Octave 3.8, jcobi/1

Percentage Accurate: 73.9% → 99.9%

Time: 3.6s

Alternatives: 15

Speedup: 1.7×

Specification

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

Math

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

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: 73.9% accurate, 1.0× speedup

Math

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

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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (/ (- (* -2.0 beta) 2.0) (* (- (- -2.0 alpha) beta) 2.0)))

C

double code(double alpha, double beta) {
	return ((-2.0 * beta) - 2.0) / (((-2.0 - alpha) - beta) * 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 = (((-2.0d0) * beta) - 2.0d0) / ((((-2.0d0) - alpha) - beta) * 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. frac-2neg N/A

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

   4. remove-double-neg N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. frac-2neg N/A

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

   9. add-to-fraction N/A

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

   10. remove-double-neg N/A

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

   11. associate-/l/ N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 74.2%

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

4. Taylor expanded in beta around 0

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 99.9%

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

6. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (* (/ -0.5 (- beta (- -2.0 alpha))) (fma beta -2.0 -2.0)))

C

double code(double alpha, double beta) {
	return (-0.5 / (beta - (-2.0 - alpha))) * fma(beta, -2.0, -2.0);
}

Julia

function code(alpha, beta)
	return Float64(Float64(-0.5 / Float64(beta - Float64(-2.0 - alpha))) * fma(beta, -2.0, -2.0))
end

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. frac-2neg N/A

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

   4. remove-double-neg N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. frac-2neg N/A

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

   9. add-to-fraction N/A

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

   10. remove-double-neg N/A

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

   11. associate-/l/ N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 74.2%

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

4. Taylor expanded in beta around 0

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 99.9%

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

6. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

8. Applied rewrites 99.9%

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

Alternative 3: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.001:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, -2, -2\right)}{\mathsf{fma}\left(\alpha, -2, -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\alpha - \left(-2 - \beta\right)}, 0.5, 0.5\right)\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 0.001)
   (/ (fma beta -2.0 -2.0) (fma alpha -2.0 -4.0))
   (fma (/ (- beta alpha) (- alpha (- -2.0 beta))) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.001) {
		tmp = fma(beta, -2.0, -2.0) / fma(alpha, -2.0, -4.0);
	} else {
		tmp = fma(((beta - alpha) / (alpha - (-2.0 - beta))), 0.5, 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) <= 0.001)
		tmp = Float64(fma(beta, -2.0, -2.0) / fma(alpha, -2.0, -4.0));
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(alpha - Float64(-2.0 - beta))), 0.5, 0.5);
	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], 0.001], N[(N[(beta * -2.0 + -2.0), $MachinePrecision] / N[(alpha * -2.0 + -4.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / N[(alpha - N[(-2.0 - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 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)) < 1e-3

   1. Initial program 73.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. add-to-fraction N/A

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

      10. remove-double-neg N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 74.2%

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

   4. Taylor expanded in beta around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 99.9%

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in beta around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 72.2%

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

   9. Applied rewrites 72.2%

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

       1. lift--.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. metadata-eval N/A

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

       5. add-flip N/A

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

       6. lift-fma.f64 72.2%

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

       7. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \color{blue}{-2}, -2\right)}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(-2 \cdot \left(2 + \alpha\right)\right)\right)} \]

       8. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \color{blue}{-2}, -2\right)}{-2 \cdot \mathsf{Rewrite=>}\left(lift-+.f64, \left(2 + \alpha\right)\right)} \]

       9. lift-fma.f64 N/A

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

       10. lift-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\beta, \color{blue}{-2}, -2\right)}{\mathsf{Rewrite=>}\left(distribute-rgt-in, \left(\alpha \cdot -2 + 2 \cdot -2\right)\right)} \]

       11. lift-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\beta, \color{blue}{-2}, -2\right)}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\alpha, -2, 2 \cdot -2\right)\right)\right)} \]

