Octave 3.8, jcobi/1

Percentage Accurate: 75.4% → 99.6%

Time: 4.6s

Alternatives: 15

Speedup: 0.6×

Specification

\[\alpha > -1 \land \beta > -1\]

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))

C

double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 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)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_] := N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]

Initial Program: 75.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))

C

double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 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)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_] := N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 5.6%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 99.9%

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

Alternative 2: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.55:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0625 \cdot \beta - 0.125, \beta, 0.25\right), \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\alpha + 1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 5e-6)
     (/ (+ 1.0 beta) alpha)
     (if (<= t_0 0.55)
       (fma (fma (- (* 0.0625 beta) 0.125) beta 0.25) beta 0.5)
       (- 1.0 (/ (+ alpha 1.0) beta))))))

C

double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 5e-6) {
		tmp = (1.0 + beta) / alpha;
	} else if (t_0 <= 0.55) {
		tmp = fma(fma(((0.0625 * beta) - 0.125), beta, 0.25), beta, 0.5);
	} else {
		tmp = 1.0 - ((alpha + 1.0) / beta);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_0 <= 5e-6)
		tmp = Float64(Float64(1.0 + beta) / alpha);
	elseif (t_0 <= 0.55)
		tmp = fma(fma(Float64(Float64(0.0625 * beta) - 0.125), beta, 0.25), beta, 0.5);
	else
		tmp = Float64(1.0 - Float64(Float64(alpha + 1.0) / beta));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-6], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.55], N[(N[(N[(N[(0.0625 * beta), $MachinePrecision] - 0.125), $MachinePrecision] * beta + 0.25), $MachinePrecision] * beta + 0.5), $MachinePrecision], N[(1.0 - N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 5.00000000000000041e-6

   1. Initial program 7.3%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in beta around 0

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 98.9

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

   8. Applied rewrites 98.9%

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

   if 5.00000000000000041e-6 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.55000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\beta \cdot \left(\frac{1}{4} + \beta \cdot \left(\frac{1}{16} \cdot \beta - \frac{1}{8}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \beta \cdot \left(\frac{1}{4} + \beta \cdot \left(\frac{1}{16} \cdot \beta - \frac{1}{8}\right)\right) + \frac{1}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{4} + \beta \cdot \left(\frac{1}{16} \cdot \beta - \frac{1}{8}\right)\right) \cdot \beta + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4} + \beta \cdot \left(\frac{1}{16} \cdot \beta - \frac{1}{8}\right), \beta, \frac{1}{2}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\beta \cdot \left(\frac{1}{16} \cdot \beta - \frac{1}{8}\right) + \frac{1}{4}, \beta, \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{16} \cdot \beta - \frac{1}{8}\right) \cdot \beta + \frac{1}{4}, \beta, \frac{1}{2}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16} \cdot \beta - \frac{1}{8}, \beta, \frac{1}{4}\right), \beta, \frac{1}{2}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16} \cdot \beta - \frac{1}{8}, \beta, \frac{1}{4}\right), \beta, \frac{1}{2}\right) \]

      8. lower-*.f64 98.4

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.0625 \cdot \beta - 0.125, \beta, 0.25\right), \beta, 0.5\right) \]

   8. Applied rewrites 98.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.0625 \cdot \beta - 0.125, \beta, 0.25\right), \color{blue}{\beta}, 0.5\right) \]

   if 0.55000000000000004 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. div-add N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. div-add N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in alpha around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-neg.f64 97.2

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

   8. Applied rewrites 97.2%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.55:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0625 \cdot \beta - 0.125, \beta, 0.25\right), \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\alpha + 1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.55:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\alpha + 1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 5e-6)
     (/ (+ 1.0 beta) alpha)
     (if (<= t_0 0.55)
       (fma (fma -0.125 beta 0.25) beta 0.5)
       (- 1.0 (/ (+ alpha 1.0) beta))))))

C

double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 5e-6) {
		tmp = (1.0 + beta) / alpha;
	} else if (t_0 <= 0.55) {
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	} else {
		tmp = 1.0 - ((alpha + 1.0) / beta);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_0 <= 5e-6)
		tmp = Float64(Float64(1.0 + beta) / alpha);
	elseif (t_0 <= 0.55)
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	else
		tmp = Float64(1.0 - Float64(Float64(alpha + 1.0) / beta));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-6], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.55], N[(N[(-0.125 * beta + 0.25), $MachinePrecision] * beta + 0.5), $MachinePrecision], N[(1.0 - N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 5.00000000000000041e-6

