Octave 3.8, jcobi/2

Percentage Accurate: 62.7% → 95.8%

Time: 4.5s

Alternatives: 14

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: 62.7% 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: 95.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2}\\ \mathbf{if}\;t\_1 \leq 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, \mathsf{fma}\left(2, \beta, 2\right)\right)}{\alpha + \alpha}\\ \mathbf{elif}\;t\_1 \leq 0.9999999999996726:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(\frac{\beta + \alpha}{\beta}, \beta, 2\right)}, 0.5, 0.5\right)\\ \end{array} \]

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 1e-16)
     (/ (fma 4.0 i (fma 2.0 beta 2.0)) (+ alpha alpha))
     (if (<= t_1 0.9999999999996726)
       t_1
       (fma
        (/ (- beta alpha) (fma (/ (+ beta alpha) 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 <= 1e-16) {
		tmp = fma(4.0, i, fma(2.0, beta, 2.0)) / (alpha + alpha);
	} else if (t_1 <= 0.9999999999996726) {
		tmp = t_1;
	} else {
		tmp = fma(((beta - alpha) / fma(((beta + alpha) / 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 <= 1e-16)
		tmp = Float64(fma(4.0, i, fma(2.0, beta, 2.0)) / Float64(alpha + alpha));
	elseif (t_1 <= 0.9999999999996726)
		tmp = t_1;
	else
		tmp = fma(Float64(Float64(beta - alpha) / fma(Float64(Float64(beta + alpha) / beta), 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, 1e-16], N[(N[(4.0 * i + N[(2.0 * beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(alpha + alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999996726], t$95$1, N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(N[(beta + alpha), $MachinePrecision] / beta), $MachinePrecision] * 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)) < 9.9999999999999998e-17

   1. Initial program 62.7%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 23.4%

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

   4. Applied rewrites 23.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. frac-times N/A

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

   6. Applied rewrites 23.4%

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

   if 9.9999999999999998e-17 < (/.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)) < 0.9999999999996726

   1. Initial program 62.7%

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

   if 0.9999999999996726 < (/.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 62.7%

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

   2. 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 66.9%

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

   4. Applied rewrites 66.9%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
   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 66.9%

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

   6. Applied rewrites 66.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sub-flip N/A

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

      4. +-commutative N/A

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

      5. sum-to-mult-rev N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 68.4%

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

      10. lift-+.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. add-to-fraction N/A

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

      13. *-lft-identity N/A

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

      14. +-commutative N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 68.4%

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

   8. Applied rewrites 68.4%

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

Alternative 2: 95.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \alpha + 2 \cdot i\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_1}}{t\_1 + 2} + 1}{2}\\ \mathbf{if}\;t\_2 \leq 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, \mathsf{fma}\left(2, \beta, 2\right)\right)}{\alpha + \alpha}\\ \mathbf{elif}\;t\_2 \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(\beta - \alpha, \frac{\alpha}{\left(2 + t\_0\right) \cdot t\_0}, 1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 1\right) + 1} + 1}{2}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ alpha (* 2.0 i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2
         (/
          (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ t_1 2.0)) 1.0)
          2.0)))
   (if (<= t_2 1e-16)
     (/ (fma 4.0 i (fma 2.0 beta 2.0)) (+ alpha alpha))
     (if (<= t_2 0.5)
       (* (fma (- beta alpha) (/ alpha (* (+ 2.0 t_0) t_0)) 1.0) 0.5)
       (/ (+ (/ (- beta alpha) (+ (+ (+ alpha beta) 1.0) 1.0)) 1.0) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = alpha + (2.0 * i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = (((((alpha + beta) * (beta - alpha)) / t_1) / (t_1 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_2 <= 1e-16) {
		tmp = fma(4.0, i, fma(2.0, beta, 2.0)) / (alpha + alpha);
	} else if (t_2 <= 0.5) {
		tmp = fma((beta - alpha), (alpha / ((2.0 + t_0) * t_0)), 1.0) * 0.5;
	} else {
		tmp = (((beta - alpha) / (((alpha + beta) + 1.0) + 1.0)) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(alpha + Float64(2.0 * i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(t_1 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_2 <= 1e-16)
		tmp = Float64(fma(4.0, i, fma(2.0, beta, 2.0)) / Float64(alpha + alpha));
	elseif (t_2 <= 0.5)
		tmp = Float64(fma(Float64(beta - alpha), Float64(alpha / Float64(Float64(2.0 + t_0) * t_0)), 1.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(Float64(alpha + beta) + 1.0) + 1.0)) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(alpha + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-16], N[(N[(4.0 * i + N[(2.0 * beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(alpha + alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.5], N[(N[(N[(beta - alpha), $MachinePrecision] * N[(alpha / N[(N[(2.0 + t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 9.9999999999999998e-17