       12. lift-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\beta, \color{blue}{-2}, -2\right)}{\mathsf{fma}\left(\alpha, -2, \mathsf{Rewrite=>}\left(metadata-eval, -4\right)\right)} \]

   11. Applied rewrites 72.2%

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

   if 1e-3 < (/.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 73.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. mult-flip N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. add-flip N/A

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

      9. associate-+l- N/A

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

      10. lower--.f64 N/A

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

      11. sub-flip-reverse N/A

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

      12. distribute-neg-out N/A

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

      13. add-flip N/A

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

      14. sub-negate N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval 73.9%

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

   3. Applied rewrites 73.9%

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

Alternative 4: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.001:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, -2, -2\right)}{\mathsf{fma}\left(\alpha, -2, -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \beta}, 0.5, 0.5\right)\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 0.001)
   (/ (fma beta -2.0 -2.0) (fma alpha -2.0 -4.0))
   (fma (/ (- beta alpha) (+ 2.0 beta)) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.001) {
		tmp = fma(beta, -2.0, -2.0) / fma(alpha, -2.0, -4.0);
	} else {
		tmp = fma(((beta - alpha) / (2.0 + beta)), 0.5, 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) <= 0.001)
		tmp = Float64(fma(beta, -2.0, -2.0) / fma(alpha, -2.0, -4.0));
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(2.0 + beta)), 0.5, 0.5);
	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], 0.001], N[(N[(beta * -2.0 + -2.0), $MachinePrecision] / N[(alpha * -2.0 + -4.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * 0.5 + 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)) < 1e-3

   1. Initial program 73.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. add-to-fraction N/A

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

      10. remove-double-neg N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 74.2%

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

   4. Taylor expanded in beta around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 99.9%

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in beta around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 72.2%

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

   9. Applied rewrites 72.2%

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

       1. lift--.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. metadata-eval N/A

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

       5. add-flip N/A

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

       6. lift-fma.f64 72.2%

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

       7. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \color{blue}{-2}, -2\right)}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(-2 \cdot \left(2 + \alpha\right)\right)\right)} \]

       8. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \color{blue}{-2}, -2\right)}{-2 \cdot \mathsf{Rewrite=>}\left(lift-+.f64, \left(2 + \alpha\right)\right)} \]

       9. lift-fma.f64 N/A

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

       10. lift-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\beta, \color{blue}{-2}, -2\right)}{\mathsf{Rewrite=>}\left(distribute-rgt-in, \left(\alpha \cdot -2 + 2 \cdot -2\right)\right)} \]

       11. lift-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\beta, \color{blue}{-2}, -2\right)}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\alpha, -2, 2 \cdot -2\right)\right)\right)} \]

       12. lift-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\beta, \color{blue}{-2}, -2\right)}{\mathsf{fma}\left(\alpha, -2, \mathsf{Rewrite=>}\left(metadata-eval, -4\right)\right)} \]

   11. Applied rewrites 72.2%

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

   if 1e-3 < (/.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 73.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. mult-flip N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. add-flip N/A

