Octave 3.8, jcobi/2

Percentage Accurate: 62.5% → 98.0%

Time: 10.5s

Alternatives: 13

Speedup: 0.9×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_1}}{t\_1 + 2} + 1}{2} \leq 0.001:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \mathsf{fma}\left(2, i, \beta\right) + \mathsf{fma}\left(0, \beta, \mathsf{fma}\left(2, i, \beta\right)\right), 4\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \alpha + \beta\right), 2, 4\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{t\_0} \cdot \frac{\beta + \alpha}{t\_0 + 2} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma i 2.0 (+ beta alpha))) (t_1 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/
         (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ t_1 2.0)) 1.0)
         2.0)
        0.001)
     (/
      (fma 2.0 (+ (fma 2.0 i beta) (fma 0.0 beta (fma 2.0 i beta))) 4.0)
      (* (fma (fma 2.0 i (+ alpha beta)) 2.0 4.0) 2.0))
     (/
      (+ (* (/ (- beta alpha) t_0) (/ (+ beta alpha) (+ t_0 2.0))) 1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (beta + alpha));
	double t_1 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((((alpha + beta) * (beta - alpha)) / t_1) / (t_1 + 2.0)) + 1.0) / 2.0) <= 0.001) {
		tmp = fma(2.0, (fma(2.0, i, beta) + fma(0.0, beta, fma(2.0, i, beta))), 4.0) / (fma(fma(2.0, i, (alpha + beta)), 2.0, 4.0) * 2.0);
	} else {
		tmp = ((((beta - alpha) / t_0) * ((beta + alpha) / (t_0 + 2.0))) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(beta + alpha))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(t_1 + 2.0)) + 1.0) / 2.0) <= 0.001)
		tmp = Float64(fma(2.0, Float64(fma(2.0, i, beta) + fma(0.0, beta, fma(2.0, i, beta))), 4.0) / Float64(fma(fma(2.0, i, Float64(alpha + beta)), 2.0, 4.0) * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) / t_0) * Float64(Float64(beta + alpha) / Float64(t_0 + 2.0))) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 0.001], N[(N[(2.0 * N[(N[(2.0 * i + beta), $MachinePrecision] + N[(0.0 * beta + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision] / N[(N[(N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(beta + alpha), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 1e-3

   1. Initial program 3.8%

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 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. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 16.4

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

   4. Applied rewrites 16.4%

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

   6. Applied rewrites 16.6%

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

   7. Taylor expanded in alpha around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 93.0

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

   9. Applied rewrites 93.0%

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

   if 1e-3 < (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 81.6%

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

      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. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 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. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \leq 0.001:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \mathsf{fma}\left(2, i, \beta\right) + \mathsf{fma}\left(0, \beta, \mathsf{fma}\left(2, i, \beta\right)\right), 4\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \alpha + \beta\right), 2, 4\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{\beta + \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2}\\ \mathbf{if}\;t\_1 \leq 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0, \beta, \mathsf{fma}\left(1, \mathsf{fma}\left(4, i, 2 \cdot \beta\right), 2\right)\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(-1, \frac{\alpha \cdot \alpha}{\left(\mathsf{fma}\left(2, i, \alpha\right) + 2\right) \cdot \mathsf{fma}\left(2, i, \alpha\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\beta - \alpha}{\left(\beta + \alpha\right) - -2}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1
         (/
          (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
          2.0)))
   (if (<= t_1 1e-14)
     (* (/ (fma 0.0 beta (fma 1.0 (fma 4.0 i (* 2.0 beta)) 2.0)) alpha) 0.5)
     (if (<= t_1 0.5)
       (*
        0.5
        (fma
         -1.0
         (/ (* alpha alpha) (* (+ (fma 2.0 i alpha) 2.0) (fma 2.0 i alpha)))
         1.0))
       (* (+ 1.0 (/ (- beta alpha) (- (+ beta alpha) -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) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 1e-14) {
		tmp = (fma(0.0, beta, fma(1.0, fma(4.0, i, (2.0 * beta)), 2.0)) / alpha) * 0.5;
	} else if (t_1 <= 0.5) {
		tmp = 0.5 * fma(-1.0, ((alpha * alpha) / ((fma(2.0, i, alpha) + 2.0) * fma(2.0, i, alpha))), 1.0);
	} else {
		tmp = (1.0 + ((beta - alpha) / ((beta + alpha) - -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(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_1 <= 1e-14)
		tmp = Float64(Float64(fma(0.0, beta, fma(1.0, fma(4.0, i, Float64(2.0 * beta)), 2.0)) / alpha) * 0.5);
	elseif (t_1 <= 0.5)
		tmp = Float64(0.5 * fma(-1.0, Float64(Float64(alpha * alpha) / Float64(Float64(fma(2.0, i, alpha) + 2.0) * fma(2.0, i, alpha))), 1.0));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) - -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[(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]}, If[LessEqual[t$95$1, 1e-14], N[(N[(N[(0.0 * beta + N[(1.0 * N[(4.0 * i + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 0.5], N[(0.5 * N[(-1.0 * N[(N[(alpha * alpha), $MachinePrecision] / N[(N[(N[(2.0 * i + alpha), $MachinePrecision] + 2.0), $MachinePrecision] * N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 9.99999999999999999e-15