   1. Initial program 62.7%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 23.4%

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

   4. Applied rewrites 23.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. frac-times N/A

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

   6. Applied rewrites 23.4%

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

   if 9.9999999999999998e-17 < (/.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 62.7%

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

   2. Taylor expanded in alpha around 0

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

   4. Applied rewrites 63.6%

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

   5. Taylor expanded in alpha around 0

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

   7. Applied rewrites 64.2%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 63.4%

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

       1. lift-/.f64 N/A

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

       2. mult-flip N/A

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

       3. metadata-eval N/A

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

       4. lower-*.f64 63.4%

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

   12. Applied rewrites 67.7%

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

   13. Taylor expanded in beta around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-+.f64 N/A

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

       7. lower-*.f64 60.4%

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

   15. Applied rewrites 60.4%

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

   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 62.7%

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

   2. 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 66.9%

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

   4. Applied rewrites 66.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 66.9%

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

   6. Applied rewrites 66.9%

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

Alternative 3: 94.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2}\\ \mathbf{if}\;t\_1 \leq 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, \mathsf{fma}\left(2, \beta, 2\right)\right)}{\alpha + \alpha}\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(\beta - \alpha, \frac{\beta}{\left(\mathsf{fma}\left(i, 2, \beta\right) - -2\right) \cdot \mathsf{fma}\left(i, 2, \beta\right)}, 1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 1\right) + 1} + 1}{2}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1
         (/
          (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
          2.0)))
   (if (<= t_1 1e-16)
     (/ (fma 4.0 i (fma 2.0 beta 2.0)) (+ alpha alpha))
     (if (<= t_1 0.5)
       (*
        (fma
         (- beta alpha)
         (/ beta (* (- (fma i 2.0 beta) -2.0) (fma i 2.0 beta)))
         1.0)
        0.5)
       (/ (+ (/ (- beta alpha) (+ (+ (+ alpha beta) 1.0) 1.0)) 1.0) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 1e-16) {
		tmp = fma(4.0, i, fma(2.0, beta, 2.0)) / (alpha + alpha);
	} else if (t_1 <= 0.5) {
		tmp = fma((beta - alpha), (beta / ((fma(i, 2.0, beta) - -2.0) * fma(i, 2.0, beta))), 1.0) * 0.5;
	} else {
		tmp = (((beta - alpha) / (((alpha + beta) + 1.0) + 1.0)) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_1 <= 1e-16)
		tmp = Float64(fma(4.0, i, fma(2.0, beta, 2.0)) / Float64(alpha + alpha));
	elseif (t_1 <= 0.5)
		tmp = Float64(fma(Float64(beta - alpha), Float64(beta / Float64(Float64(fma(i, 2.0, beta) - -2.0) * fma(i, 2.0, beta))), 1.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(Float64(alpha + beta) + 1.0) + 1.0)) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-16], N[(N[(4.0 * i + N[(2.0 * beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(alpha + alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.5], N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta / N[(N[(N[(i * 2.0 + beta), $MachinePrecision] - -2.0), $MachinePrecision] * N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 9.9999999999999998e-17

   1. Initial program 62.7%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 23.4%

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

   4. Applied rewrites 23.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. frac-times N/A

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

   6. Applied rewrites 23.4%

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

   if 9.9999999999999998e-17 < (/.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 62.7%