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

      9. associate-+l- N/A

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

      10. lower--.f64 N/A

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

      11. sub-flip-reverse N/A

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

      12. distribute-neg-out N/A

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

      13. add-flip N/A

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

      14. sub-negate N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval 73.9%

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

   3. Applied rewrites 73.9%

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

   4. Taylor expanded in alpha around 0

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

      1. lower-+.f64 71.5%

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

   6. Applied rewrites 71.5%

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

Alternative 5: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.500002:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, -2, -2\right)}{\mathsf{fma}\left(\alpha, -2, -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\beta, \frac{-0.5}{\left(-2 - \alpha\right) - \beta}, 0.5\right)\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 0.500002)
   (/ (fma beta -2.0 -2.0) (fma alpha -2.0 -4.0))
   (fma beta (/ -0.5 (- (- -2.0 alpha) beta)) 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.500002) {
		tmp = fma(beta, -2.0, -2.0) / fma(alpha, -2.0, -4.0);
	} else {
		tmp = fma(beta, (-0.5 / ((-2.0 - alpha) - beta)), 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) <= 0.500002)
		tmp = Float64(fma(beta, -2.0, -2.0) / fma(alpha, -2.0, -4.0));
	else
		tmp = fma(beta, Float64(-0.5 / Float64(Float64(-2.0 - alpha) - beta)), 0.5);
	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], 0.500002], N[(N[(beta * -2.0 + -2.0), $MachinePrecision] / N[(alpha * -2.0 + -4.0), $MachinePrecision]), $MachinePrecision], N[(beta * N[(-0.5 / N[(N[(-2.0 - alpha), $MachinePrecision] - beta), $MachinePrecision]), $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)) < 0.50000199999999995

   1. Initial program 73.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. add-to-fraction N/A

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

      10. remove-double-neg N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 74.2%

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

   4. Taylor expanded in beta around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 99.9%

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in beta around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 72.2%

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

   9. Applied rewrites 72.2%

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

       1. lift--.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. metadata-eval N/A

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

       5. add-flip N/A

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

       6. lift-fma.f64 72.2%

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

       7. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \color{blue}{-2}, -2\right)}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(-2 \cdot \left(2 + \alpha\right)\right)\right)} \]

       8. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \color{blue}{-2}, -2\right)}{-2 \cdot \mathsf{Rewrite=>}\left(lift-+.f64, \left(2 + \alpha\right)\right)} \]

       9. lift-fma.f64 N/A

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

       10. lift-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\beta, \color{blue}{-2}, -2\right)}{\mathsf{Rewrite=>}\left(distribute-rgt-in, \left(\alpha \cdot -2 + 2 \cdot -2\right)\right)} \]

       11. lift-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\beta, \color{blue}{-2}, -2\right)}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\alpha, -2, 2 \cdot -2\right)\right)\right)} \]

       12. lift-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\beta, \color{blue}{-2}, -2\right)}{\mathsf{fma}\left(\alpha, -2, \mathsf{Rewrite=>}\left(metadata-eval, -4\right)\right)} \]

   11. Applied rewrites 72.2%

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

   if 0.50000199999999995 < (/.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 73.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. mult-flip N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. add-flip N/A

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

      9. associate-+l- N/A

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

      10. lower--.f64 N/A

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

      11. sub-flip-reverse N/A

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

      12. distribute-neg-out N/A

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

      13. add-flip N/A

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

      14. sub-negate N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval 73.9%

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

   3. Applied rewrites 73.9%

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

   4. Taylor expanded in alpha around 0

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

   6. Applied rewrites 72.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lift--.f64 N/A

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

      9. sub-flip N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. associate--l+ N/A

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

      13. sub-negate-rev N/A

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

      14. associate--l- N/A

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

      15. lift--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. metadata-eval 72.1%

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

   8. Applied rewrites 72.1%

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

Alternative 6: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.500002:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, -2, -2\right)}{\mathsf{fma}\left(\alpha, -2, -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\beta - -2}, 0.5, 0.5\right)\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 0.500002)
   (/ (fma beta -2.0 -2.0) (fma alpha -2.0 -4.0))
   (fma (/ beta (- beta -2.0)) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.500002) {
		tmp = fma(beta, -2.0, -2.0) / fma(alpha, -2.0, -4.0);
	} else {
		tmp = fma((beta / (beta - -2.0)), 0.5, 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) <= 0.500002)
		tmp = Float64(fma(beta, -2.0, -2.0) / fma(alpha, -2.0, -4.0));
	else
		tmp = fma(Float64(beta / Float64(beta - -2.0)), 0.5, 0.5);
	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], 0.500002], N[(N[(beta * -2.0 + -2.0), $MachinePrecision] / N[(alpha * -2.0 + -4.0), $MachinePrecision]), $MachinePrecision], N[(N[(beta / N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + 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)) < 0.50000199999999995

