Octave 3.8, jcobi/2

Percentage Accurate: 62.9% → 98.0%

Time: 6.3s

Alternatives: 14

Speedup: 2.6×

Specification

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

Math

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

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: 62.9% accurate, 1.0× speedup

Math

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

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

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \leq 10^{-12}:\\ \;\;\;\;0.5 \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \mathsf{fma}\left(2, \beta, 4 \cdot i\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, \frac{\beta + \alpha}{i + \left(i + \left(\left(\alpha + \beta\right) - -2\right)\right)}, 1\right)}{2}\\ \end{array} \]

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)
       1e-12)
    (*
     0.5
     (/
      (-
       (+ beta (* -1.0 beta))
       (* -1.0 (+ 2.0 (fma 2.0 beta (* 4.0 i)))))
      alpha))
    (/
     (fma
      (/ (- beta alpha) (fma i 2.0 (+ beta alpha)))
      (/ (+ beta alpha) (+ i (+ i (- (+ alpha beta) -2.0))))
      1.0)
     2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0) <= 1e-12) {
		tmp = 0.5 * (((beta + (-1.0 * beta)) - (-1.0 * (2.0 + fma(2.0, beta, (4.0 * i))))) / alpha);
	} else {
		tmp = fma(((beta - alpha) / fma(i, 2.0, (beta + alpha))), ((beta + alpha) / (i + (i + ((alpha + beta) - -2.0)))), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0) <= 1e-12)
		tmp = Float64(0.5 * Float64(Float64(Float64(beta + Float64(-1.0 * beta)) - Float64(-1.0 * Float64(2.0 + fma(2.0, beta, Float64(4.0 * i))))) / alpha));
	else
		tmp = Float64(fma(Float64(Float64(beta - alpha) / fma(i, 2.0, Float64(beta + alpha))), Float64(Float64(beta + alpha) / Float64(i + Float64(i + Float64(Float64(alpha + beta) - -2.0)))), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(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], 1e-12], N[(0.5 * N[(N[(N[(beta + N[(-1.0 * beta), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(2.0 + N[(2.0 * beta + N[(4.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + alpha), $MachinePrecision] / N[(i + N[(i + N[(N[(alpha + beta), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.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)) < 9.9999999999999998e-13

   1. Initial program 62.9%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 23.2%

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

   4. Applied rewrites 23.2%

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

   if 9.9999999999999998e-13 < (/.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 62.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 80.6%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. add-flip-rev N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. associate-+l+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower-+.f64 80.6%

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. add-flip-rev N/A

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

      19. metadata-eval N/A

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

      20. lower--.f64 80.6%

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

   5. Applied rewrites 80.6%

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

Alternative 2: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_1}}{t\_1 + 2} + 1}{2} \leq 10^{-12}:\\ \;\;\;\;0.5 \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \mathsf{fma}\left(2, \beta, 4 \cdot i\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{t\_0}, \frac{\beta + \alpha}{t\_0 - -2}, 1\right)}{2}\\ \end{array} \]

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)
       1e-12)
    (*
     0.5
     (/
      (-
       (+ beta (* -1.0 beta))
       (* -1.0 (+ 2.0 (fma 2.0 beta (* 4.0 i)))))
      alpha))
    (/
     (fma (/ (- 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) <= 1e-12) {
		tmp = 0.5 * (((beta + (-1.0 * beta)) - (-1.0 * (2.0 + fma(2.0, beta, (4.0 * i))))) / alpha);
	} else {
		tmp = fma(((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) <= 1e-12)
		tmp = Float64(0.5 * Float64(Float64(Float64(beta + Float64(-1.0 * beta)) - Float64(-1.0 * Float64(2.0 + fma(2.0, beta, Float64(4.0 * i))))) / alpha));
	else
		tmp = Float64(fma(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], 1e-12], N[(0.5 * N[(N[(N[(beta + N[(-1.0 * beta), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(2.0 + N[(2.0 * beta + N[(4.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(beta + alpha), $MachinePrecision] / N[(t$95$0 - -2.0), $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)) < 9.9999999999999998e-13

