Octave 3.8, jcobi/3

Percentage Accurate: 94.2% → 99.8%

Time: 6.8s

Alternatives: 17

Speedup: 1.2×

Specification

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

Math

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

FPCore

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

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 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
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * 1.0d0)
    code = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / t_0) / t_0) / (t_0 + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 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
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * 1.0d0)
    code = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / t_0) / t_0) / (t_0 + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\alpha \leq 30500:\\ \;\;\;\;\frac{\frac{\frac{-1 - \mathsf{fma}\left(\alpha - -1, \beta, \alpha\right)}{\alpha - \left(-2 - \beta\right)}}{\left(\alpha + \beta\right) - -3}}{\left(-2 - \alpha\right) - \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\beta, \frac{\alpha}{\beta + \alpha}, 1\right), \frac{\beta + \alpha}{\left(\beta + \alpha\right) - -2}, \frac{1}{\alpha}\right)}{t\_0}}{t\_0 + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= alpha 30500.0)
     (/
      (/
       (/ (- -1.0 (fma (- alpha -1.0) beta alpha)) (- alpha (- -2.0 beta)))
       (- (+ alpha beta) -3.0))
      (- (- -2.0 alpha) beta))
     (/
      (/
       (fma
        (fma beta (/ alpha (+ beta alpha)) 1.0)
        (/ (+ beta alpha) (- (+ beta alpha) -2.0))
        (/ 1.0 alpha))
       t_0)
      (+ t_0 1.0)))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (alpha <= 30500.0) {
		tmp = (((-1.0 - fma((alpha - -1.0), beta, alpha)) / (alpha - (-2.0 - beta))) / ((alpha + beta) - -3.0)) / ((-2.0 - alpha) - beta);
	} else {
		tmp = (fma(fma(beta, (alpha / (beta + alpha)), 1.0), ((beta + alpha) / ((beta + alpha) - -2.0)), (1.0 / alpha)) / t_0) / (t_0 + 1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (alpha <= 30500.0)
		tmp = Float64(Float64(Float64(Float64(-1.0 - fma(Float64(alpha - -1.0), beta, alpha)) / Float64(alpha - Float64(-2.0 - beta))) / Float64(Float64(alpha + beta) - -3.0)) / Float64(Float64(-2.0 - alpha) - beta));
	else
		tmp = Float64(Float64(fma(fma(beta, Float64(alpha / Float64(beta + alpha)), 1.0), Float64(Float64(beta + alpha) / Float64(Float64(beta + alpha) - -2.0)), Float64(1.0 / alpha)) / t_0) / Float64(t_0 + 1.0));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[alpha, 30500.0], N[(N[(N[(N[(-1.0 - N[(N[(alpha - -1.0), $MachinePrecision] * beta + alpha), $MachinePrecision]), $MachinePrecision] / N[(alpha - N[(-2.0 - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision] / N[(N[(-2.0 - alpha), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(beta * N[(alpha / N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(beta + alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 30500

   1. Initial program 94.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-+.f64 N/A

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

      5. sum-to-mult N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-1 - \mathsf{fma}\left(\alpha - -1, \beta, \alpha\right)}{\alpha - \left(-2 - \beta\right)}}{\left(\alpha + \beta\right) - -3}}{\left(-2 - \alpha\right) - \beta}} \]

   if 30500 < alpha

   1. Initial program 94.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-+.f64 N/A

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

      5. sum-to-mult N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in alpha around inf