   1. Initial program 73.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. add-to-fraction N/A

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

      10. remove-double-neg N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 74.2%

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

   4. Taylor expanded in beta around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 99.9%

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in beta around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 72.2%

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

   9. Applied rewrites 72.2%

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

       1. lift--.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. metadata-eval N/A

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

       5. add-flip N/A

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

       6. lift-fma.f64 72.2%

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

       7. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \color{blue}{-2}, -2\right)}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(-2 \cdot \left(2 + \alpha\right)\right)\right)} \]

       8. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\beta, \color{blue}{-2}, -2\right)}{-2 \cdot \mathsf{Rewrite=>}\left(lift-+.f64, \left(2 + \alpha\right)\right)} \]

       9. lift-fma.f64 N/A

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

       10. lift-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\beta, \color{blue}{-2}, -2\right)}{\mathsf{Rewrite=>}\left(distribute-rgt-in, \left(\alpha \cdot -2 + 2 \cdot -2\right)\right)} \]

       11. lift-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\beta, \color{blue}{-2}, -2\right)}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\alpha, -2, 2 \cdot -2\right)\right)\right)} \]

       12. lift-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\beta, \color{blue}{-2}, -2\right)}{\mathsf{fma}\left(\alpha, -2, \mathsf{Rewrite=>}\left(metadata-eval, -4\right)\right)} \]

   11. Applied rewrites 72.2%

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

   if 0.50000199999999995 < (/.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 73.9%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 72.1%

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

   4. Applied rewrites 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 72.1%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. add-flip-rev N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 72.1%

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

   6. Applied rewrites 72.1%

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

Alternative 7: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.001:\\ \;\;\;\;\frac{-2 \cdot \beta - 2}{-2 \cdot \alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\beta - -2}, 0.5, 0.5\right)\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 0.001)
   (/ (- (* -2.0 beta) 2.0) (* -2.0 alpha))
   (fma (/ beta (- beta -2.0)) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.001) {
		tmp = ((-2.0 * beta) - 2.0) / (-2.0 * alpha);
	} else {
		tmp = fma((beta / (beta - -2.0)), 0.5, 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) <= 0.001)
		tmp = Float64(Float64(Float64(-2.0 * beta) - 2.0) / Float64(-2.0 * alpha));
	else
		tmp = fma(Float64(beta / Float64(beta - -2.0)), 0.5, 0.5);
	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], 0.001], N[(N[(N[(-2.0 * beta), $MachinePrecision] - 2.0), $MachinePrecision] / N[(-2.0 * alpha), $MachinePrecision]), $MachinePrecision], N[(N[(beta / N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + 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)) < 1e-3

   1. Initial program 73.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. add-to-fraction N/A

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

      10. remove-double-neg N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 74.2%

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

   4. Taylor expanded in beta around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 99.9%

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in alpha around inf

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

      1. lower-*.f64 30.1%

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

   9. Applied rewrites 30.1%

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

   if 1e-3 < (/.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 73.9%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 72.1%

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

   4. Applied rewrites 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 72.1%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. add-flip-rev N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 72.1%

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

   6. Applied rewrites 72.1%

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

Alternative 8: 93.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\alpha - -2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\beta - -2}, 0.5, 0.5\right)\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.5e-6)
   (/ 1.0 (- alpha -2.0))
   (fma (/ beta (- beta -2.0)) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.5e-6) {
		tmp = 1.0 / (alpha - -2.0);
	} else {
		tmp = fma((beta / (beta - -2.0)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.5e-6)
		tmp = Float64(1.0 / Float64(alpha - -2.0));
	else
		tmp = fma(Float64(beta / Float64(beta - -2.0)), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 8.5e-6], N[(1.0 / N[(alpha - -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(beta / N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.4999999999999999e-6