   1. Initial program 62.9%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 23.2%

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

   4. Applied rewrites 23.2%

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

   if 9.9999999999999998e-13 < (/.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 62.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 80.6%

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

Alternative 3: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \leq 10^{-12}:\\ \;\;\;\;0.5 \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \mathsf{fma}\left(2, \beta, 4 \cdot i\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{\frac{\alpha + \beta}{\mathsf{fma}\left(i + 1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}\\ \end{array} \]

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)
       1e-12)
    (*
     0.5
     (/
      (-
       (+ beta (* -1.0 beta))
       (* -1.0 (+ 2.0 (fma 2.0 beta (* 4.0 i)))))
      alpha))
    (/
     (fma
      (- beta alpha)
      (/
       (/ (+ alpha beta) (fma (+ i 1.0) 2.0 (+ alpha beta)))
       (fma 2.0 i (+ alpha beta)))
      1.0)
     2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0) <= 1e-12) {
		tmp = 0.5 * (((beta + (-1.0 * beta)) - (-1.0 * (2.0 + fma(2.0, beta, (4.0 * i))))) / alpha);
	} else {
		tmp = fma((beta - alpha), (((alpha + beta) / fma((i + 1.0), 2.0, (alpha + beta))) / fma(2.0, i, (alpha + beta))), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0) <= 1e-12)
		tmp = Float64(0.5 * Float64(Float64(Float64(beta + Float64(-1.0 * beta)) - Float64(-1.0 * Float64(2.0 + fma(2.0, beta, Float64(4.0 * i))))) / alpha));
	else
		tmp = Float64(fma(Float64(beta - alpha), Float64(Float64(Float64(alpha + beta) / fma(Float64(i + 1.0), 2.0, Float64(alpha + beta))) / fma(2.0, i, Float64(alpha + beta))), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(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], 1e-12], N[(0.5 * N[(N[(N[(beta + N[(-1.0 * beta), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(2.0 + N[(2.0 * beta + N[(4.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(N[(alpha + beta), $MachinePrecision] / N[(N[(i + 1.0), $MachinePrecision] * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $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)) < 9.9999999999999998e-13

   1. Initial program 62.9%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 23.2%

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

   4. Applied rewrites 23.2%

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

   if 9.9999999999999998e-13 < (/.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 62.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 80.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 80.4%

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

Alternative 4: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_1}}{t\_1 + 2} + 1}{2}\\ \mathbf{if}\;t\_2 \leq 10^{-12}:\\ \;\;\;\;0.5 \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \mathsf{fma}\left(2, \beta, 4 \cdot i\right)\right)}{\alpha}\\ \mathbf{elif}\;t\_2 \leq 0.5077661419928561:\\ \;\;\;\;\mathsf{fma}\left(\left(\beta - \alpha\right) \cdot \left(\beta + \alpha\right), \frac{1}{\left(\left(i + \left(2 + \left(\alpha + i\right)\right)\right) \cdot t\_0\right) \cdot 2}, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{t\_0}, 1, 1\right)}{2}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
  :precision binary64
  (let* ((t_0 (fma i 2.0 (+ beta alpha)))
       (t_1 (+ (+ alpha beta) (* 2.0 i)))
       (t_2
        (/
         (+
          (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ t_1 2.0))
          1.0)
         2.0)))
  (if (<= t_2 1e-12)
    (*
     0.5
     (/
      (-
       (+ beta (* -1.0 beta))
       (* -1.0 (+ 2.0 (fma 2.0 beta (* 4.0 i)))))
      alpha))
    (if (<= t_2 0.5077661419928561)
      (fma
       (* (- beta alpha) (+ beta alpha))
       (/ 1.0 (* (* (+ i (+ 2.0 (+ alpha i))) t_0) 2.0))
       0.5)
      (/ (fma (/ (- beta alpha) t_0) 1.0 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 t_2 = (((((alpha + beta) * (beta - alpha)) / t_1) / (t_1 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_2 <= 1e-12) {
		tmp = 0.5 * (((beta + (-1.0 * beta)) - (-1.0 * (2.0 + fma(2.0, beta, (4.0 * i))))) / alpha);
	} else if (t_2 <= 0.5077661419928561) {
		tmp = fma(((beta - alpha) * (beta + alpha)), (1.0 / (((i + (2.0 + (alpha + i))) * t_0) * 2.0)), 0.5);
	} else {
		tmp = fma(((beta - alpha) / t_0), 1.0, 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))
	t_2 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(t_1 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_2 <= 1e-12)
		tmp = Float64(0.5 * Float64(Float64(Float64(beta + Float64(-1.0 * beta)) - Float64(-1.0 * Float64(2.0 + fma(2.0, beta, Float64(4.0 * i))))) / alpha));
	elseif (t_2 <= 0.5077661419928561)
		tmp = fma(Float64(Float64(beta - alpha) * Float64(beta + alpha)), Float64(1.0 / Float64(Float64(Float64(i + Float64(2.0 + Float64(alpha + i))) * t_0) * 2.0)), 0.5);
	else
		tmp = Float64(fma(Float64(Float64(beta - alpha) / t_0), 1.0, 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]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, 1e-12], N[(0.5 * N[(N[(N[(beta + N[(-1.0 * beta), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(2.0 + N[(2.0 * beta + N[(4.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.5077661419928561], N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(i + N[(2.0 + N[(alpha + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * 1.0 + 1.0), $MachinePrecision] / 2.0), $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)) < 9.9999999999999998e-13

