Octave 3.8, jcobi/2

Percentage Accurate: 62.8% → 97.8%

Time: 20.4s

Alternatives: 4

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 + 2\\ \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_1} - -1}{2} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0, \beta, \mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{t\_1} - -1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))) (t_1 (+ t_0 2.0)))
   (if (<=
        (/ (- (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1) -1.0) 2.0)
        5e-11)
     (/ (/ (fma 0.0 beta (+ (fma 4.0 i (* 2.0 beta)) 2.0)) alpha) 2.0)
     (/
      (-
       (/ (* (+ beta alpha) (/ (- beta alpha) (fma 2.0 i (+ beta alpha)))) t_1)
       -1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 + 2.0;
	double tmp;
	if (((((((alpha + beta) * (beta - alpha)) / t_0) / t_1) - -1.0) / 2.0) <= 5e-11) {
		tmp = (fma(0.0, beta, (fma(4.0, i, (2.0 * beta)) + 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((((beta + alpha) * ((beta - alpha) / fma(2.0, i, (beta + alpha)))) / t_1) - -1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 + 2.0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / t_1) - -1.0) / 2.0) <= 5e-11)
		tmp = Float64(Float64(fma(0.0, beta, Float64(fma(4.0, i, Float64(2.0 * beta)) + 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta + alpha) * Float64(Float64(beta - alpha) / fma(2.0, i, Float64(beta + alpha)))) / t_1) - -1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], 5e-11], N[(N[(N[(0.0 * beta + N[(N[(4.0 * i + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(beta + alpha), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 5.00000000000000018e-11

   1. Initial program 2.3%

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

   3. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

   if 5.00000000000000018e-11 < (/.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 83.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - -1}{2} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0, \beta, \mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - -1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_1}}{t\_1 + 2} - -1}{2} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0, \beta, \mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{\beta}{t\_0}\right) - \frac{\alpha}{t\_0}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))) (t_1 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/
         (- (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ t_1 2.0)) -1.0)
         2.0)
        5e-11)
     (/ (/ (fma 0.0 beta (+ (fma 4.0 i (* 2.0 beta)) 2.0)) alpha) 2.0)
     (/ (- (+ 1.0 (/ beta t_0)) (/ alpha t_0)) 2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = 2.0 + (alpha + beta);
	double t_1 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((((alpha + beta) * (beta - alpha)) / t_1) / (t_1 + 2.0)) - -1.0) / 2.0) <= 5e-11) {
		tmp = (fma(0.0, beta, (fma(4.0, i, (2.0 * beta)) + 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((1.0 + (beta / t_0)) - (alpha / t_0)) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(t_1 + 2.0)) - -1.0) / 2.0) <= 5e-11)
		tmp = Float64(Float64(fma(0.0, beta, Float64(fma(4.0, i, Float64(2.0 * beta)) + 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(beta / t_0)) - Float64(alpha / t_0)) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], 5e-11], N[(N[(N[(0.0 * beta + N[(N[(4.0 * i + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(beta / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 5.00000000000000018e-11

   1. Initial program 2.3%

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

   3. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

   if 5.00000000000000018e-11 < (/.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 83.7%