   1. Initial program 2.5%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0, \beta, \mathsf{fma}\left(1, \mathsf{fma}\left(4, i, 2 \cdot \beta\right), 2\right)\right)}{\alpha} \cdot 0.5} \]

   if 9.99999999999999999e-15 < (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.5

   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. 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. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 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. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in beta around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if 0.5 < (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 40.8%

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

   3. Taylor expanded in i around 0

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

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 95.8

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

   5. Applied rewrites 95.8%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \leq 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0, \beta, \mathsf{fma}\left(1, \mathsf{fma}\left(4, i, 2 \cdot \beta\right), 2\right)\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \leq 0.5:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(-1, \frac{\alpha \cdot \alpha}{\left(\mathsf{fma}\left(2, i, \alpha\right) + 2\right) \cdot \mathsf{fma}\left(2, i, \alpha\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\beta - \alpha}{\left(\beta + \alpha\right) - -2}\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2}\\ \mathbf{if}\;t\_1 \leq 0.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(0, \beta, \mathsf{fma}\left(1, \mathsf{fma}\left(4, i, 2 \cdot \beta\right), 2\right)\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(\beta \cdot \beta, \frac{0.5}{\left(\mathsf{fma}\left(i, 2, \beta\right) + 2\right) \cdot \mathsf{fma}\left(i, 2, \beta\right)}, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\beta - \alpha}{\left(\beta + \alpha\right) - -2}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1
         (/
          (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
          2.0)))
   (if (<= t_1 0.2)
     (* (/ (fma 0.0 beta (fma 1.0 (fma 4.0 i (* 2.0 beta)) 2.0)) alpha) 0.5)
     (if (<= t_1 0.6)
       (fma
        (* beta beta)
        (/ 0.5 (* (+ (fma i 2.0 beta) 2.0) (fma i 2.0 beta)))
        0.5)
       (* (+ 1.0 (/ (- beta alpha) (- (+ beta alpha) -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) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 0.2) {
		tmp = (fma(0.0, beta, fma(1.0, fma(4.0, i, (2.0 * beta)), 2.0)) / alpha) * 0.5;
	} else if (t_1 <= 0.6) {
		tmp = fma((beta * beta), (0.5 / ((fma(i, 2.0, beta) + 2.0) * fma(i, 2.0, beta))), 0.5);
	} else {
		tmp = (1.0 + ((beta - alpha) / ((beta + alpha) - -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(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_1 <= 0.2)
		tmp = Float64(Float64(fma(0.0, beta, fma(1.0, fma(4.0, i, Float64(2.0 * beta)), 2.0)) / alpha) * 0.5);
	elseif (t_1 <= 0.6)
		tmp = fma(Float64(beta * beta), Float64(0.5 / Float64(Float64(fma(i, 2.0, beta) + 2.0) * fma(i, 2.0, beta))), 0.5);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) - -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[(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]}, If[LessEqual[t$95$1, 0.2], N[(N[(N[(0.0 * beta + N[(1.0 * N[(4.0 * i + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 0.6], N[(N[(beta * beta), $MachinePrecision] * N[(0.5 / N[(N[(N[(i * 2.0 + beta), $MachinePrecision] + 2.0), $MachinePrecision] * N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.20000000000000001