   1. Initial program 62.9%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 23.2%

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

   4. Applied rewrites 23.2%

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

   if 9.9999999999999998e-13 < (/.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)) < 0.50776614199285608

   1. Initial program 62.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. associate-/l/ N/A

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

      8. mult-flip N/A

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

   3. Applied rewrites 62.2%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. add-flip-rev N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. associate-+l+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower-+.f64 62.2%

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. add-flip-rev N/A

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

      19. metadata-eval N/A

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

      20. lower--.f64 62.2%

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

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in beta around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 54.8%

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

   8. Applied rewrites 54.8%

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

   if 0.50776614199285608 < (/.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 62.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 80.6%

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

   4. Taylor expanded in alpha around inf

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

   6. Applied rewrites 65.2%

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

Alternative 5: 96.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_1}}{t\_1 + 2} + 1}{2}\\ \mathbf{if}\;t\_2 \leq 10^{-12}:\\ \;\;\;\;0.5 \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \mathsf{fma}\left(2, \beta, 4 \cdot i\right)\right)}{\alpha}\\ \mathbf{elif}\;t\_2 \leq 0.5077661419928561:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \frac{\beta + \alpha}{i + \left(i + \left(2 + \alpha\right)\right)}, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, 1, 1\right)}{2}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
  :precision binary64
  (let* ((t_0 (/ (- beta alpha) (fma i 2.0 (+ beta alpha))))
       (t_1 (+ (+ alpha beta) (* 2.0 i)))
       (t_2
        (/
         (+
          (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ t_1 2.0))
          1.0)
         2.0)))
  (if (<= t_2 1e-12)
    (*
     0.5
     (/
      (-
       (+ beta (* -1.0 beta))
       (* -1.0 (+ 2.0 (fma 2.0 beta (* 4.0 i)))))
      alpha))
    (if (<= t_2 0.5077661419928561)
      (/
       (fma t_0 (/ (+ beta alpha) (+ i (+ i (+ 2.0 alpha)))) 1.0)
       2.0)
      (/ (fma t_0 1.0 1.0) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (beta - alpha) / fma(i, 2.0, (beta + alpha));
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = (((((alpha + beta) * (beta - alpha)) / t_1) / (t_1 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_2 <= 1e-12) {
		tmp = 0.5 * (((beta + (-1.0 * beta)) - (-1.0 * (2.0 + fma(2.0, beta, (4.0 * i))))) / alpha);
	} else if (t_2 <= 0.5077661419928561) {
		tmp = fma(t_0, ((beta + alpha) / (i + (i + (2.0 + alpha)))), 1.0) / 2.0;
	} else {
		tmp = fma(t_0, 1.0, 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(beta - alpha) / fma(i, 2.0, Float64(beta + alpha)))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(t_1 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_2 <= 1e-12)
		tmp = Float64(0.5 * Float64(Float64(Float64(beta + Float64(-1.0 * beta)) - Float64(-1.0 * Float64(2.0 + fma(2.0, beta, Float64(4.0 * i))))) / alpha));
	elseif (t_2 <= 0.5077661419928561)
		tmp = Float64(fma(t_0, Float64(Float64(beta + alpha) / Float64(i + Float64(i + Float64(2.0 + alpha)))), 1.0) / 2.0);
	else
		tmp = Float64(fma(t_0, 1.0, 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, 1e-12], N[(0.5 * N[(N[(N[(beta + N[(-1.0 * beta), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(2.0 + N[(2.0 * beta + N[(4.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.5077661419928561], N[(N[(t$95$0 * N[(N[(beta + alpha), $MachinePrecision] / N[(i + N[(i + N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 * 1.0 + 1.0), $MachinePrecision] / 2.0), $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)) < 9.9999999999999998e-13