   1. Initial program 73.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. add-to-fraction N/A

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

      10. remove-double-neg N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 74.2%

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

   4. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 69.8%

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

   6. Applied rewrites 69.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip-rev N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 69.8%

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

   8. Applied rewrites 69.8%

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

   if 8.4999999999999999e-6 < beta

   1. Initial program 73.9%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 72.1%

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

   4. Applied rewrites 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 72.1%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. add-flip-rev N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 72.1%

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

   6. Applied rewrites 72.1%

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

Alternative 9: 92.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2300:\\ \;\;\;\;\frac{1}{\alpha - -2}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2300.0) (/ 1.0 (- alpha -2.0)) (- 1.0 (/ 1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2300.0) {
		tmp = 1.0 / (alpha - -2.0);
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	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 <= 2300.0d0) then
        tmp = 1.0d0 / (alpha - (-2.0d0))
    else
        tmp = 1.0d0 - (1.0d0 / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2300.0) {
		tmp = 1.0 / (alpha - -2.0);
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2300.0:
		tmp = 1.0 / (alpha - -2.0)
	else:
		tmp = 1.0 - (1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2300.0)
		tmp = Float64(1.0 / Float64(alpha - -2.0));
	else
		tmp = Float64(1.0 - Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2300.0)
		tmp = 1.0 / (alpha - -2.0);
	else
		tmp = 1.0 - (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2300.0], N[(1.0 / N[(alpha - -2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2300

   1. Initial program 73.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. add-to-fraction N/A

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

      10. remove-double-neg N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 74.2%

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

   4. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 69.8%

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

   6. Applied rewrites 69.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip-rev N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 69.8%

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

   8. Applied rewrites 69.8%

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

   if 2300 < beta

   1. Initial program 73.9%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 72.1%

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

   4. Applied rewrites 72.1%

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

   5. Taylor expanded in beta around inf

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 30.1%

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

   7. Applied rewrites 30.1%

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

Alternative 10: 92.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 0.001:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.6:\\ \;\;\;\;0.5 + -0.25 \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 0.001)
     (/ 1.0 alpha)
     (if (<= t_0 0.6) (+ 0.5 (* -0.25 alpha)) (- 1.0 (/ 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 <= 0.001) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.6) {
		tmp = 0.5 + (-0.25 * alpha);
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
    if (t_0 <= 0.001d0) then
        tmp = 1.0d0 / alpha
    else if (t_0 <= 0.6d0) then
        tmp = 0.5d0 + ((-0.25d0) * alpha)
    else
        tmp = 1.0d0 - (1.0d0 / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 0.001) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.6) {
		tmp = 0.5 + (-0.25 * alpha);
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0
	tmp = 0
	if t_0 <= 0.001:
		tmp = 1.0 / alpha
	elif t_0 <= 0.6:
		tmp = 0.5 + (-0.25 * alpha)
	else:
		tmp = 1.0 - (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 <= 0.001)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.6)
		tmp = Float64(0.5 + Float64(-0.25 * alpha));
	else
		tmp = Float64(1.0 - Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	tmp = 0.0;
	if (t_0 <= 0.001)
		tmp = 1.0 / alpha;
	elseif (t_0 <= 0.6)
		tmp = 0.5 + (-0.25 * alpha);
	else
		tmp = 1.0 - (1.0 / beta);
	end
	tmp_2 = 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, 0.001], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.6], N[(0.5 + N[(-0.25 * alpha), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(1.0 / 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)) < 1e-3

   1. Initial program 73.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. add-to-fraction N/A

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

      10. remove-double-neg N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 74.2%

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

   4. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 69.8%

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

   6. Applied rewrites 69.8%

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

   7. Taylor expanded in alpha around inf

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

      1. lower-/.f64 24.8%

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

   9. Applied rewrites 24.8%

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

   if 1e-3 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.59999999999999998

   1. Initial program 73.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. add-to-fraction N/A

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

      10. remove-double-neg N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 74.2%