   1. Initial program 7.3%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in beta around 0

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 98.9

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

   8. Applied rewrites 98.9%

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

   if 5.00000000000000041e-6 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.55000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

         \[\leadsto \beta \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot \beta\right) + \frac{1}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{4} + \frac{-1}{8} \cdot \beta\right) \cdot \beta + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4} + \frac{-1}{8} \cdot \beta, \beta, \frac{1}{2}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{8} \cdot \beta + \frac{1}{4}, \beta, \frac{1}{2}\right) \]

      5. lower-fma.f64 98.4

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \beta, 0.5\right) \]

   8. Applied rewrites 98.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \color{blue}{\beta}, 0.5\right) \]

   if 0.55000000000000004 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. div-add N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. div-add N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in alpha around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-neg.f64 97.2

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

   8. Applied rewrites 97.2%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.55:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\alpha + 1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.55:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{\beta}, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 5e-6)
     (/ (+ 1.0 beta) alpha)
     (if (<= t_0 0.55)
       (fma (fma -0.125 beta 0.25) beta 0.5)
       (fma (/ 2.0 beta) -0.5 1.0)))))

C

double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 5e-6) {
		tmp = (1.0 + beta) / alpha;
	} else if (t_0 <= 0.55) {
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	} else {
		tmp = fma((2.0 / beta), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_0 <= 5e-6)
		tmp = Float64(Float64(1.0 + beta) / alpha);
	elseif (t_0 <= 0.55)
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	else
		tmp = fma(Float64(2.0 / beta), -0.5, 1.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-6], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.55], N[(N[(-0.125 * beta + 0.25), $MachinePrecision] * beta + 0.5), $MachinePrecision], N[(N[(2.0 / beta), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 5.00000000000000041e-6

   1. Initial program 7.3%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in beta around 0

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 98.9

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

   8. Applied rewrites 98.9%

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

   if 5.00000000000000041e-6 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.55000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

         \[\leadsto \beta \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot \beta\right) + \frac{1}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{4} + \frac{-1}{8} \cdot \beta\right) \cdot \beta + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4} + \frac{-1}{8} \cdot \beta, \beta, \frac{1}{2}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{8} \cdot \beta + \frac{1}{4}, \beta, \frac{1}{2}\right) \]

      5. lower-fma.f64 98.4

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \beta, 0.5\right) \]

   8. Applied rewrites 98.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \color{blue}{\beta}, 0.5\right) \]

   if 0.55000000000000004 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. div-add N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. div-add N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 96.6%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.55:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{\beta}, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.55:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 5e-6)
     (/ (+ 1.0 beta) alpha)
     (if (<= t_0 0.55) (fma (fma -0.125 beta 0.25) beta 0.5) 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 5e-6) {
		tmp = (1.0 + beta) / alpha;
	} else if (t_0 <= 0.55) {
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_0 <= 5e-6)
		tmp = Float64(Float64(1.0 + beta) / alpha);
	elseif (t_0 <= 0.55)
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-6], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.55], N[(N[(-0.125 * beta + 0.25), $MachinePrecision] * beta + 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 5.00000000000000041e-6

   1. Initial program 7.3%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in beta around 0

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 98.9

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

   8. Applied rewrites 98.9%

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

   if 5.00000000000000041e-6 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.55000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

         \[\leadsto \beta \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot \beta\right) + \frac{1}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{4} + \frac{-1}{8} \cdot \beta\right) \cdot \beta + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4} + \frac{-1}{8} \cdot \beta, \beta, \frac{1}{2}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{8} \cdot \beta + \frac{1}{4}, \beta, \frac{1}{2}\right) \]

      5. lower-fma.f64 98.4

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \beta, 0.5\right) \]

   8. Applied rewrites 98.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \color{blue}{\beta}, 0.5\right) \]

   if 0.55000000000000004 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 95.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.55:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.55:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 5e-6)
     (/ beta alpha)
     (if (<= t_0 0.55) (fma (fma -0.125 beta 0.25) beta 0.5) 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 5e-6) {
		tmp = beta / alpha;
	} else if (t_0 <= 0.55) {
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_0 <= 5e-6)
		tmp = Float64(beta / alpha);
	elseif (t_0 <= 0.55)
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-6], N[(beta / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.55], N[(N[(-0.125 * beta + 0.25), $MachinePrecision] * beta + 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 5.00000000000000041e-6

   1. Initial program 7.3%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in beta around inf

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

      1. lower-/.f64 25.8

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

   8. Applied rewrites 25.8%

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

   if 5.00000000000000041e-6 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.55000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

         \[\leadsto \beta \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot \beta\right) + \frac{1}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{4} + \frac{-1}{8} \cdot \beta\right) \cdot \beta + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4} + \frac{-1}{8} \cdot \beta, \beta, \frac{1}{2}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{8} \cdot \beta + \frac{1}{4}, \beta, \frac{1}{2}\right) \]