   1. Initial program 62.9%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 23.2%

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

   4. Applied rewrites 23.2%

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

   if 9.9999999999999998e-13 < (/.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)) < 0.50776614199285608

   1. Initial program 62.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 80.6%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. add-flip-rev N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. associate-+l+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower-+.f64 80.6%

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. add-flip-rev N/A

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

      19. metadata-eval N/A

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

      20. lower--.f64 80.6%

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in beta around 0

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

      1. lower-+.f64 59.8%

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

   8. Applied rewrites 59.8%

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

   if 0.50776614199285608 < (/.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 62.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 80.6%

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

   4. Taylor expanded in alpha around inf

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

   6. Applied rewrites 65.2%

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

Alternative 6: 96.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := \frac{\frac{\frac{t\_0}{t\_1}}{t\_1 + 2} + 1}{2}\\ \mathbf{if}\;t\_2 \leq 10^{-5}:\\ \;\;\;\;0.5 \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \mathsf{fma}\left(2, \beta, 4 \cdot i\right)\right)}{\alpha}\\ \mathbf{elif}\;t\_2 \leq 0.5077661419928561:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0}{\left(\left(\alpha + \beta\right) - -2\right) \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right)}, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, 1, 1\right)}{2}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
  :precision binary64
  (let* ((t_0 (* (+ alpha beta) (- beta alpha)))
       (t_1 (+ (+ alpha beta) (* 2.0 i)))
       (t_2 (/ (+ (/ (/ t_0 t_1) (+ t_1 2.0)) 1.0) 2.0)))
  (if (<= t_2 1e-5)
    (*
     0.5
     (/
      (-
       (+ beta (* -1.0 beta))
       (* -1.0 (+ 2.0 (fma 2.0 beta (* 4.0 i)))))
      alpha))
    (if (<= t_2 0.5077661419928561)
      (fma
       (/ t_0 (* (- (+ alpha beta) -2.0) (fma 2.0 i (+ alpha beta))))
       0.5
       0.5)
      (/
       (fma (/ (- beta alpha) (fma i 2.0 (+ beta alpha))) 1.0 1.0)
       2.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) * (beta - alpha);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = (((t_0 / t_1) / (t_1 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_2 <= 1e-5) {
		tmp = 0.5 * (((beta + (-1.0 * beta)) - (-1.0 * (2.0 + fma(2.0, beta, (4.0 * i))))) / alpha);
	} else if (t_2 <= 0.5077661419928561) {
		tmp = fma((t_0 / (((alpha + beta) - -2.0) * fma(2.0, i, (alpha + beta)))), 0.5, 0.5);
	} else {
		tmp = fma(((beta - alpha) / fma(i, 2.0, (beta + alpha))), 1.0, 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) * Float64(beta - alpha))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(Float64(Float64(Float64(t_0 / t_1) / Float64(t_1 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_2 <= 1e-5)
		tmp = Float64(0.5 * Float64(Float64(Float64(beta + Float64(-1.0 * beta)) - Float64(-1.0 * Float64(2.0 + fma(2.0, beta, Float64(4.0 * i))))) / alpha));
	elseif (t_2 <= 0.5077661419928561)
		tmp = fma(Float64(t_0 / Float64(Float64(Float64(alpha + beta) - -2.0) * fma(2.0, i, Float64(alpha + beta)))), 0.5, 0.5);
	else
		tmp = Float64(fma(Float64(Float64(beta - alpha) / fma(i, 2.0, Float64(beta + alpha))), 1.0, 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(t$95$0 / t$95$1), $MachinePrecision] / N[(t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-5], N[(0.5 * N[(N[(N[(beta + N[(-1.0 * beta), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(2.0 + N[(2.0 * beta + N[(4.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.5077661419928561], N[(N[(t$95$0 / N[(N[(N[(alpha + beta), $MachinePrecision] - -2.0), $MachinePrecision] * N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0 + 1.0), $MachinePrecision] / 2.0), $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)) < 1.0000000000000001e-5

