) (+ alpha (+ beta (fma 2.0 i 2.0))))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.999995) {
		tmp = ((2.0 + fma(beta, 2.0, (i * 4.0))) * 0.5) / alpha;
	} else {
		tmp = fma(((beta - alpha) / (alpha + fma(2.0, i, beta))), ((alpha + beta) / (alpha + (beta + fma(2.0, i, 2.0)))), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.999995)
		tmp = Float64(Float64(Float64(2.0 + fma(beta, 2.0, Float64(i * 4.0))) * 0.5) / alpha);
	else
		tmp = Float64(fma(Float64(Float64(beta - alpha) / Float64(alpha + fma(2.0, i, beta))), Float64(Float64(alpha + beta) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.999995], N[(N[(N[(2.0 + N[(beta * 2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.99999499999999997

   1. Initial program 2.6%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. mul0-lft N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 90.2

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

   5. Applied rewrites 90.2%

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

   if -0.99999499999999997 < (/.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 83.5%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999995:\\ \;\;\;\;\frac{\left(2 + \mathsf{fma}\left(\beta, 2, i \cdot 4\right)\right) \cdot 0.5}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, \frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.999995:\\ \;\;\;\;\frac{\left(2 + \mathsf{fma}\left(\beta, 2, i \cdot 4\right)\right) \cdot 0.5}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha \cdot \alpha}{\mathsf{fma}\left(2, i, \alpha\right) \cdot \left(-2 - \mathsf{fma}\left(2, i, \alpha\right)\right)}, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\beta - \alpha, 0.5 \cdot \frac{1}{\left(\alpha + \beta\right) + 2}, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -0.999995)
     (/ (* (+ 2.0 (fma beta 2.0 (* i 4.0))) 0.5) alpha)
     (if (<= t_1 5e-60)
       (fma
        (/ (* alpha alpha) (* (fma 2.0 i alpha) (- -2.0 (fma 2.0 i alpha))))
        0.5
        0.5)
       (fma (- beta alpha) (* 0.5 (/ 1.0 (+ (+ alpha beta) 2.0))) 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -0.999995) {
		tmp = ((2.0 + fma(beta, 2.0, (i * 4.0))) * 0.5) / alpha;
	} else if (t_1 <= 5e-60) {
		tmp = fma(((alpha * alpha) / (fma(2.0, i, alpha) * (-2.0 - fma(2.0, i, alpha)))), 0.5, 0.5);
	} else {
		tmp = fma((beta - alpha), (0.5 * (1.0 / ((alpha + beta) + 2.0))), 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0))
	tmp = 0.0
	if (t_1 <= -0.999995)
		tmp = Float64(Float64(Float64(2.0 + fma(beta, 2.0, Float64(i * 4.0))) * 0.5) / alpha);
	elseif (t_1 <= 5e-60)
		tmp = fma(Float64(Float64(alpha * alpha) / Float64(fma(2.0, i, alpha) * Float64(-2.0 - fma(2.0, i, alpha)))), 0.5, 0.5);
	else
		tmp = fma(Float64(beta - alpha), Float64(0.5 * Float64(1.0 / Float64(Float64(alpha + beta) + 2.0))), 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.999995], N[(N[(N[(2.0 + N[(beta * 2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 5e-60], N[(N[(N[(alpha * alpha), $MachinePrecision] / N[(N[(2.0 * i + alpha), $MachinePrecision] * N[(-2.0 - N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision], N[(N[(beta - alpha), $MachinePrecision] * N[(0.5 * N[(1.0 / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.99999499999999997

   1. Initial program 3.5%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. mul0-lft N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 90.3

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

   5. Applied rewrites 90.3%

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

   if -0.99999499999999997 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 5.0000000000000001e-60

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. distribute-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 99.3%

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

   if 5.0000000000000001e-60 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 46.7%

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

   3. Taylor expanded in i around 0

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 91.5

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

   5. Applied rewrites 91.5%

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

   7. Applied rewrites 91.5%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999995:\\ \;\;\;\;\frac{\left(2 + \mathsf{fma}\left(\beta, 2, i \cdot 4\right)\right) \cdot 0.5}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha \cdot \alpha}{\mathsf{fma}\left(2, i, \alpha\right) \cdot \left(-2 - \mathsf{fma}\left(2, i, \alpha\right)\right)}, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\beta - \alpha, 0.5 \cdot \frac{1}{\left(\alpha + \beta\right) + 2}, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(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))