Octave 3.8, jcobi/2

Percentage Accurate: 62.9% → 97.4%

Time: 6.1s

Alternatives: 11

Speedup: 0.9×

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: 97.4% accurate, 0.5× speedup

Math

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

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)
        2e-14)
     (/ (* (- -2.0 (fma 4.0 i (+ beta beta))) -0.5) alpha)
     (fma
      (/ (- alpha beta) (fma 2.0 i (+ alpha beta)))
      (/ (* (+ alpha beta) 0.5) (- (- -2.0 (fma 2.0 i beta)) alpha))
      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) <= 2e-14) {
		tmp = ((-2.0 - fma(4.0, i, (beta + beta))) * -0.5) / alpha;
	} else {
		tmp = fma(((alpha - beta) / fma(2.0, i, (alpha + beta))), (((alpha + beta) * 0.5) / ((-2.0 - fma(2.0, i, beta)) - alpha)), 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) <= 2e-14)
		tmp = Float64(Float64(Float64(-2.0 - fma(4.0, i, Float64(beta + beta))) * -0.5) / alpha);
	else
		tmp = fma(Float64(Float64(alpha - beta) / fma(2.0, i, Float64(alpha + beta))), Float64(Float64(Float64(alpha + beta) * 0.5) / Float64(Float64(-2.0 - fma(2.0, i, beta)) - alpha)), 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, 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], 2e-14], N[(N[(N[(-2.0 - N[(4.0 * i + N[(beta + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(alpha - beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(alpha + beta), $MachinePrecision] * 0.5), $MachinePrecision] / N[(N[(-2.0 - N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] - alpha), $MachinePrecision]), $MachinePrecision] + 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)) < 2e-14

   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

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 80.8%

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

   4. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 23.0%

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

   6. Applied rewrites 23.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 23.0%

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

   if 2e-14 < (/.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

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 80.8%

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

   5. Applied rewrites 80.8%

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

Alternative 2: 95.3% accurate, 0.3× speedup

Math

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

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 2e-14)
     (/ (* (- -2.0 (fma 4.0 i (+ beta beta))) -0.5) alpha)
     (if (<= t_1 0.5)
       (/
        (+
         (*
          (/ (- beta alpha) (fma i 2.0 (+ beta alpha)))
          (/ alpha (+ 2.0 (+ alpha (* 2.0 i)))))
         1.0)
        2.0)
       (fma (/ (- beta alpha) (- (+ alpha beta) -2.0)) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 2e-14) {
		tmp = ((-2.0 - fma(4.0, i, (beta + beta))) * -0.5) / alpha;
	} else if (t_1 <= 0.5) {
		tmp = ((((beta - alpha) / fma(i, 2.0, (beta + alpha))) * (alpha / (2.0 + (alpha + (2.0 * i))))) + 1.0) / 2.0;
	} else {
		tmp = fma(((beta - alpha) / ((alpha + beta) - -2.0)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_1 <= 2e-14)
		tmp = Float64(Float64(Float64(-2.0 - fma(4.0, i, Float64(beta + beta))) * -0.5) / alpha);
	elseif (t_1 <= 0.5)
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) / fma(i, 2.0, Float64(beta + alpha))) * Float64(alpha / Float64(2.0 + Float64(alpha + Float64(2.0 * i))))) + 1.0) / 2.0);
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) - -2.0)), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-14], N[(N[(N[(-2.0 - N[(4.0 * i + N[(beta + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 0.5], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(alpha / N[(2.0 + N[(alpha + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 2e-14

   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

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 80.8%

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

   4. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 23.0%

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

   6. Applied rewrites 23.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 23.0%

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

   if 2e-14 < (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.5

   1. Initial program 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

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 80.8%

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

   3. Applied rewrites 80.8%

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

   4. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 65.9%

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

   6. Applied rewrites 65.9%

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

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

   1. Initial program 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 67.7%

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

   4. Applied rewrites 67.7%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
   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 67.7%