   1. Initial program 5.2%

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

   3. Taylor expanded in alpha around 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. lower-*.f64 N/A

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

   5. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0, \beta, \mathsf{fma}\left(1, \mathsf{fma}\left(4, i, 2 \cdot \beta\right), 2\right)\right)}{\alpha} \cdot 0.5} \]

   if 0.20000000000000001 < (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.599999999999999978

   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. 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. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 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. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in alpha around 0

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 99.3

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

   7. Applied rewrites 99.3%

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

   9. Applied rewrites 99.3%

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

   if 0.599999999999999978 < (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 36.5%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 97.0

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

   5. Applied rewrites 97.0%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \leq 0.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(0, \beta, \mathsf{fma}\left(1, \mathsf{fma}\left(4, i, 2 \cdot \beta\right), 2\right)\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(\beta \cdot \beta, \frac{0.5}{\left(\mathsf{fma}\left(i, 2, \beta\right) + 2\right) \cdot \mathsf{fma}\left(i, 2, \beta\right)}, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\beta - \alpha}{\left(\beta + \alpha\right) - -2}\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2}\\ \mathbf{if}\;t\_1 \leq 0.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(0, \beta, \mathsf{fma}\left(1, \mathsf{fma}\left(4, i, 2 \cdot \beta\right), 2\right)\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\beta - \alpha}{\left(\beta + \alpha\right) - -2}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1
         (/
          (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
          2.0)))
   (if (<= t_1 0.2)
     (* (/ (fma 0.0 beta (fma 1.0 (fma 4.0 i (* 2.0 beta)) 2.0)) alpha) 0.5)
     (if (<= t_1 0.5)
       0.5
       (* (+ 1.0 (/ (- beta alpha) (- (+ beta alpha) -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) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 0.2) {
		tmp = (fma(0.0, beta, fma(1.0, fma(4.0, i, (2.0 * beta)), 2.0)) / alpha) * 0.5;
	} else if (t_1 <= 0.5) {
		tmp = 0.5;
	} else {
		tmp = (1.0 + ((beta - alpha) / ((beta + alpha) - -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(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_1 <= 0.2)
		tmp = Float64(Float64(fma(0.0, beta, fma(1.0, fma(4.0, i, Float64(2.0 * beta)), 2.0)) / alpha) * 0.5);
	elseif (t_1 <= 0.5)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) - -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[(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]}, If[LessEqual[t$95$1, 0.2], N[(N[(N[(0.0 * beta + N[(1.0 * N[(4.0 * i + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 0.5], 0.5, N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.20000000000000001

   1. Initial program 5.2%

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

   3. Taylor expanded in alpha around 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. lower-*.f64 N/A

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

   5. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0, \beta, \mathsf{fma}\left(1, \mathsf{fma}\left(4, i, 2 \cdot \beta\right), 2\right)\right)}{\alpha} \cdot 0.5} \]

   if 0.20000000000000001 < (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 99.1%

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

   if 0.5 < (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 40.8%

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

   3. Taylor expanded in i around 0

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

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 95.8

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

   5. Applied rewrites 95.8%

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

Alternative 5: 91.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2}\\ \mathbf{if}\;t\_1 \leq 0.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\beta - \alpha}{\left(\beta + \alpha\right) - -2}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1
         (/
          (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
          2.0)))
   (if (<= t_1 0.2)
     (* (/ (fma 4.0 i 2.0) alpha) 0.5)
     (if (<= t_1 0.5)
       0.5
       (* (+ 1.0 (/ (- beta alpha) (- (+ beta alpha) -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) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 0.2) {
		tmp = (fma(4.0, i, 2.0) / alpha) * 0.5;
	} else if (t_1 <= 0.5) {
		tmp = 0.5;
	} else {
		tmp = (1.0 + ((beta - alpha) / ((beta + alpha) - -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(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_1 <= 0.2)
		tmp = Float64(Float64(fma(4.0, i, 2.0) / alpha) * 0.5);
	elseif (t_1 <= 0.5)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) - -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[(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]}, If[LessEqual[t$95$1, 0.2], N[(N[(N[(4.0 * i + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 0.5], 0.5, N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.20000000000000001