   1. Initial program 62.9%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 23.2%

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

   4. Applied rewrites 23.2%

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

   if 1.0000000000000001e-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)) < 0.50776614199285608

   1. Initial program 62.9%

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

   2. Taylor expanded in i around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 62.4%

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

   4. Applied rewrites 62.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 62.4%

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

   6. Applied rewrites 61.7%

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

   if 0.50776614199285608 < (/.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 62.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 80.6%

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

   4. Taylor expanded in alpha around inf

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

   6. Applied rewrites 65.2%

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

Alternative 7: 96.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2}\\ \mathbf{if}\;t\_1 \leq 10^{-12}:\\ \;\;\;\;0.5 \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \mathsf{fma}\left(2, \beta, 4 \cdot i\right)\right)}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 0.5077661419928561:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{\alpha}{\left(\alpha + 2 \cdot i\right) \cdot \left(\alpha + 2 \cdot \left(1 + i\right)\right)}, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, 1, 1\right)}{2}\\ \end{array} \]

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-12)
    (*
     0.5
     (/
      (-
       (+ beta (* -1.0 beta))
       (* -1.0 (+ 2.0 (fma 2.0 beta (* 4.0 i)))))
      alpha))
    (if (<= t_1 0.5077661419928561)
      (/
       (fma
        (- beta alpha)
        (/ alpha (* (+ alpha (* 2.0 i)) (+ alpha (* 2.0 (+ 1.0 i)))))
        1.0)
       2.0)
      (/
       (fma (/ (- beta alpha) (fma i 2.0 (+ beta alpha))) 1.0 1.0)
       2.0)))))

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-12) {
		tmp = 0.5 * (((beta + (-1.0 * beta)) - (-1.0 * (2.0 + fma(2.0, beta, (4.0 * i))))) / alpha);
	} else if (t_1 <= 0.5077661419928561) {
		tmp = fma((beta - alpha), (alpha / ((alpha + (2.0 * i)) * (alpha + (2.0 * (1.0 + i))))), 1.0) / 2.0;
	} else {
		tmp = fma(((beta - alpha) / fma(i, 2.0, (beta + alpha))), 1.0, 1.0) / 2.0;
	}
	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-12)
		tmp = Float64(0.5 * Float64(Float64(Float64(beta + Float64(-1.0 * beta)) - Float64(-1.0 * Float64(2.0 + fma(2.0, beta, Float64(4.0 * i))))) / alpha));
	elseif (t_1 <= 0.5077661419928561)
		tmp = Float64(fma(Float64(beta - alpha), Float64(alpha / Float64(Float64(alpha + Float64(2.0 * i)) * Float64(alpha + Float64(2.0 * Float64(1.0 + i))))), 1.0) / 2.0);
	else
		tmp = Float64(fma(Float64(Float64(beta - alpha) / fma(i, 2.0, Float64(beta + alpha))), 1.0, 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, 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-12], N[(0.5 * N[(N[(N[(beta + N[(-1.0 * beta), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(2.0 + N[(2.0 * beta + N[(4.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.5077661419928561], N[(N[(N[(beta - alpha), $MachinePrecision] * N[(alpha / N[(N[(alpha + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(alpha + N[(2.0 * N[(1.0 + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0 + 1.0), $MachinePrecision] / 2.0), $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)) < 9.9999999999999998e-13