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

   2. Taylor expanded in alpha around 0

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

   4. Applied rewrites 63.6%

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

   5. Taylor expanded in alpha around 0

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

   7. Applied rewrites 64.2%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 63.4%

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

       1. lift-/.f64 N/A

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

       2. mult-flip N/A

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

       3. metadata-eval N/A

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

       4. lower-*.f64 63.4%

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

   12. Applied rewrites 67.7%

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

   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 62.7%

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

   2. 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 66.9%

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

   4. Applied rewrites 66.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 66.9%

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

   6. Applied rewrites 66.9%

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

Alternative 4: 94.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2}\\ \mathbf{if}\;t\_1 \leq 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, \mathsf{fma}\left(2, \beta, 2\right)\right)}{\alpha + \alpha}\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 1\right) + 1} + 1}{2}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1
         (/
          (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
          2.0)))
   (if (<= t_1 1e-16)
     (/ (fma 4.0 i (fma 2.0 beta 2.0)) (+ alpha alpha))
     (if (<= t_1 0.5)
       0.5
       (/ (+ (/ (- beta alpha) (+ (+ (+ alpha beta) 1.0) 1.0)) 1.0) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 1e-16) {
		tmp = fma(4.0, i, fma(2.0, beta, 2.0)) / (alpha + alpha);
	} else if (t_1 <= 0.5) {
		tmp = 0.5;
	} else {
		tmp = (((beta - alpha) / (((alpha + beta) + 1.0) + 1.0)) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_1 <= 1e-16)
		tmp = Float64(fma(4.0, i, fma(2.0, beta, 2.0)) / Float64(alpha + alpha));
	elseif (t_1 <= 0.5)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(Float64(alpha + beta) + 1.0) + 1.0)) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-16], N[(N[(4.0 * i + N[(2.0 * beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(alpha + alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.5], 0.5, N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 9.9999999999999998e-17

   1. Initial program 62.7%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 23.4%

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

   4. Applied rewrites 23.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. frac-times N/A

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

   6. Applied rewrites 23.4%

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

   if 9.9999999999999998e-17 < (/.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 62.7%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 61.2%

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

   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 62.7%

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

   2. 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 66.9%

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

   4. Applied rewrites 66.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 66.9%

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

   6. Applied rewrites 66.9%

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

Alternative 5: 94.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2}\\ \mathbf{if}\;t\_1 \leq 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, \mathsf{fma}\left(2, \beta, 2\right)\right)}{\alpha + \alpha}\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) - -2}, 0.5, 0.5\right)\\ \end{array} \]

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 1e-16)
     (/ (fma 4.0 i (fma 2.0 beta 2.0)) (+ alpha 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 <= 1e-16) {
		tmp = fma(4.0, i, fma(2.0, beta, 2.0)) / (alpha + 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 <= 1e-16)
		tmp = Float64(fma(4.0, i, fma(2.0, beta, 2.0)) / Float64(alpha + 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, 1e-16], N[(N[(4.0 * i + N[(2.0 * beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(alpha + 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)) < 9.9999999999999998e-17

   1. Initial program 62.7%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 23.4%

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

   4. Applied rewrites 23.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. frac-times N/A

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

   6. Applied rewrites 23.4%

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

   if 9.9999999999999998e-17 < (/.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 62.7%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 61.2%

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

   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 62.7%

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

   2. 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 66.9%

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

   4. Applied rewrites 66.9%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
   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 66.9%

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

   6. Applied rewrites 66.9%

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

Alternative 6: 94.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2}\\ \mathbf{if}\;t\_1 \leq 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, i, \mathsf{fma}\left(2, \beta, 2\right)\right)}{\alpha + \alpha}\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + 1\right) + 1} + 1}{2}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1
         (/
          (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
          2.0)))
   (if (<= t_1 1e-16)
     (/ (fma 4.0 i (fma 2.0 beta 2.0)) (+ alpha alpha))
     (if (<= t_1 0.5) 0.5 (/ (+ (/ beta (+ (+ beta 1.0) 1.0)) 1.0) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 1e-16) {
		tmp = fma(4.0, i, fma(2.0, beta, 2.0)) / (alpha + alpha);
	} else if (t_1 <= 0.5) {
		tmp = 0.5;
	} else {
		tmp = ((beta / ((beta + 1.0) + 1.0)) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_1 <= 1e-16)
		tmp = Float64(fma(4.0, i, fma(2.0, beta, 2.0)) / Float64(alpha + alpha));
	elseif (t_1 <= 0.5)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(beta / Float64(Float64(beta + 1.0) + 1.0)) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-16], N[(N[(4.0 * i + N[(2.0 * beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(alpha + alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.5], 0.5, N[(N[(N[(beta / N[(N[(beta + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 9.9999999999999998e-17