   1. Initial program 5.2%

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

   3. Taylor expanded in alpha around 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. lower-*.f64 N/A

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

   5. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0, \beta, \mathsf{fma}\left(1, \mathsf{fma}\left(4, i, 2 \cdot \beta\right), 2\right)\right)}{\alpha} \cdot 0.5} \]

   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 77.4%

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

   if 0.20000000000000001 < (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 99.1%

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

   if 0.5 < (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 40.8%

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

   3. Taylor expanded in i around 0

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

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 95.8

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

   5. Applied rewrites 95.8%

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

Alternative 6: 91.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2}\\ \mathbf{if}\;t\_1 \leq 0.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{2 + \beta}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1
         (/
          (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
          2.0)))
   (if (<= t_1 0.2)
     (* (/ (fma 4.0 i 2.0) alpha) 0.5)
     (if (<= t_1 0.5) 0.5 (fma (/ beta (+ 2.0 beta)) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 0.2) {
		tmp = (fma(4.0, i, 2.0) / alpha) * 0.5;
	} else if (t_1 <= 0.5) {
		tmp = 0.5;
	} else {
		tmp = fma((beta / (2.0 + beta)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_1 <= 0.2)
		tmp = Float64(Float64(fma(4.0, i, 2.0) / alpha) * 0.5);
	elseif (t_1 <= 0.5)
		tmp = 0.5;
	else
		tmp = fma(Float64(beta / Float64(2.0 + beta)), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(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]}, If[LessEqual[t$95$1, 0.2], N[(N[(N[(4.0 * i + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 0.5], 0.5, N[(N[(beta / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.20000000000000001

   1. Initial program 5.2%

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

   3. Taylor expanded in alpha around 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. lower-*.f64 N/A

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

   5. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0, \beta, \mathsf{fma}\left(1, \mathsf{fma}\left(4, i, 2 \cdot \beta\right), 2\right)\right)}{\alpha} \cdot 0.5} \]

   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 77.4%

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

   if 0.20000000000000001 < (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 99.1%

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

   if 0.5 < (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 40.8%

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 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. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in alpha around 0

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 36.8

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

   7. Applied rewrites 36.8%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 92.4%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \leq 0.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{2 + \beta}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2}\\ \mathbf{if}\;t\_1 \leq 0.001:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \beta, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{2 + \beta}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1
         (/
          (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
          2.0)))
   (if (<= t_1 0.001)
     (* (/ (fma 2.0 beta 2.0) alpha) 0.5)
     (if (<= t_1 0.5) 0.5 (fma (/ beta (+ 2.0 beta)) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 0.001) {
		tmp = (fma(2.0, beta, 2.0) / alpha) * 0.5;
	} else if (t_1 <= 0.5) {
		tmp = 0.5;
	} else {
		tmp = fma((beta / (2.0 + beta)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_1 <= 0.001)
		tmp = Float64(Float64(fma(2.0, beta, 2.0) / alpha) * 0.5);
	elseif (t_1 <= 0.5)
		tmp = 0.5;
	else
		tmp = fma(Float64(beta / Float64(2.0 + beta)), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(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]}, If[LessEqual[t$95$1, 0.001], N[(N[(N[(2.0 * beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 0.5], 0.5, N[(N[(beta / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 1e-3

   1. Initial program 3.8%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

   5. Applied rewrites 89.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0, \beta, \mathsf{fma}\left(1, \mathsf{fma}\left(4, i, 2 \cdot \beta\right), 2\right)\right)}{\alpha} \cdot 0.5} \]

   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 62.1%

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

   if 1e-3 < (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 98.4%

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

   if 0.5 < (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 40.8%

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 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. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in alpha around 0

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 36.8

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

   7. Applied rewrites 36.8%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 92.4%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \leq 0.001:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \beta, 2\right)}{\alpha} \cdot 0.5\\ \mathbf{elif}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{2 + \beta}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2}\\ \mathbf{if}\;t\_1 \leq 10^{-14}:\\ \;\;\;\;2 \cdot \frac{i}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{2 + \beta}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1
         (/
          (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
          2.0)))
   (if (<= t_1 1e-14)
     (* 2.0 (/ i alpha))
     (if (<= t_1 0.5) 0.5 (fma (/ beta (+ 2.0 beta)) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 1e-14) {
		tmp = 2.0 * (i / alpha);
	} else if (t_1 <= 0.5) {
		tmp = 0.5;
	} else {
		tmp = fma((beta / (2.0 + beta)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_1 <= 1e-14)
		tmp = Float64(2.0 * Float64(i / alpha));
	elseif (t_1 <= 0.5)
		tmp = 0.5;
	else
		tmp = fma(Float64(beta / Float64(2.0 + beta)), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(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]}, If[LessEqual[t$95$1, 1e-14], N[(2.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.5], 0.5, N[(N[(beta / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 9.99999999999999999e-15

