Octave 3.8, jcobi/2

Percentage Accurate: 62.5% → 97.1%

Time: 12.1s

Alternatives: 14

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.5% 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: 97.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ t_1 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_2 := \mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2\\ \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_1}}{t\_1 + 2} \leq -0.98:\\ \;\;\;\;\frac{\frac{\frac{\beta \cdot \beta}{\alpha} - \left(\mathsf{fma}\left(-2 - \mathsf{fma}\left(i, 2, \beta\right), \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\alpha}, \frac{t\_2}{\alpha} \cdot t\_2\right) - t\_2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\beta + \alpha}{\frac{t\_0}{\alpha - \beta} \cdot \left(t\_0 + 2\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha)))
        (t_1 (+ (* i 2.0) (+ beta alpha)))
        (t_2 (+ (fma 4.0 i (* 2.0 beta)) 2.0)))
   (if (<= (/ (/ (* (- beta alpha) (+ beta alpha)) t_1) (+ t_1 2.0)) -0.98)
     (/
      (/
       (-
        (/ (* beta beta) alpha)
        (-
         (fma
          (- -2.0 (fma i 2.0 beta))
          (/ (fma i 2.0 beta) alpha)
          (* (/ t_2 alpha) t_2))
         t_2))
       alpha)
      2.0)
     (/
      (- 1.0 (/ (+ beta alpha) (* (/ t_0 (- alpha beta)) (+ t_0 2.0))))
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double t_1 = (i * 2.0) + (beta + alpha);
	double t_2 = fma(4.0, i, (2.0 * beta)) + 2.0;
	double tmp;
	if (((((beta - alpha) * (beta + alpha)) / t_1) / (t_1 + 2.0)) <= -0.98) {
		tmp = ((((beta * beta) / alpha) - (fma((-2.0 - fma(i, 2.0, beta)), (fma(i, 2.0, beta) / alpha), ((t_2 / alpha) * t_2)) - t_2)) / alpha) / 2.0;
	} else {
		tmp = (1.0 - ((beta + alpha) / ((t_0 / (alpha - beta)) * (t_0 + 2.0)))) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	t_1 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	t_2 = Float64(fma(4.0, i, Float64(2.0 * beta)) + 2.0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta + alpha)) / t_1) / Float64(t_1 + 2.0)) <= -0.98)
		tmp = Float64(Float64(Float64(Float64(Float64(beta * beta) / alpha) - Float64(fma(Float64(-2.0 - fma(i, 2.0, beta)), Float64(fma(i, 2.0, beta) / alpha), Float64(Float64(t_2 / alpha) * t_2)) - t_2)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(beta + alpha) / Float64(Float64(t_0 / Float64(alpha - beta)) * Float64(t_0 + 2.0)))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * i + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision], -0.98], N[(N[(N[(N[(N[(beta * beta), $MachinePrecision] / alpha), $MachinePrecision] - N[(N[(N[(-2.0 - N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(N[(i * 2.0 + beta), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(t$95$2 / alpha), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 - N[(N[(beta + alpha), $MachinePrecision] / N[(N[(t$95$0 / N[(alpha - beta), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < -0.97999999999999998

   1. Initial program 3.7%

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

   3. Taylor expanded in alpha around inf

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

   5. Applied rewrites 91.1%

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

   if -0.97999999999999998 < (/.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 79.3%

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

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. div-inv N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. lower-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

   6. Applied rewrites 100.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.98:\\ \;\;\;\;\frac{\frac{\frac{\beta \cdot \beta}{\alpha} - \left(\mathsf{fma}\left(-2 - \mathsf{fma}\left(i, 2, \beta\right), \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\alpha}, \frac{\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2}{\alpha} \cdot \left(\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2\right)\right) - \left(\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\beta + \alpha}{\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right)}{\alpha - \beta} \cdot \left(\mathsf{fma}\left(2, i, \beta + \alpha\right) + 2\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{t\_0}}{t\_0 + 2}\\ \mathbf{if}\;t\_1 \leq -0.98:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(\left(\alpha \cdot \alpha\right) \cdot 0.5, \frac{1}{\left(-2 - \mathsf{fma}\left(i, 2, \alpha\right)\right) \cdot \mathsf{fma}\left(i, 2, \alpha\right)}, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\left(2 + \beta\right) + \alpha}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

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 -0.98)
     (* 0.5 (/ (+ (fma 4.0 i (* 2.0 beta)) 2.0) alpha))
     (if (<= t_1 2e-24)
       (fma
        (* (* alpha alpha) 0.5)
        (/ 1.0 (* (- -2.0 (fma i 2.0 alpha)) (fma i 2.0 alpha)))
        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 <= -0.98) {
		tmp = 0.5 * ((fma(4.0, i, (2.0 * beta)) + 2.0) / alpha);
	} else if (t_1 <= 2e-24) {
		tmp = fma(((alpha * alpha) * 0.5), (1.0 / ((-2.0 - fma(i, 2.0, alpha)) * fma(i, 2.0, alpha))), 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 <= -0.98)
		tmp = Float64(0.5 * Float64(Float64(fma(4.0, i, Float64(2.0 * beta)) + 2.0) / alpha));
	elseif (t_1 <= 2e-24)
		tmp = fma(Float64(Float64(alpha * alpha) * 0.5), Float64(1.0 / Float64(Float64(-2.0 - fma(i, 2.0, alpha)) * fma(i, 2.0, alpha))), 0.5);
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(Float64(2.0 + 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, -0.98], N[(0.5 * N[(N[(N[(4.0 * i + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-24], N[(N[(N[(alpha * alpha), $MachinePrecision] * 0.5), $MachinePrecision] * N[(1.0 / N[(N[(-2.0 - N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision] * N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] + alpha), $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))) < -0.97999999999999998

   1. Initial program 4.4%

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

   3. Taylor expanded in alpha around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. neg-sub0 N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 90.9

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

   5. Applied rewrites 90.9%

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

   if -0.97999999999999998 < (/.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))) < 1.99999999999999985e-24

   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 beta around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

   if 1.99999999999999985e-24 < (/.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 39.1%

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

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 36.3

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

   5. Applied rewrites 36.3%

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

   6. Taylor expanded in beta around inf

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 85.1%

      \[\leadsto 1 \]

   9. Taylor expanded in i around 0

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

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

       9. lower-+.f64 N/A

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

       10. +-commutative N/A

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

       11. lower-+.f64 91.9

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

   11. Applied rewrites 91.9%

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

   13. Applied rewrites 91.9%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq -0.98:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right)}{i \cdot 2 + \left(\beta + \alpha\right)}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) + 2} \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(\left(\alpha \cdot \alpha\right) \cdot 0.5, \frac{1}{\left(-2 - \mathsf{fma}\left(i, 2, \alpha\right)\right) \cdot \mathsf{fma}\left(i, 2, \alpha\right)}, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\left(2 + \beta\right) + \alpha}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024234 
(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))