   1. Initial program 62.7%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 23.4%

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

   4. Applied rewrites 23.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. frac-times N/A

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

   6. Applied rewrites 23.4%

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

   if 9.9999999999999998e-17 < (/.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 62.7%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 61.2%

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

   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 62.7%

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

   2. 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 66.9%

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

   4. Applied rewrites 66.9%

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

   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 71.8%

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

   7. Applied rewrites 71.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 71.8%

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

   9. Applied rewrites 71.8%

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

Alternative 7: 90.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2}\\ \mathbf{if}\;t\_1 \leq 10^{-16}:\\ \;\;\;\;0.5 \cdot \frac{2 + 4 \cdot i}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + 1\right) + 1} + 1}{2}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1
         (/
          (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
          2.0)))
   (if (<= t_1 1e-16)
     (* 0.5 (/ (+ 2.0 (* 4.0 i)) alpha))
     (if (<= t_1 0.5) 0.5 (/ (+ (/ beta (+ (+ beta 1.0) 1.0)) 1.0) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 1e-16) {
		tmp = 0.5 * ((2.0 + (4.0 * i)) / alpha);
	} else if (t_1 <= 0.5) {
		tmp = 0.5;
	} else {
		tmp = ((beta / ((beta + 1.0) + 1.0)) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0
    if (t_1 <= 1d-16) then
        tmp = 0.5d0 * ((2.0d0 + (4.0d0 * i)) / alpha)
    else if (t_1 <= 0.5d0) then
        tmp = 0.5d0
    else
        tmp = ((beta / ((beta + 1.0d0) + 1.0d0)) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 1e-16) {
		tmp = 0.5 * ((2.0 + (4.0 * i)) / alpha);
	} else if (t_1 <= 0.5) {
		tmp = 0.5;
	} else {
		tmp = ((beta / ((beta + 1.0) + 1.0)) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0
	tmp = 0
	if t_1 <= 1e-16:
		tmp = 0.5 * ((2.0 + (4.0 * i)) / alpha)
	elif t_1 <= 0.5:
		tmp = 0.5
	else:
		tmp = ((beta / ((beta + 1.0) + 1.0)) + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_1 <= 1e-16)
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(4.0 * i)) / alpha));
	elseif (t_1 <= 0.5)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(beta / Float64(Float64(beta + 1.0) + 1.0)) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	tmp = 0.0;
	if (t_1 <= 1e-16)
		tmp = 0.5 * ((2.0 + (4.0 * i)) / alpha);
	elseif (t_1 <= 0.5)
		tmp = 0.5;
	else
		tmp = ((beta / ((beta + 1.0) + 1.0)) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-16], 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 / N[(N[(beta + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 9.9999999999999998e-17

   1. Initial program 62.7%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 23.4%

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

   4. Applied rewrites 23.4%

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

   5. Taylor expanded in beta around 0

      \[\leadsto 0.5 \cdot \frac{2 + 4 \cdot i}{\color{blue}{\alpha}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 20.0%

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

   7. Applied rewrites 20.0%

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

   if 9.9999999999999998e-17 < (/.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 62.7%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 61.2%

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

   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 62.7%

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

   2. 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 66.9%

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

   4. Applied rewrites 66.9%

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

   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 71.8%

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

   7. Applied rewrites 71.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 71.8%

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

   9. Applied rewrites 71.8%

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

Alternative 8: 90.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2}\\ \mathbf{if}\;t\_1 \leq 10^{-16}:\\ \;\;\;\;0.5 \cdot \frac{2 + 4 \cdot i}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\beta - -2}, 0.5, 0.5\right)\\ \end{array} \]