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

   6. Applied rewrites 67.7%

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

Alternative 3: 95.2% accurate, 0.3× speedup

Math

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

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 2e-14)
     (/ (* (- -2.0 (fma 4.0 i (+ beta beta))) -0.5) alpha)
     (if (<= t_1 0.5)
       (fma
        (/ (/ alpha (+ alpha (* 2.0 i))) (- -2.0 (fma i 2.0 (+ beta alpha))))
        (/ (+ beta alpha) 2.0)
        0.5)
       (fma (/ (- beta alpha) (- (+ alpha beta) -2.0)) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 2e-14) {
		tmp = ((-2.0 - fma(4.0, i, (beta + beta))) * -0.5) / alpha;
	} else if (t_1 <= 0.5) {
		tmp = fma(((alpha / (alpha + (2.0 * i))) / (-2.0 - fma(i, 2.0, (beta + alpha)))), ((beta + alpha) / 2.0), 0.5);
	} else {
		tmp = fma(((beta - alpha) / ((alpha + beta) - -2.0)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_1 <= 2e-14)
		tmp = Float64(Float64(Float64(-2.0 - fma(4.0, i, Float64(beta + beta))) * -0.5) / alpha);
	elseif (t_1 <= 0.5)
		tmp = fma(Float64(Float64(alpha / Float64(alpha + Float64(2.0 * i))) / Float64(-2.0 - fma(i, 2.0, Float64(beta + alpha)))), Float64(Float64(beta + alpha) / 2.0), 0.5);
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) - -2.0)), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-14], N[(N[(N[(-2.0 - N[(4.0 * i + N[(beta + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 0.5], N[(N[(N[(alpha / N[(alpha + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-2.0 - N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + alpha), $MachinePrecision] / 2.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 2e-14

   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

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 80.8%

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

   4. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 23.0%

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

   6. Applied rewrites 23.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 23.0%

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

   if 2e-14 < (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.5

   1. Initial program 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

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

   3. Applied rewrites 80.7%

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

   4. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 61.2%

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

   6. Applied rewrites 61.2%

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

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

   1. Initial program 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 67.7%

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

   4. Applied rewrites 67.7%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
   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 67.7%

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

   6. Applied rewrites 67.7%

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

Alternative 4: 94.5% accurate, 0.4× speedup

Math

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

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 2e-14)
     (/ (* (- -2.0 (fma 4.0 i (+ beta beta))) -0.5) alpha)
     (if (<= t_1 0.5)
       0.5
       (fma (/ (- beta alpha) (- (+ alpha beta) -2.0)) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 2e-14) {
		tmp = ((-2.0 - fma(4.0, i, (beta + beta))) * -0.5) / alpha;
	} else if (t_1 <= 0.5) {
		tmp = 0.5;
	} else {
		tmp = fma(((beta - alpha) / ((alpha + beta) - -2.0)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_1 <= 2e-14)
		tmp = Float64(Float64(Float64(-2.0 - fma(4.0, i, Float64(beta + beta))) * -0.5) / alpha);
	elseif (t_1 <= 0.5)
		tmp = 0.5;
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) - -2.0)), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-14], N[(N[(N[(-2.0 - N[(4.0 * i + N[(beta + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$1, 0.5], 0.5, N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 2e-14

   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

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 80.8%

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

   4. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 23.0%

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

   6. Applied rewrites 23.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 23.0%

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

   if 2e-14 < (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.5

   1. Initial program 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 61.5%

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

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

   1. Initial program 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 67.7%

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

   4. Applied rewrites 67.7%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
   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 67.7%

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

   6. Applied rewrites 67.7%

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

Alternative 5: 90.9% accurate, 0.4× speedup

Math

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

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 2e-14)
     (* 0.5 (/ (+ 2.0 (* 4.0 i)) alpha))
     (if (<= t_1 0.5)
       0.5
       (fma (/ (- beta alpha) (- (+ alpha beta) -2.0)) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 2e-14) {
		tmp = 0.5 * ((2.0 + (4.0 * i)) / alpha);
	} else if (t_1 <= 0.5) {
		tmp = 0.5;
	} else {
		tmp = fma(((beta - alpha) / ((alpha + beta) - -2.0)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_1 <= 2e-14)
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(4.0 * i)) / alpha));
	elseif (t_1 <= 0.5)
		tmp = 0.5;
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) - -2.0)), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-14], N[(0.5 * N[(N[(2.0 + N[(4.0 * i), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.5], 0.5, N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 2e-14

   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

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 80.8%

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

   4. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 23.0%

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

   6. Applied rewrites 23.0%

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

   7. Taylor expanded in beta around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{2 + 4 \cdot i}{\color{blue}{\alpha}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{2 + 4 \cdot i}{\alpha} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{2 + 4 \cdot i}{\alpha} \]