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

   3. Taylor expanded in alpha around inf

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

   5. Applied rewrites 1.9%

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

   6. Taylor expanded in i around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 79.8

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

   8. Applied rewrites 79.8%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - -1}{2} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0, \beta, \mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{\beta}{2 + \left(\alpha + \beta\right)}\right) - \frac{\alpha}{2 + \left(\alpha + \beta\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ t_1 := \mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2\\ t_2 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_2}}{t\_2 + 2} - -1}{2} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0, \beta, \frac{\beta \cdot \beta}{\alpha}\right) + t\_1}{\alpha} - \frac{\mathsf{fma}\left(t\_1, \frac{\mathsf{fma}\left(0, \beta, t\_1\right)}{\alpha}, \mathsf{fma}\left(-1, \frac{\beta \cdot \left(2 + \beta\right)}{\alpha}, i \cdot \mathsf{fma}\left(-4, \frac{i}{\alpha}, -\mathsf{fma}\left(2, \frac{\beta}{\alpha}, 2 \cdot \frac{2 + \beta}{\alpha}\right)\right)\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{\beta}{t\_0}\right) - \frac{\alpha}{t\_0}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta)))
        (t_1 (+ (fma 4.0 i (* 2.0 beta)) 2.0))
        (t_2 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/
         (- (/ (/ (* (+ alpha beta) (- beta alpha)) t_2) (+ t_2 2.0)) -1.0)
         2.0)
        5e-9)
     (/
      (-
       (/ (+ (fma 0.0 beta (/ (* beta beta) alpha)) t_1) alpha)
       (/
        (fma
         t_1
         (/ (fma 0.0 beta t_1) alpha)
         (fma
          -1.0
          (/ (* beta (+ 2.0 beta)) alpha)
          (*
           i
           (fma
            -4.0
            (/ i alpha)
            (- (fma 2.0 (/ beta alpha) (* 2.0 (/ (+ 2.0 beta) alpha))))))))
        alpha))
      2.0)
     (/ (- (+ 1.0 (/ beta t_0)) (/ alpha t_0)) 2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = 2.0 + (alpha + beta);
	double t_1 = fma(4.0, i, (2.0 * beta)) + 2.0;
	double t_2 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((((alpha + beta) * (beta - alpha)) / t_2) / (t_2 + 2.0)) - -1.0) / 2.0) <= 5e-9) {
		tmp = (((fma(0.0, beta, ((beta * beta) / alpha)) + t_1) / alpha) - (fma(t_1, (fma(0.0, beta, t_1) / alpha), fma(-1.0, ((beta * (2.0 + beta)) / alpha), (i * fma(-4.0, (i / alpha), -fma(2.0, (beta / alpha), (2.0 * ((2.0 + beta) / alpha))))))) / alpha)) / 2.0;
	} else {
		tmp = ((1.0 + (beta / t_0)) - (alpha / t_0)) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	t_1 = Float64(fma(4.0, i, Float64(2.0 * beta)) + 2.0)
	t_2 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_2) / Float64(t_2 + 2.0)) - -1.0) / 2.0) <= 5e-9)
		tmp = Float64(Float64(Float64(Float64(fma(0.0, beta, Float64(Float64(beta * beta) / alpha)) + t_1) / alpha) - Float64(fma(t_1, Float64(fma(0.0, beta, t_1) / alpha), fma(-1.0, Float64(Float64(beta * Float64(2.0 + beta)) / alpha), Float64(i * fma(-4.0, Float64(i / alpha), Float64(-fma(2.0, Float64(beta / alpha), Float64(2.0 * Float64(Float64(2.0 + beta) / alpha)))))))) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(beta / t_0)) - Float64(alpha / t_0)) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * i + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = 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$2), $MachinePrecision] / N[(t$95$2 + 2.0), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], 5e-9], N[(N[(N[(N[(N[(0.0 * beta + N[(N[(beta * beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] / alpha), $MachinePrecision] - N[(N[(t$95$1 * N[(N[(0.0 * beta + t$95$1), $MachinePrecision] / alpha), $MachinePrecision] + N[(-1.0 * N[(N[(beta * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] + N[(i * N[(-4.0 * N[(i / alpha), $MachinePrecision] + (-N[(2.0 * N[(beta / alpha), $MachinePrecision] + N[(2.0 * N[(N[(2.0 + beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(beta / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 5.0000000000000001e-9

   1. Initial program 3.5%

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

   3. Taylor expanded in alpha around inf

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

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in i around 0

      \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(0, \beta, \frac{\beta \cdot \beta}{\alpha}\right) - \left(-\left(\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2\right)\right)\right) - \mathsf{fma}\left(\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2, \frac{\mathsf{fma}\left(0, \beta, 1 \cdot \left(\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2\right)\right)}{\alpha}, -1 \cdot \frac{\beta \cdot \left(2 + \beta\right)}{\alpha} + i \cdot \left(-4 \cdot \frac{i}{\alpha} + -1 \cdot \left(2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{2 + \beta}{\alpha}\right)\right)\right)}{\alpha}}{2} \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 75.6

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

   8. Applied rewrites 75.6%

      \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(0, \beta, \frac{\beta \cdot \beta}{\alpha}\right) - \left(-\left(\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2\right)\right)\right) - \mathsf{fma}\left(\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2, \frac{\mathsf{fma}\left(0, \beta, 1 \cdot \left(\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2\right)\right)}{\alpha}, \mathsf{fma}\left(-1, \frac{\beta \cdot \left(2 + \beta\right)}{\alpha}, i \cdot \mathsf{fma}\left(-4, \frac{i}{\alpha}, -1 \cdot \mathsf{fma}\left(2, \frac{\beta}{\alpha}, 2 \cdot \frac{2 + \beta}{\alpha}\right)\right)\right)\right)}{\alpha}}{2} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

   10. Applied rewrites 75.6%

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

   if 5.0000000000000001e-9 < (/.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 83.7%

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

   3. Taylor expanded in alpha around inf

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

   5. Applied rewrites 1.4%

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

   6. Taylor expanded in i around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 79.8

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

   8. Applied rewrites 79.8%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - -1}{2} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0, \beta, \frac{\beta \cdot \beta}{\alpha}\right) + \left(\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2\right)}{\alpha} - \frac{\mathsf{fma}\left(\mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2, \frac{\mathsf{fma}\left(0, \beta, \mathsf{fma}\left(4, i, 2 \cdot \beta\right) + 2\right)}{\alpha}, \mathsf{fma}\left(-1, \frac{\beta \cdot \left(2 + \beta\right)}{\alpha}, i \cdot \mathsf{fma}\left(-4, \frac{i}{\alpha}, -\mathsf{fma}\left(2, \frac{\beta}{\alpha}, 2 \cdot \frac{2 + \beta}{\alpha}\right)\right)\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{\beta}{2 + \left(\alpha + \beta\right)}\right) - \frac{\alpha}{2 + \left(\alpha + \beta\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.5% accurate, N/A× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta, double i) {
	double t_0 = 2.0 + (alpha + beta);
	return ((1.0 + (beta / t_0)) - (alpha / t_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 = 2.0d0 + (alpha + beta)
    code = ((1.0d0 + (beta / t_0)) - (alpha / t_0)) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

3. Taylor expanded in alpha around inf

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

5. Applied rewrites 16.6%

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

6. Taylor expanded in i around 0

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lift-+.f64 63.4

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

8. Applied rewrites 63.4%

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

Reproduce

herbie shell --seed 2025057 
(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))