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 1e-16)
     (* 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 <= 1e-16) {
		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 <= 1e-16)
		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, 1e-16], 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)) < 9.9999999999999998e-17

   1. Initial program 62.7%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 23.4%

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

   4. Applied rewrites 23.4%

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

   5. Taylor expanded in beta around 0

      \[\leadsto 0.5 \cdot \frac{2 + 4 \cdot i}{\color{blue}{\alpha}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 20.0%

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

   7. Applied rewrites 20.0%

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

   if 9.9999999999999998e-17 < (/.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 62.7%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 61.2%

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

   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 62.7%

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

   2. 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 66.9%

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

   4. Applied rewrites 66.9%

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

   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 71.8%

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

   7. Applied rewrites 71.8%

      \[\leadsto \frac{\frac{\beta}{\color{blue}{2 + \beta}} + 1}{2} \]
   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 71.8%

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

Alternative 9: 80.7% 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 62.7%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 23.4%

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

   4. Applied rewrites 23.4%

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

   5. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 9.9%

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

   7. Applied rewrites 9.9%

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

   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 62.7%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 61.2%

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

   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 62.7%

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

   2. 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 66.9%

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

   4. Applied rewrites 66.9%

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

   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 71.8%

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

   7. Applied rewrites 71.8%

      \[\leadsto \frac{\frac{\beta}{\color{blue}{2 + \beta}} + 1}{2} \]
   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 71.8%

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

Alternative 10: 80.4% accurate, 0.4× speedup

Math

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

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 (- (/ (- alpha) beta) -1.0)))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0
    if (t_1 <= 0.0d0) then
        tmp = 2.0d0 * (i / alpha)
    else if (t_1 <= 0.6d0) then
        tmp = 0.5d0
    else
        tmp = (-alpha / beta) - (-1.0d0)
    end if
    code = tmp
end function

Java

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

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0
	tmp = 0
	if t_1 <= 0.0:
		tmp = 2.0 * (i / alpha)
	elif t_1 <= 0.6:
		tmp = 0.5
	else:
		tmp = (-alpha / beta) - -1.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(2.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.6], 0.5, N[(N[((-alpha) / beta), $MachinePrecision] - -1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 62.7%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 23.4%

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

   4. Applied rewrites 23.4%

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

   5. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 9.9%

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

   7. Applied rewrites 9.9%

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

   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 62.7%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 61.2%

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

   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 62.7%

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

   2. Taylor expanded in beta around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 22.7%

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

   4. Applied rewrites 22.7%

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

   5. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 24.8%

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

   7. Applied rewrites 24.8%

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

   8. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 27.4%

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

   10. Applied rewrites 27.4%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. add-flip N/A

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

       4. metadata-eval N/A

          \[\leadsto -1 \cdot \frac{\alpha}{\beta} - -1 \]

       5. lower--.f64 27.4%

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

       6. lift-*.f64 N/A

          \[\leadsto -1 \cdot \frac{\alpha}{\beta} - -1 \]

       7. lift-/.f64 N/A

          \[\leadsto -1 \cdot \frac{\alpha}{\beta} - -1 \]

       8. associate-*r/ N/A

          \[\leadsto \frac{-1 \cdot \alpha}{\beta} - -1 \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{-1 \cdot \alpha}{\beta} - -1 \]

       10. mul-1-neg N/A

           \[\leadsto \frac{\mathsf{neg}\left(\alpha\right)}{\beta} - -1 \]

       11. lower-neg.f64 27.4%

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

   12. Applied rewrites 27.4%

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

Alternative 11: 80.3% accurate, 0.5× speedup

Math

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

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 1.0))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0
    if (t_1 <= 0.0d0) then
        tmp = 2.0d0 * (i / alpha)
    else if (t_1 <= 0.6d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0
	tmp = 0
	if t_1 <= 0.0:
		tmp = 2.0 * (i / alpha)
	elif t_1 <= 0.6:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(2.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.6], 0.5, 1.0]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 62.7%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 23.4%