      4. lower-*.f64 19.5%

         \[\leadsto 0.5 \cdot \frac{2 + 4 \cdot i}{\alpha} \]

   9. Applied rewrites 19.5%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{2 + 4 \cdot i}{\alpha}} \]

   if 2e-14 < (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.5

   1. Initial program 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 61.5%

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

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

   1. Initial program 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 67.7%

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

   4. Applied rewrites 67.7%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
   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 67.7%

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

   6. Applied rewrites 67.7%

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

Alternative 6: 90.6% accurate, 0.4× speedup

Math

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

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 2e-14)
     (* 0.5 (/ (+ 2.0 (* 4.0 i)) alpha))
     (if (<= t_1 0.5) 0.5 (fma (/ (- beta alpha) (- beta -2.0)) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 2e-14) {
		tmp = 0.5 * ((2.0 + (4.0 * i)) / alpha);
	} else if (t_1 <= 0.5) {
		tmp = 0.5;
	} else {
		tmp = fma(((beta - alpha) / (beta - -2.0)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_1 <= 2e-14)
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(4.0 * i)) / alpha));
	elseif (t_1 <= 0.5)
		tmp = 0.5;
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(beta - -2.0)), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-14], N[(0.5 * N[(N[(2.0 + N[(4.0 * i), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.5], 0.5, N[(N[(N[(beta - alpha), $MachinePrecision] / N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 2e-14

   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

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 80.8%

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

   4. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 23.0%

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

   6. Applied rewrites 23.0%

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

   7. Taylor expanded in beta around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{2 + 4 \cdot i}{\color{blue}{\alpha}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{2 + 4 \cdot i}{\alpha} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{2 + 4 \cdot i}{\alpha} \]

      4. lower-*.f64 19.5%

         \[\leadsto 0.5 \cdot \frac{2 + 4 \cdot i}{\alpha} \]

   9. Applied rewrites 19.5%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{2 + 4 \cdot i}{\alpha}} \]

   if 2e-14 < (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.5

   1. Initial program 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 61.5%

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

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

   1. Initial program 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 67.7%

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

   4. Applied rewrites 67.7%

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

   5. Taylor expanded in beta around 0

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

      1. lower-+.f64 46.5%

         \[\leadsto \frac{\frac{\beta - \alpha}{2 + \alpha} + 1}{2} \]

   7. Applied rewrites 46.5%

      \[\leadsto \frac{\frac{\beta - \alpha}{2 + \color{blue}{\alpha}} + 1}{2} \]

   8. Taylor expanded in alpha around 0

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

      1. lower-+.f64 65.8%

         \[\leadsto \frac{\frac{\beta - \alpha}{2 + \beta} + 1}{2} \]

   10. Applied rewrites 65.8%

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

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{2 + \beta} + 1}{2}} \]

       2. lift-+.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{2 + \beta} + 1}}{2} \]

       3. div-add N/A

          \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{2 + \beta}}{2} + \frac{1}{2}} \]

       4. mult-flip N/A

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

       5. metadata-eval N/A

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

       6. metadata-eval N/A

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

   12. Applied rewrites 65.8%

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

Alternative 7: 90.5% accurate, 0.4× speedup

Math

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

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 2e-14)
     (* 0.5 (/ (+ 2.0 (* 4.0 i)) alpha))
     (if (<= t_1 0.5) 0.5 (fma (/ beta (- beta -2.0)) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 2e-14) {
		tmp = 0.5 * ((2.0 + (4.0 * i)) / alpha);
	} else if (t_1 <= 0.5) {
		tmp = 0.5;
	} else {
		tmp = fma((beta / (beta - -2.0)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_1 <= 2e-14)
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(4.0 * i)) / alpha));
	elseif (t_1 <= 0.5)
		tmp = 0.5;
	else
		tmp = fma(Float64(beta / Float64(beta - -2.0)), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-14], N[(0.5 * N[(N[(2.0 + N[(4.0 * i), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.5], 0.5, N[(N[(beta / N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 2e-14

   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

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 80.8%

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

   4. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 23.0%

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

   6. Applied rewrites 23.0%

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

   7. Taylor expanded in beta around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{2 + 4 \cdot i}{\color{blue}{\alpha}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{2 + 4 \cdot i}{\alpha} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{2 + 4 \cdot i}{\alpha} \]