   1. Initial program 2.5%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0, \beta, \mathsf{fma}\left(1, \mathsf{fma}\left(4, i, 2 \cdot \beta\right), 2\right)\right)}{\alpha} \cdot 0.5} \]

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 32.0%

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

   if 9.99999999999999999e-15 < (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.5

   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 i around inf

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

   5. Applied rewrites 97.8%

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

   if 0.5 < (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 40.8%

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 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. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in alpha around 0

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 36.8

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

   7. Applied rewrites 36.8%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 92.4%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \leq 10^{-14}:\\ \;\;\;\;2 \cdot \frac{i}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{2 + \beta}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2}\\ \mathbf{if}\;t\_1 \leq 10^{-14}:\\ \;\;\;\;2 \cdot \frac{i}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 0.6:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1
         (/
          (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
          2.0)))
   (if (<= t_1 1e-14) (* 2.0 (/ i alpha)) (if (<= t_1 0.6) 0.5 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 1e-14) {
		tmp = 2.0 * (i / alpha);
	} else if (t_1 <= 0.6) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0
    if (t_1 <= 1d-14) then
        tmp = 2.0d0 * (i / alpha)
    else if (t_1 <= 0.6d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 1e-14) {
		tmp = 2.0 * (i / alpha);
	} else if (t_1 <= 0.6) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0
	tmp = 0
	if t_1 <= 1e-14:
		tmp = 2.0 * (i / alpha)
	elif t_1 <= 0.6:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_1 <= 1e-14)
		tmp = Float64(2.0 * Float64(i / alpha));
	elseif (t_1 <= 0.6)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	tmp = 0.0;
	if (t_1 <= 1e-14)
		tmp = 2.0 * (i / alpha);
	elseif (t_1 <= 0.6)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, 1e-14], N[(2.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.6], 0.5, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 9.99999999999999999e-15

   1. Initial program 2.5%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0, \beta, \mathsf{fma}\left(1, \mathsf{fma}\left(4, i, 2 \cdot \beta\right), 2\right)\right)}{\alpha} \cdot 0.5} \]

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 32.0%

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

   if 9.99999999999999999e-15 < (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.599999999999999978

   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 i around inf

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

   5. Applied rewrites 96.7%

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

   if 0.599999999999999978 < (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 36.5%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 93.6%

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

Alternative 10: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \leq 0.49999999999993167:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \mathsf{fma}\left(2, i, \beta\right) + \mathsf{fma}\left(0, \beta, \mathsf{fma}\left(2, i, \beta\right)\right), 4\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \alpha + \beta\right), 2, 4\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(\frac{\beta}{\mathsf{fma}\left(2, i, \beta\right) - -2}, \frac{\beta}{\mathsf{fma}\left(2, i, \beta\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/
         (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
         2.0)
        0.49999999999993167)
     (/
      (fma 2.0 (+ (fma 2.0 i beta) (fma 0.0 beta (fma 2.0 i beta))) 4.0)
      (* (fma (fma 2.0 i (+ alpha beta)) 2.0 4.0) 2.0))
     (*
      0.5
      (fma
       (/ beta (- (fma 2.0 i beta) -2.0))
       (/ beta (fma 2.0 i beta))
       1.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0) <= 0.49999999999993167) {
		tmp = fma(2.0, (fma(2.0, i, beta) + fma(0.0, beta, fma(2.0, i, beta))), 4.0) / (fma(fma(2.0, i, (alpha + beta)), 2.0, 4.0) * 2.0);
	} else {
		tmp = 0.5 * fma((beta / (fma(2.0, i, beta) - -2.0)), (beta / fma(2.0, i, beta)), 1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0) <= 0.49999999999993167)
		tmp = Float64(fma(2.0, Float64(fma(2.0, i, beta) + fma(0.0, beta, fma(2.0, i, beta))), 4.0) / Float64(fma(fma(2.0, i, Float64(alpha + beta)), 2.0, 4.0) * 2.0));
	else
		tmp = Float64(0.5 * fma(Float64(beta / Float64(fma(2.0, i, beta) - -2.0)), Float64(beta / fma(2.0, i, beta)), 1.0));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], 0.49999999999993167], N[(N[(2.0 * N[(N[(2.0 * i + beta), $MachinePrecision] + N[(0.0 * beta + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision] / N[(N[(N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * 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]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.499999999999931666