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

   4. Applied rewrites 23.4%

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

   5. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 9.9%

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

   7. Applied rewrites 9.9%

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

   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 62.7%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 61.2%

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

   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 62.7%

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

   2. Taylor expanded in alpha around 0

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

   4. Applied rewrites 63.6%

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

   5. Taylor expanded in alpha around 0

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

   7. Applied rewrites 64.2%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 63.4%

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

       1. lift-/.f64 N/A

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

       2. mult-flip N/A

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

       3. metadata-eval N/A

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

       4. lower-*.f64 63.4%

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

   12. Applied rewrites 67.7%

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

   13. Taylor expanded in beta around inf

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

   15. Applied rewrites 32.5%

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

Alternative 12: 77.1% accurate, 0.5× speedup

Math

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

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 (- INFINITY)) (/ beta alpha) (if (<= t_1 0.6) 0.5 1.0))))

C

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

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 <= -Double.POSITIVE_INFINITY) {
		tmp = beta / alpha;
	} else if (t_1 <= 0.6) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < -inf.0

   1. Initial program 62.7%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 23.4%

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

   4. Applied rewrites 23.4%

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

   5. Taylor expanded in beta around inf

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

      1. lower-/.f64 7.3%

         \[\leadsto \frac{\beta}{\alpha} \]

   7. Applied rewrites 7.3%

      \[\leadsto \frac{\beta}{\color{blue}{\alpha}} \]

   if -inf.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 62.7%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 61.2%

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

   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 62.7%

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

   2. Taylor expanded in alpha around 0

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

   4. Applied rewrites 63.6%

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

   5. Taylor expanded in alpha around 0

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

   7. Applied rewrites 64.2%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 63.4%

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

       1. lift-/.f64 N/A

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

       2. mult-flip N/A

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

       3. metadata-eval N/A

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

       4. lower-*.f64 63.4%

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

   12. Applied rewrites 67.7%

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

   13. Taylor expanded in beta around inf

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

   15. Applied rewrites 32.5%

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

Alternative 13: 76.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \leq 0.98:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

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)
        0.98)
     0.5
     1.0)))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0) <= 0.98) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0) <= 0.98d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0) <= 0.98) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0) <= 0.98:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0) <= 0.98)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0) <= 0.98)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 0.98], 0.5, 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.97999999999999998

   1. Initial program 62.7%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 61.2%

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

   if 0.97999999999999998 < (/.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 62.7%

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

   2. Taylor expanded in alpha around 0

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

   4. Applied rewrites 63.6%

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

   5. Taylor expanded in alpha around 0

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

   7. Applied rewrites 64.2%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 63.4%

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

       1. lift-/.f64 N/A

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

       2. mult-flip N/A

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

       3. metadata-eval N/A

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

       4. lower-*.f64 63.4%

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

   12. Applied rewrites 67.7%

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

   13. Taylor expanded in beta around inf

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

   15. Applied rewrites 32.5%

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

Alternative 14: 32.5% accurate, 42.2× speedup

Math

\[1 \]

FPCore

(FPCore (alpha beta i) :precision binary64 1.0)

C

double code(double alpha, double beta, double i) {
	return 1.0;
}

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
    code = 1.0d0
end function

Java

public static double code(double alpha, double beta, double i) {
	return 1.0;
}

Python

def code(alpha, beta, i):
	return 1.0

Julia

function code(alpha, beta, i)
	return 1.0
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 1.0;
end

Wolfram

code[alpha_, beta_, i_] := 1.0

Derivation

1. Initial program 62.7%

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

2. Taylor expanded in alpha around 0

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

4. Applied rewrites 63.6%

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

5. Taylor expanded in alpha around 0

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

7. Applied rewrites 64.2%

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

8. Taylor expanded in alpha around 0

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

10. Applied rewrites 63.4%

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

    1. lift-/.f64 N/A

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

    2. mult-flip N/A

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

    3. metadata-eval N/A

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

    4. lower-*.f64 63.4%

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

12. Applied rewrites 67.7%

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

13. Taylor expanded in beta around inf

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

15. Applied rewrites 32.5%

    \[\leadsto \color{blue}{1} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2025207 
(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))