      4. lower-*.f64 19.5%

         \[\leadsto 0.5 \cdot \frac{2 + 4 \cdot i}{\alpha} \]

   9. Applied rewrites 19.5%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{2 + 4 \cdot i}{\alpha}} \]

   if 2e-14 < (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.5

   1. Initial program 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 61.5%

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

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

   1. Initial program 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 67.7%

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

   4. Applied rewrites 67.7%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\beta}{2 + \color{blue}{\beta}} + 1}{2} \]

      2. lower-+.f64 72.2%

         \[\leadsto \frac{\frac{\beta}{2 + \beta} + 1}{2} \]

   7. Applied rewrites 72.2%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\beta}{2 + \beta} + 1}{2}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta} + 1}}{2} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\frac{\beta}{2 + \beta}}{2} + \frac{1}{2}} \]

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

   9. Applied rewrites 72.2%

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

Alternative 8: 80.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1
         (/
          (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
          2.0)))
   (if (<= t_1 0.0)
     (* 2.0 (/ i alpha))
     (if (<= t_1 0.5) 0.5 (fma (/ beta (- beta -2.0)) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 2.0 * (i / alpha);
	} else if (t_1 <= 0.5) {
		tmp = 0.5;
	} else {
		tmp = fma((beta / (beta - -2.0)), 0.5, 0.5);
	}
	return tmp;
}

Julia

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

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(2.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.5], 0.5, N[(N[(beta / N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.0

   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

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 80.8%

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

   4. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 23.0%

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

   6. Applied rewrites 23.0%

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

   7. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 9.5%

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

   9. Applied rewrites 9.5%

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

   if 0.0 < (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.5

   1. Initial program 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 61.5%

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

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

   1. Initial program 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 67.7%

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

   4. Applied rewrites 67.7%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\beta}{2 + \color{blue}{\beta}} + 1}{2} \]

      2. lower-+.f64 72.2%

         \[\leadsto \frac{\frac{\beta}{2 + \beta} + 1}{2} \]

   7. Applied rewrites 72.2%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\beta}{2 + \beta} + 1}{2}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta} + 1}}{2} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\frac{\beta}{2 + \beta}}{2} + \frac{1}{2}} \]

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

   9. Applied rewrites 72.2%

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

Alternative 9: 80.1% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1
         (/
          (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
          2.0)))
   (if (<= t_1 0.0) (* 2.0 (/ i alpha)) (if (<= t_1 0.6) 0.5 (* 2.0 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 2.0 * (i / alpha);
	} else if (t_1 <= 0.6) {
		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) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0
    if (t_1 <= 0.0d0) then
        tmp = 2.0d0 * (i / alpha)
    else if (t_1 <= 0.6d0) 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 t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 2.0 * (i / alpha);
	} else if (t_1 <= 0.6) {
		tmp = 0.5;
	} else {
		tmp = 2.0 * 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0
	tmp = 0
	if t_1 <= 0.0:
		tmp = 2.0 * (i / alpha)
	elif t_1 <= 0.6:
		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))
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(2.0 * Float64(i / alpha));
	elseif (t_1 <= 0.6)
		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);
	t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = 2.0 * (i / alpha);
	elseif (t_1 <= 0.6)
		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]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(2.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.6], 0.5, N[(2.0 * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.0

   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

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 80.8%

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

   4. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 23.0%

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

   6. Applied rewrites 23.0%

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

   7. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 9.5%

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

   9. Applied rewrites 9.5%

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

   if 0.0 < (/.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)) < 0.59999999999999998

   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 61.5%

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

   if 0.59999999999999998 < (/.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 32.6%

      \[\leadsto \frac{\color{blue}{2}}{2} \]
   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 32.6%

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

   6. Applied rewrites 32.6%

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

Alternative 10: 76.7% accurate, 0.9× speedup

Math

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

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)
        0.6)
     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) <= 0.6) {
		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) <= 0.6d0) 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) <= 0.6) {
		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) <= 0.6:
		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) <= 0.6)
		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) <= 0.6)
		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], 0.6], 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)) < 0.59999999999999998

   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 61.5%

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

   if 0.59999999999999998 < (/.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 32.6%

      \[\leadsto \frac{\color{blue}{2}}{2} \]
   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 32.6%

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

   6. Applied rewrites 32.6%

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

Alternative 11: 61.5% 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 61.5%

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

Reproduce

herbie shell --seed 2025205 
(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))