   1. Initial program 62.9%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 23.2%

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

   4. Applied rewrites 23.2%

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

   if 9.9999999999999998e-13 < (/.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)) < 0.50776614199285608

   1. Initial program 62.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 80.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 80.4%

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

   6. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 60.3%

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

   8. Applied rewrites 60.3%

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

   if 0.50776614199285608 < (/.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 62.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 80.6%

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

   4. Taylor expanded in alpha around inf

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

   6. Applied rewrites 65.2%

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

Alternative 8: 96.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_1}}{t\_1 + 2} + 1}{2}\\ \mathbf{if}\;t\_2 \leq 10^{-5}:\\ \;\;\;\;0.5 \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \mathsf{fma}\left(2, \beta, 4 \cdot i\right)\right)}{\alpha}\\ \mathbf{elif}\;t\_2 \leq 0.5077661419928561:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \frac{\beta + \alpha}{t\_0 - -2}, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{t\_0}, 1, 1\right)}{2}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
  :precision binary64
  (let* ((t_0 (fma i 2.0 (+ beta alpha)))
       (t_1 (+ (+ alpha beta) (* 2.0 i)))
       (t_2
        (/
         (+
          (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ t_1 2.0))
          1.0)
         2.0)))
  (if (<= t_2 1e-5)
    (*
     0.5
     (/
      (-
       (+ beta (* -1.0 beta))
       (* -1.0 (+ 2.0 (fma 2.0 beta (* 4.0 i)))))
      alpha))
    (if (<= t_2 0.5077661419928561)
      (/ (fma -1.0 (/ (+ beta alpha) (- t_0 -2.0)) 1.0) 2.0)
      (/ (fma (/ (- beta alpha) t_0) 1.0 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 t_2 = (((((alpha + beta) * (beta - alpha)) / t_1) / (t_1 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_2 <= 1e-5) {
		tmp = 0.5 * (((beta + (-1.0 * beta)) - (-1.0 * (2.0 + fma(2.0, beta, (4.0 * i))))) / alpha);
	} else if (t_2 <= 0.5077661419928561) {
		tmp = fma(-1.0, ((beta + alpha) / (t_0 - -2.0)), 1.0) / 2.0;
	} else {
		tmp = fma(((beta - alpha) / t_0), 1.0, 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))
	t_2 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(t_1 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_2 <= 1e-5)
		tmp = Float64(0.5 * Float64(Float64(Float64(beta + Float64(-1.0 * beta)) - Float64(-1.0 * Float64(2.0 + fma(2.0, beta, Float64(4.0 * i))))) / alpha));
	elseif (t_2 <= 0.5077661419928561)
		tmp = Float64(fma(-1.0, Float64(Float64(beta + alpha) / Float64(t_0 - -2.0)), 1.0) / 2.0);
	else
		tmp = Float64(fma(Float64(Float64(beta - alpha) / t_0), 1.0, 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]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, 1e-5], N[(0.5 * N[(N[(N[(beta + N[(-1.0 * beta), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(2.0 + N[(2.0 * beta + N[(4.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.5077661419928561], N[(N[(-1.0 * N[(N[(beta + alpha), $MachinePrecision] / N[(t$95$0 - -2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * 1.0 + 1.0), $MachinePrecision] / 2.0), $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)) < 1.0000000000000001e-5

   1. Initial program 62.9%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 23.2%

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

   4. Applied rewrites 23.2%

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

   if 1.0000000000000001e-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)) < 0.50776614199285608

   1. Initial program 62.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 80.6%

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

   4. Taylor expanded in alpha around inf

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

   6. Applied rewrites 58.1%

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

   if 0.50776614199285608 < (/.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 62.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 80.6%