      5. lower-fma.f64 98.4

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \beta, 0.5\right) \]

   8. Applied rewrites 98.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \color{blue}{\beta}, 0.5\right) \]

   if 0.55000000000000004 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 95.0%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.55:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 4 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\alpha}, 2, \frac{2}{\alpha}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\frac{2 + \alpha}{\beta} + 1\right) \cdot \beta} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 4e-15)
   (* (fma (/ beta alpha) 2.0 (/ 2.0 alpha)) 0.5)
   (/ (+ (/ (- beta alpha) (* (+ (/ (+ 2.0 alpha) beta) 1.0) beta)) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 4e-15) {
		tmp = fma((beta / alpha), 2.0, (2.0 / alpha)) * 0.5;
	} else {
		tmp = (((beta - alpha) / ((((2.0 + alpha) / beta) + 1.0) * beta)) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0) <= 4e-15)
		tmp = Float64(fma(Float64(beta / alpha), 2.0, Float64(2.0 / alpha)) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(Float64(Float64(2.0 + alpha) / beta) + 1.0) * beta)) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 4e-15], N[(N[(N[(beta / alpha), $MachinePrecision] * 2.0 + N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(N[(N[(2.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] + 1.0), $MachinePrecision] * beta), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 4.0000000000000003e-15

   1. Initial program 6.1%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 4.0000000000000003e-15 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 4 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\alpha}, 2, \frac{2}{\alpha}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\frac{2 + \alpha}{\beta} + 1\right) \cdot \beta} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.55:\\ \;\;\;\;\mathsf{fma}\left(0.25, \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 5e-6)
     (/ beta alpha)
     (if (<= t_0 0.55) (fma 0.25 beta 0.5) 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 5e-6) {
		tmp = beta / alpha;
	} else if (t_0 <= 0.55) {
		tmp = fma(0.25, beta, 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_0 <= 5e-6)
		tmp = Float64(beta / alpha);
	elseif (t_0 <= 0.55)
		tmp = fma(0.25, beta, 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-6], N[(beta / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.55], N[(0.25 * beta + 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 5.00000000000000041e-6

   1. Initial program 7.3%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in beta around inf

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

      1. lower-/.f64 25.8

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

   8. Applied rewrites 25.8%

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

   if 5.00000000000000041e-6 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.55000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 97.9

         \[\leadsto \mathsf{fma}\left(0.25, \beta, 0.5\right) \]

   8. Applied rewrites 97.9%

      \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{\beta}, 0.5\right) \]

   if 0.55000000000000004 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 95.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \mathbf{elif}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.55:\\ \;\;\;\;\mathsf{fma}\left(0.25, \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\alpha}, 2, \frac{2}{\alpha}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 4e-15) (* (fma (/ beta alpha) 2.0 (/ 2.0 alpha)) 0.5) t_0)))

C

double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 4e-15) {
		tmp = fma((beta / alpha), 2.0, (2.0 / alpha)) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_0 <= 4e-15)
		tmp = Float64(fma(Float64(beta / alpha), 2.0, Float64(2.0 / alpha)) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-15], N[(N[(N[(beta / alpha), $MachinePrecision] * 2.0 + N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 4.0000000000000003e-15

   1. Initial program 6.1%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 4.0000000000000003e-15 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 4 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\alpha}, 2, \frac{2}{\alpha}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\alpha}, 2, \frac{2}{\alpha}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\beta + \left(2 + \beta\right)\right)}{2 + \beta} \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 5e-6)
   (* (fma (/ beta alpha) 2.0 (/ 2.0 alpha)) 0.5)
   (* (/ (* 2.0 (+ beta (+ 2.0 beta))) (+ 2.0 beta)) 0.25)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 5e-6) {
		tmp = fma((beta / alpha), 2.0, (2.0 / alpha)) * 0.5;
	} else {
		tmp = ((2.0 * (beta + (2.0 + beta))) / (2.0 + beta)) * 0.25;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0) <= 5e-6)
		tmp = Float64(fma(Float64(beta / alpha), 2.0, Float64(2.0 / alpha)) * 0.5);
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(beta + Float64(2.0 + beta))) / Float64(2.0 + beta)) * 0.25);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 5e-6], N[(N[(N[(beta / alpha), $MachinePrecision] * 2.0 + N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(2.0 * N[(beta + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 5.00000000000000041e-6

   1. Initial program 7.3%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 98.9

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

   7. Applied rewrites 98.9%

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

   if 5.00000000000000041e-6 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 98.1

         \[\leadsto \frac{2 \cdot \left(\beta + \left(2 + \beta\right)\right)}{2 + \beta} \cdot 0.25 \]