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

      1. lower-/.f64 40.4

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

   10. Applied rewrites 40.4%

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

Alternative 2: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. div-add N/A

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

   4. lift-+.f64 N/A

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

   5. sum-to-mult N/A

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

   6. associate-/l* N/A

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

   7. lower-fma.f64 N/A

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

3. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. metadata-eval 99.8

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

5. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. metadata-eval 99.8

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

7. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. sum-to-mult N/A

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

   7. times-frac N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 99.8

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

9. Applied rewrites 99.8%

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

Alternative 3: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. div-add N/A

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

   4. lift-+.f64 N/A

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

   5. sum-to-mult N/A

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

   6. associate-/l* N/A

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

   7. lower-fma.f64 N/A

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

3. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. metadata-eval 99.8

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

5. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. metadata-eval 99.8

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

7. Applied rewrites 99.8%

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

Alternative 4: 97.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot 1\\ \mathbf{if}\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}}{t\_0 + 1} \leq 0.1:\\ \;\;\;\;\frac{\frac{\frac{-1 - \mathsf{fma}\left(\alpha - -1, \beta, \alpha\right)}{\alpha - \left(-2 - \beta\right)}}{\left(\alpha + \beta\right) - -3}}{\left(-2 - \alpha\right) - \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha - -1}{\beta}}{1 - \frac{-3}{\alpha + \beta}}}{\alpha + \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 1.0))))
   (if (<=
        (/
         (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_0) t_0)
         (+ t_0 1.0))
        0.1)
     (/
      (/
       (/ (- -1.0 (fma (- alpha -1.0) beta alpha)) (- alpha (- -2.0 beta)))
       (- (+ alpha beta) -3.0))
      (- (- -2.0 alpha) beta))
     (/
      (/ (/ (- alpha -1.0) beta) (- 1.0 (/ -3.0 (+ alpha beta))))
      (+ alpha beta)))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	double tmp;
	if (((((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0)) <= 0.1) {
		tmp = (((-1.0 - fma((alpha - -1.0), beta, alpha)) / (alpha - (-2.0 - beta))) / ((alpha + beta) - -3.0)) / ((-2.0 - alpha) - beta);
	} else {
		tmp = (((alpha - -1.0) / beta) / (1.0 - (-3.0 / (alpha + beta)))) / (alpha + beta);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / t_0) / t_0) / Float64(t_0 + 1.0)) <= 0.1)
		tmp = Float64(Float64(Float64(Float64(-1.0 - fma(Float64(alpha - -1.0), beta, alpha)) / Float64(alpha - Float64(-2.0 - beta))) / Float64(Float64(alpha + beta) - -3.0)) / Float64(Float64(-2.0 - alpha) - beta));
	else
		tmp = Float64(Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(1.0 - Float64(-3.0 / Float64(alpha + beta)))) / Float64(alpha + beta));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], 0.1], N[(N[(N[(N[(-1.0 - N[(N[(alpha - -1.0), $MachinePrecision] * beta + alpha), $MachinePrecision]), $MachinePrecision] / N[(alpha - N[(-2.0 - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision] / N[(N[(-2.0 - alpha), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 - N[(-3.0 / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) #s(literal 1 binary64)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64))) #s(literal 1 binary64))) < 0.10000000000000001

   1. Initial program 94.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-+.f64 N/A

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

      5. sum-to-mult N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-1 - \mathsf{fma}\left(\alpha - -1, \beta, \alpha\right)}{\alpha - \left(-2 - \beta\right)}}{\left(\alpha + \beta\right) - -3}}{\left(-2 - \alpha\right) - \beta}} \]

   if 0.10000000000000001 < (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) #s(literal 1 binary64)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64))) #s(literal 1 binary64)))

   1. Initial program 94.2%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.2

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

   4. Applied rewrites 29.2%

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

   6. Applied rewrites 29.2%

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

Alternative 5: 96.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{-1 - \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{\left(\beta + \alpha\right) - -2}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) - -3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha - -1}{\beta}}{1 - \frac{-3}{\alpha + \beta}}}{\alpha + \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5e+139)
   (/
    (/ (- -1.0 (fma beta alpha (+ beta alpha))) (- (+ beta alpha) -2.0))
    (* (- -2.0 (+ beta alpha)) (- (+ beta alpha) -3.0)))
   (/
    (/ (/ (- alpha -1.0) beta) (- 1.0 (/ -3.0 (+ alpha beta))))
    (+ alpha beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e+139) {
		tmp = ((-1.0 - fma(beta, alpha, (beta + alpha))) / ((beta + alpha) - -2.0)) / ((-2.0 - (beta + alpha)) * ((beta + alpha) - -3.0));
	} else {
		tmp = (((alpha - -1.0) / beta) / (1.0 - (-3.0 / (alpha + beta)))) / (alpha + beta);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5e+139)
		tmp = Float64(Float64(Float64(-1.0 - fma(beta, alpha, Float64(beta + alpha))) / Float64(Float64(beta + alpha) - -2.0)) / Float64(Float64(-2.0 - Float64(beta + alpha)) * Float64(Float64(beta + alpha) - -3.0)));
	else
		tmp = Float64(Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(1.0 - Float64(-3.0 / Float64(alpha + beta)))) / Float64(alpha + beta));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 5e+139], N[(N[(N[(-1.0 - N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(-2.0 - N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + alpha), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 - N[(-3.0 / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.0000000000000003e139

   1. Initial program 94.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 92.5%

      \[\leadsto \color{blue}{\frac{\frac{-1 - \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{\left(\beta + \alpha\right) - -2}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) - -3\right)}} \]

   if 5.0000000000000003e139 < beta

   1. Initial program 94.2%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.2

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

   4. Applied rewrites 29.2%

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

   6. Applied rewrites 29.2%

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

Alternative 6: 92.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) - -2\\ \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{t\_0 \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha - -1}{\beta}}{1 - \frac{-3}{\alpha + \beta}}}{\alpha + \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- (+ beta alpha) -2.0)))
   (if (<= beta 1.8e+16)
     (/
      (- (fma beta alpha (+ beta alpha)) -1.0)
      (* t_0 (* (- (+ beta alpha) -3.0) t_0)))
     (/
      (/ (/ (- alpha -1.0) beta) (- 1.0 (/ -3.0 (+ alpha beta))))
      (+ alpha beta)))))