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

   4. Taylor expanded in alpha around inf

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

   6. Applied rewrites 65.2%

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

Alternative 9: 79.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;i \leq 7 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\left(\alpha + \beta\right) - -2} \cdot \left(\alpha - \beta\right), 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, 1, 1\right)}{2}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
  :precision binary64
  (if (<= i 7e-11)
  (fma (* (/ -1.0 (- (+ alpha beta) -2.0)) (- alpha beta)) 0.5 0.5)
  (/ (fma (/ (- beta alpha) (fma i 2.0 (+ beta alpha))) 1.0 1.0) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 7e-11) {
		tmp = fma(((-1.0 / ((alpha + beta) - -2.0)) * (alpha - beta)), 0.5, 0.5);
	} else {
		tmp = fma(((beta - alpha) / fma(i, 2.0, (beta + alpha))), 1.0, 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 7e-11)
		tmp = fma(Float64(Float64(-1.0 / Float64(Float64(alpha + beta) - -2.0)) * Float64(alpha - beta)), 0.5, 0.5);
	else
		tmp = Float64(fma(Float64(Float64(beta - alpha) / fma(i, 2.0, Float64(beta + alpha))), 1.0, 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 7e-11], N[(N[(N[(-1.0 / N[(N[(alpha + beta), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] * N[(alpha - beta), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 7.0000000000000004e-11

   1. Initial program 62.9%

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

   2. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 68.0%

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

   4. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 68.0%

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

   6. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. frac-2neg N/A

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

      6. metadata-eval N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 68.1%

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

   8. Applied rewrites 68.1%

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

   if 7.0000000000000004e-11 < i

   1. Initial program 62.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 80.6%

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

   4. Taylor expanded in alpha around inf

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

   6. Applied rewrites 65.2%

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

Alternative 10: 76.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;i \leq 2.9 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\left(\alpha + \beta\right) - -2} \cdot \left(\alpha - \beta\right), 0.5, 0.5\right)\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{+193}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{2 + \beta}, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
  :precision binary64
  (if (<= i 2.9e-9)
  (fma (* (/ -1.0 (- (+ alpha beta) -2.0)) (- alpha beta)) 0.5 0.5)
  (if (<= i 2.5e+193) (fma (/ beta (+ 2.0 beta)) 0.5 0.5) 0.5)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 2.9e-9) {
		tmp = fma(((-1.0 / ((alpha + beta) - -2.0)) * (alpha - beta)), 0.5, 0.5);
	} else if (i <= 2.5e+193) {
		tmp = fma((beta / (2.0 + beta)), 0.5, 0.5);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 2.9e-9)
		tmp = fma(Float64(Float64(-1.0 / Float64(Float64(alpha + beta) - -2.0)) * Float64(alpha - beta)), 0.5, 0.5);
	elseif (i <= 2.5e+193)
		tmp = fma(Float64(beta / Float64(2.0 + beta)), 0.5, 0.5);
	else
		tmp = 0.5;
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 2.9e-9], N[(N[(N[(-1.0 / N[(N[(alpha + beta), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] * N[(alpha - beta), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision], If[LessEqual[i, 2.5e+193], N[(N[(beta / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision], 0.5]]