   7. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\beta + \left(2 + \beta\right)\right)}{2 + \beta} \cdot 0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\alpha}, 2, \frac{2}{\alpha}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\beta + \left(2 + \beta\right)\right)}{2 + \beta} \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\beta + \left(2 + \beta\right)\right)}{2 + \beta} \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 5e-6)
   (/ (+ 1.0 beta) alpha)
   (* (/ (* 2.0 (+ beta (+ 2.0 beta))) (+ 2.0 beta)) 0.25)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 5e-6) {
		tmp = (1.0 + beta) / alpha;
	} else {
		tmp = ((2.0 * (beta + (2.0 + beta))) / (2.0 + beta)) * 0.25;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (((((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0) <= 5d-6) then
        tmp = (1.0d0 + beta) / alpha
    else
        tmp = ((2.0d0 * (beta + (2.0d0 + beta))) / (2.0d0 + beta)) * 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 5e-6) {
		tmp = (1.0 + beta) / alpha;
	} else {
		tmp = ((2.0 * (beta + (2.0 + beta))) / (2.0 + beta)) * 0.25;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 5e-6:
		tmp = (1.0 + beta) / alpha
	else:
		tmp = ((2.0 * (beta + (2.0 + beta))) / (2.0 + beta)) * 0.25
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0) <= 5e-6)
		tmp = Float64(Float64(1.0 + beta) / alpha);
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(beta + Float64(2.0 + beta))) / Float64(2.0 + beta)) * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 5e-6)
		tmp = (1.0 + beta) / alpha;
	else
		tmp = ((2.0 * (beta + (2.0 + beta))) / (2.0 + beta)) * 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 5e-6], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(2.0 * N[(beta + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 5.00000000000000041e-6

   1. Initial program 7.3%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in beta around 0

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 98.9

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

   8. Applied rewrites 98.9%

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

   if 5.00000000000000041e-6 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 98.1

         \[\leadsto \frac{2 \cdot \left(\beta + \left(2 + \beta\right)\right)}{2 + \beta} \cdot 0.25 \]

   7. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\beta + \left(2 + \beta\right)\right)}{2 + \beta} \cdot 0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\beta + \left(2 + \beta\right)\right)}{2 + \beta} \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\beta}{2 + \beta} + 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 5e-6)
   (/ (+ 1.0 beta) alpha)
   (* (+ (/ beta (+ 2.0 beta)) 1.0) 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 5e-6) {
		tmp = (1.0 + beta) / alpha;
	} else {
		tmp = ((beta / (2.0 + beta)) + 1.0) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (((((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0) <= 5d-6) then
        tmp = (1.0d0 + beta) / alpha
    else
        tmp = ((beta / (2.0d0 + beta)) + 1.0d0) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 5e-6) {
		tmp = (1.0 + beta) / alpha;
	} else {
		tmp = ((beta / (2.0 + beta)) + 1.0) * 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 5e-6:
		tmp = (1.0 + beta) / alpha
	else:
		tmp = ((beta / (2.0 + beta)) + 1.0) * 0.5
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0) <= 5e-6)
		tmp = Float64(Float64(1.0 + beta) / alpha);
	else
		tmp = Float64(Float64(Float64(beta / Float64(2.0 + beta)) + 1.0) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 5e-6)
		tmp = (1.0 + beta) / alpha;
	else
		tmp = ((beta / (2.0 + beta)) + 1.0) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 5e-6], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 5.00000000000000041e-6

   1. Initial program 7.3%

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

   3. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in beta around 0

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 98.9

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

   8. Applied rewrites 98.9%

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

   if 5.00000000000000041e-6 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 100.0%

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

   3. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\beta}{2 + \beta} + 1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 0.6], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 70.1%

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

   3. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 67.9

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

   5. Applied rewrites 67.9%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.6%

      \[\leadsto 0.5 \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 96.0%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.6:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\mathsf{fma}\left(0.25, \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0) (fma 0.25 beta 0.5) 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = fma(0.25, beta, 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = fma(0.25, beta, 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(0.25 * beta + 0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 74.7%

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

   3. Taylor expanded in alpha around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 72.8

         \[\leadsto \mathsf{fma}\left(0.25, \beta, 0.5\right) \]

   8. Applied rewrites 72.8%

      \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{\beta}, 0.5\right) \]

   if 2 < beta

   1. Initial program 87.4%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 82.6%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\mathsf{fma}\left(0.25, \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.0% accurate, 35.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 1.0)

C

double code(double alpha, double beta) {
	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)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 1.0d0
end function

Java

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

Python

def code(alpha, beta):
	return 1.0

Julia

function code(alpha, beta)
	return 1.0
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 1.0

Derivation

1. Initial program 79.3%

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

3. Taylor expanded in beta around inf

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

5. Applied rewrites 39.5%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 39.5%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))