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

   4. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 69.8%

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

   6. Applied rewrites 69.8%

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

   7. Taylor expanded in alpha around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 46.9%

         \[\leadsto 0.5 + -0.25 \cdot \alpha \]

   9. Applied rewrites 46.9%

      \[\leadsto 0.5 + \color{blue}{-0.25 \cdot \alpha} \]

   if 0.59999999999999998 < (/.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 73.9%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 72.1%

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

   4. Applied rewrites 72.1%

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

   5. Taylor expanded in beta around inf

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 30.1%

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

   7. Applied rewrites 30.1%

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

Alternative 11: 92.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 0.001:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.6:\\ \;\;\;\;0.5 + -0.25 \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 0.001)
     (/ 1.0 alpha)
     (if (<= t_0 0.6) (+ 0.5 (* -0.25 alpha)) 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 <= 0.001) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.6) {
		tmp = 0.5 + (-0.25 * alpha);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
    if (t_0 <= 0.001d0) then
        tmp = 1.0d0 / alpha
    else if (t_0 <= 0.6d0) then
        tmp = 0.5d0 + ((-0.25d0) * alpha)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 0.001) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.6) {
		tmp = 0.5 + (-0.25 * alpha);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0
	tmp = 0
	if t_0 <= 0.001:
		tmp = 1.0 / alpha
	elif t_0 <= 0.6:
		tmp = 0.5 + (-0.25 * alpha)
	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 <= 0.001)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.6)
		tmp = Float64(0.5 + Float64(-0.25 * alpha));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	tmp = 0.0;
	if (t_0 <= 0.001)
		tmp = 1.0 / alpha;
	elseif (t_0 <= 0.6)
		tmp = 0.5 + (-0.25 * alpha);
	else
		tmp = 1.0;
	end
	tmp_2 = 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, 0.001], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.6], N[(0.5 + N[(-0.25 * alpha), $MachinePrecision]), $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)) < 1e-3

   1. Initial program 73.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. add-to-fraction N/A

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

      10. remove-double-neg N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 74.2%

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

   4. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 69.8%

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

   6. Applied rewrites 69.8%

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

   7. Taylor expanded in alpha around inf

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

      1. lower-/.f64 24.8%

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

   9. Applied rewrites 24.8%

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

   if 1e-3 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.59999999999999998

   1. Initial program 73.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. add-to-fraction N/A

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

      10. remove-double-neg N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 74.2%

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

   4. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 69.8%

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

   6. Applied rewrites 69.8%

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

   7. Taylor expanded in alpha around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 46.9%

         \[\leadsto 0.5 + -0.25 \cdot \alpha \]

   9. Applied rewrites 46.9%

      \[\leadsto 0.5 + \color{blue}{-0.25 \cdot \alpha} \]

   if 0.59999999999999998 < (/.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 73.9%

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

   2. Taylor expanded in beta around inf

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

   4. Applied rewrites 37.3%

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

Alternative 12: 91.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 0.001:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.6:\\ \;\;\;\;0.5 + 0.25 \cdot \beta\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 0.001)
     (/ 1.0 alpha)
     (if (<= t_0 0.6) (+ 0.5 (* 0.25 beta)) 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 <= 0.001) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.6) {
		tmp = 0.5 + (0.25 * beta);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
    if (t_0 <= 0.001d0) then
        tmp = 1.0d0 / alpha
    else if (t_0 <= 0.6d0) then
        tmp = 0.5d0 + (0.25d0 * beta)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 0.001) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.6) {
		tmp = 0.5 + (0.25 * beta);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0
	tmp = 0
	if t_0 <= 0.001:
		tmp = 1.0 / alpha
	elif t_0 <= 0.6:
		tmp = 0.5 + (0.25 * beta)
	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 <= 0.001)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.6)
		tmp = Float64(0.5 + Float64(0.25 * beta));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	tmp = 0.0;
	if (t_0 <= 0.001)
		tmp = 1.0 / alpha;
	elseif (t_0 <= 0.6)
		tmp = 0.5 + (0.25 * beta);
	else
		tmp = 1.0;
	end
	tmp_2 = 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, 0.001], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.6], N[(0.5 + N[(0.25 * beta), $MachinePrecision]), $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)) < 1e-3