   1. Initial program 9.2%

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

      1. lift-/.f64 N/A

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

      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. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 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. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 21.1

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

   4. Applied rewrites 21.1%

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

   6. Applied rewrites 21.4%

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

   7. Taylor expanded in alpha around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 92.3

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

   9. Applied rewrites 92.3%

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

   if 0.499999999999931666 < (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 81.2%

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

   3. Taylor expanded in alpha around 0

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

      1. 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. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. 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%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \leq 0.49999999999993167:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \mathsf{fma}\left(2, i, \beta\right) + \mathsf{fma}\left(0, \beta, \mathsf{fma}\left(2, i, \beta\right)\right), 4\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \alpha + \beta\right), 2, 4\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(\frac{\beta}{\mathsf{fma}\left(2, i, \beta\right) - -2}, \frac{\beta}{\mathsf{fma}\left(2, i, \beta\right)}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \leq 0.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(0, \beta, \mathsf{fma}\left(1, \mathsf{fma}\left(4, i, 2 \cdot \beta\right), 2\right)\right)}{\alpha} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(\frac{\beta}{\mathsf{fma}\left(2, i, \beta\right) - -2}, \frac{\beta}{\mathsf{fma}\left(2, i, \beta\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/
         (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
         2.0)
        0.2)
     (* (/ (fma 0.0 beta (fma 1.0 (fma 4.0 i (* 2.0 beta)) 2.0)) alpha) 0.5)
     (*
      0.5
      (fma
       (/ beta (- (fma 2.0 i beta) -2.0))
       (/ beta (fma 2.0 i beta))
       1.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0) <= 0.2) {
		tmp = (fma(0.0, beta, fma(1.0, fma(4.0, i, (2.0 * beta)), 2.0)) / alpha) * 0.5;
	} else {
		tmp = 0.5 * fma((beta / (fma(2.0, i, beta) - -2.0)), (beta / fma(2.0, i, beta)), 1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0) <= 0.2)
		tmp = Float64(Float64(fma(0.0, beta, fma(1.0, fma(4.0, i, Float64(2.0 * beta)), 2.0)) / alpha) * 0.5);
	else
		tmp = Float64(0.5 * fma(Float64(beta / Float64(fma(2.0, i, beta) - -2.0)), Float64(beta / fma(2.0, i, beta)), 1.0));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], 0.2], N[(N[(N[(0.0 * beta + N[(1.0 * N[(4.0 * i + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * 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]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.20000000000000001

   1. Initial program 5.2%

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

   3. Taylor expanded in alpha around 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. lower-*.f64 N/A

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

   5. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0, \beta, \mathsf{fma}\left(1, \mathsf{fma}\left(4, i, 2 \cdot \beta\right), 2\right)\right)}{\alpha} \cdot 0.5} \]

   if 0.20000000000000001 < (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 81.5%

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

   3. Taylor expanded in alpha around 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. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

Alternative 12: 76.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \leq 0.6:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/
         (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
         2.0)
        0.6)
     0.5
     1.0)))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0) <= 0.6) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0) <= 0.6d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0) <= 0.6) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0) <= 0.6:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0) <= 0.6)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0) <= 0.6)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(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], 0.6], 0.5, 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.599999999999999978

   1. Initial program 68.4%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 69.7%

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

   if 0.599999999999999978 < (/.f64 (+.f64 (/.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))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 36.5%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 93.6%

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

Alternative 13: 61.1% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    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 61.5%

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

3. Taylor expanded in i around inf

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

5. Applied rewrites 59.2%

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

Reproduce

herbie shell --seed 2024362 
(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))