Derivation

1. Split input into 3 regimes

2. if i < 2.8999999999999999e-9

   1. Initial program 62.9%

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

   2. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 68.0%

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

   4. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 68.0%

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

   6. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. frac-2neg N/A

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

      6. metadata-eval N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 68.1%

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

   8. Applied rewrites 68.1%

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

   if 2.8999999999999999e-9 < i < 2.4999999999999999e193

   1. Initial program 62.9%

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

   2. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 68.0%

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

   4. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 68.0%

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

   6. Applied rewrites 68.0%

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

   7. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 72.2%

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

   9. Applied rewrites 72.2%

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

   if 2.4999999999999999e193 < i

   1. Initial program 62.9%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 60.9%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 76.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;i \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha - \beta}{-2 - \left(\alpha + \beta\right)}, 0.5, 0.5\right)\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{+193}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{2 + \beta}, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
  :precision binary64
  (if (<= i 2.8e-9)
  (fma (/ (- alpha beta) (- -2.0 (+ alpha beta))) 0.5 0.5)
  (if (<= i 2.5e+193) (fma (/ beta (+ 2.0 beta)) 0.5 0.5) 0.5)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 2.8e-9) {
		tmp = fma(((alpha - beta) / (-2.0 - (alpha + beta))), 0.5, 0.5);
	} else if (i <= 2.5e+193) {
		tmp = fma((beta / (2.0 + beta)), 0.5, 0.5);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 2.8e-9)
		tmp = fma(Float64(Float64(alpha - beta) / Float64(-2.0 - Float64(alpha + beta))), 0.5, 0.5);
	elseif (i <= 2.5e+193)
		tmp = fma(Float64(beta / Float64(2.0 + beta)), 0.5, 0.5);
	else
		tmp = 0.5;
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 2.8e-9], N[(N[(N[(alpha - beta), $MachinePrecision] / N[(-2.0 - N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision], If[LessEqual[i, 2.5e+193], N[(N[(beta / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision], 0.5]]

Derivation

1. Split input into 3 regimes

2. if i < 2.7999999999999998e-9

   1. Initial program 62.9%

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

   2. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 68.0%

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

   4. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 68.0%

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

   6. Applied rewrites 68.0%

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

   if 2.7999999999999998e-9 < i < 2.4999999999999999e193

   1. Initial program 62.9%

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

   2. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 68.0%

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

   4. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 68.0%

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

   6. Applied rewrites 68.0%

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

   7. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 72.2%

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

   9. Applied rewrites 72.2%

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

   if 2.4999999999999999e193 < i

   1. Initial program 62.9%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 60.9%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 76.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;i \leq 2.5 \cdot 10^{+193}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{2 + \beta}, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
  :precision binary64
  (if (<= i 2.5e+193) (fma (/ beta (+ 2.0 beta)) 0.5 0.5) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 2.5e+193) {
		tmp = fma((beta / (2.0 + beta)), 0.5, 0.5);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 2.5e+193)
		tmp = fma(Float64(beta / Float64(2.0 + beta)), 0.5, 0.5);
	else
		tmp = 0.5;
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 2.5e+193], N[(N[(beta / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if i < 2.4999999999999999e193

   1. Initial program 62.9%

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

   2. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 68.0%

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

   4. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 68.0%

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

   6. Applied rewrites 68.0%

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

   7. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 72.2%

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

   9. Applied rewrites 72.2%

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

   if 2.4999999999999999e193 < i

   1. Initial program 62.9%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 60.9%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \leq 0.75:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot 0.5\\ \end{array} \]

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)
       0.75)
    0.5
    (* 2.0 0.5))))

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) <= 0.75) {
		tmp = 0.5;
	} else {
		tmp = 2.0 * 0.5;
	}
	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) <= 0.75d0) then
        tmp = 0.5d0
    else
        tmp = 2.0d0 * 0.5d0
    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) <= 0.75) {
		tmp = 0.5;
	} else {
		tmp = 2.0 * 0.5;
	}
	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) <= 0.75:
		tmp = 0.5
	else:
		tmp = 2.0 * 0.5
	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) <= 0.75)
		tmp = 0.5;
	else
		tmp = Float64(2.0 * 0.5);
	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) <= 0.75)
		tmp = 0.5;
	else
		tmp = 2.0 * 0.5;
	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], 0.75], 0.5, N[(2.0 * 0.5), $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)) < 0.75

   1. Initial program 62.9%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 60.9%

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

   if 0.75 < (/.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 62.9%

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

   2. Taylor expanded in beta around inf

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

   4. Applied rewrites 32.8%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{2}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{2}} \]

      3. metadata-eval N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{1}{2}} \]

      4. lower-*.f64 32.8%

         \[\leadsto \color{blue}{2 \cdot 0.5} \]

   6. Applied rewrites 32.8%

      \[\leadsto \color{blue}{2 \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 60.9% accurate, 42.2× speedup

Math

\[0.5 \]

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 62.9%

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

2. Taylor expanded in i around inf

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

4. Applied rewrites 60.9%

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

Reproduce

herbie shell --seed 2025213 
(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))