   1. Initial program 73.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. add-to-fraction N/A

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

      10. remove-double-neg N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 74.2%

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

   4. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 69.8%

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

   6. Applied rewrites 69.8%

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

   7. Taylor expanded in alpha around inf

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

      1. lower-/.f64 24.8%

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

   9. Applied rewrites 24.8%

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

   if 1e-3 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.59999999999999998

   1. Initial program 73.9%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 72.1%

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

   4. Applied rewrites 72.1%

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

   5. Taylor expanded in beta around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 45.0%

         \[\leadsto 0.5 + 0.25 \cdot \beta \]

   7. Applied rewrites 45.0%

      \[\leadsto 0.5 + \color{blue}{0.25 \cdot \beta} \]

   if 0.59999999999999998 < (/.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 73.9%

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

   2. Taylor expanded in beta around inf

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

   4. Applied rewrites 37.3%

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

Alternative 13: 91.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 0.001:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.6:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 0.001) (/ 1.0 alpha) (if (<= t_0 0.6) 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 <= 0.001) {
		tmp = 1.0 / alpha;
	} else if (t_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) :: t_0
    real(8) :: tmp
    t_0 = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
    if (t_0 <= 0.001d0) then
        tmp = 1.0d0 / alpha
    else if (t_0 <= 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 t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 0.001) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.6) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0
	tmp = 0
	if t_0 <= 0.001:
		tmp = 1.0 / alpha
	elif t_0 <= 0.6:
		tmp = 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 <= 0.001)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.6)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	tmp = 0.0;
	if (t_0 <= 0.001)
		tmp = 1.0 / alpha;
	elseif (t_0 <= 0.6)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = 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, 0.001], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.6], 0.5, 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)) < 1e-3

   1. Initial program 73.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. add-to-fraction N/A

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

      10. remove-double-neg N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 74.2%

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

   4. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 69.8%

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

   6. Applied rewrites 69.8%

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

   7. Taylor expanded in alpha around inf

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

      1. lower-/.f64 24.8%

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

   9. Applied rewrites 24.8%

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

   if 1e-3 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.59999999999999998

   1. Initial program 73.9%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 72.1%

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

   4. Applied rewrites 72.1%

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

   5. Taylor expanded in beta around 0

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

   7. Applied rewrites 48.5%

      \[\leadsto 0.5 \]

   if 0.59999999999999998 < (/.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 73.9%

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

   2. Taylor expanded in beta around inf

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

   4. Applied rewrites 37.3%

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

Alternative 14: 70.8% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 0.98)
   0.5
   1.0))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta)
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) <= 0.98d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.98) {
		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) <= 0.98:
		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) <= 0.98)
		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) <= 0.98)
		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], 0.98], 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)) < 0.97999999999999998

   1. Initial program 73.9%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 72.1%

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

   4. Applied rewrites 72.1%

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

   5. Taylor expanded in beta around 0

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

   7. Applied rewrites 48.5%

      \[\leadsto 0.5 \]

   if 0.97999999999999998 < (/.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 73.9%

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

   2. Taylor expanded in beta around inf

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

   4. Applied rewrites 37.3%

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

Alternative 15: 37.3% accurate, 18.5× speedup

Math

\[1 \]

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 73.9%

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

2. Taylor expanded in beta around inf

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

4. Applied rewrites 37.3%

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

Reproduce

herbie shell --seed 2025191 
(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))