C

double code(double alpha, double beta) {
	double t_0 = (beta + alpha) - -2.0;
	double tmp;
	if (beta <= 1.8e+16) {
		tmp = (fma(beta, alpha, (beta + alpha)) - -1.0) / (t_0 * (((beta + alpha) - -3.0) * t_0));
	} else {
		tmp = (((alpha - -1.0) / beta) / (1.0 - (-3.0 / (alpha + beta)))) / (alpha + beta);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) - -2.0)
	tmp = 0.0
	if (beta <= 1.8e+16)
		tmp = Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) - -1.0) / Float64(t_0 * Float64(Float64(Float64(beta + alpha) - -3.0) * t_0)));
	else
		tmp = Float64(Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(1.0 - Float64(-3.0 / Float64(alpha + beta)))) / Float64(alpha + beta));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] - -2.0), $MachinePrecision]}, If[LessEqual[beta, 1.8e+16], N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / N[(t$95$0 * N[(N[(N[(beta + alpha), $MachinePrecision] - -3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 - N[(-3.0 / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.8e16

   1. Initial program 94.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 84.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\left(\beta + \alpha\right) - -2\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\left(\beta + \alpha\right) - -2\right)\right)}} \]

   if 1.8e16 < beta

   1. Initial program 94.2%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.2

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

   4. Applied rewrites 29.2%

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

   6. Applied rewrites 29.2%

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

Alternative 7: 92.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. div-add N/A

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

   4. lift-+.f64 N/A

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

   5. sum-to-mult N/A

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

   6. associate-/l* N/A

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

   7. lower-fma.f64 N/A

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

3. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. metadata-eval 99.8

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

5. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. metadata-eval 99.8

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

7. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. sum-to-mult N/A

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

   7. times-frac N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 99.8

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

9. Applied rewrites 99.8%

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

10. Taylor expanded in alpha around 0

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

12. Applied rewrites 85.9%

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

13. Taylor expanded in alpha around 0

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

15. Applied rewrites 88.4%

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

16. Taylor expanded in alpha around 0

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

18. Applied rewrites 89.7%

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

Alternative 8: 91.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{-1 - \mathsf{fma}\left(\beta, \alpha, \beta\right)}{\beta - -2}}{\left(-2 - \beta\right) \cdot \left(\beta - -3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha - -1}{\beta}}{1 - \frac{-3}{\alpha + \beta}}}{\alpha + \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.5e+15)
   (/
    (/ (- -1.0 (fma beta alpha beta)) (- beta -2.0))
    (* (- -2.0 beta) (- beta -3.0)))
   (/
    (/ (/ (- alpha -1.0) beta) (- 1.0 (/ -3.0 (+ alpha beta))))
    (+ alpha beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.5e+15) {
		tmp = ((-1.0 - fma(beta, alpha, beta)) / (beta - -2.0)) / ((-2.0 - beta) * (beta - -3.0));
	} else {
		tmp = (((alpha - -1.0) / beta) / (1.0 - (-3.0 / (alpha + beta)))) / (alpha + beta);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.5e+15)
		tmp = Float64(Float64(Float64(-1.0 - fma(beta, alpha, beta)) / Float64(beta - -2.0)) / Float64(Float64(-2.0 - beta) * Float64(beta - -3.0)));
	else
		tmp = Float64(Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(1.0 - Float64(-3.0 / Float64(alpha + beta)))) / Float64(alpha + beta));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 9.5e+15], N[(N[(N[(-1.0 - N[(beta * alpha + beta), $MachinePrecision]), $MachinePrecision] / N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(-2.0 - beta), $MachinePrecision] * N[(beta - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 - N[(-3.0 / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.5e15

   1. Initial program 94.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 92.5%

      \[\leadsto \color{blue}{\frac{\frac{-1 - \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{\left(\beta + \alpha\right) - -2}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) - -3\right)}} \]

   4. Taylor expanded in alpha around 0

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

   6. Applied rewrites 81.4%

      \[\leadsto \frac{\frac{-1 - \mathsf{fma}\left(\beta, \alpha, \color{blue}{\beta}\right)}{\left(\beta + \alpha\right) - -2}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) - -3\right)} \]

   7. Taylor expanded in alpha around 0

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

   9. Applied rewrites 81.1%

      \[\leadsto \frac{\frac{-1 - \mathsf{fma}\left(\beta, \alpha, \beta\right)}{\color{blue}{\beta} - -2}}{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) - -3\right)} \]

   10. Taylor expanded in alpha around 0

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

   12. Applied rewrites 69.3%

       \[\leadsto \frac{\frac{-1 - \mathsf{fma}\left(\beta, \alpha, \beta\right)}{\beta - -2}}{\left(-2 - \color{blue}{\beta}\right) \cdot \left(\left(\beta + \alpha\right) - -3\right)} \]

   13. Taylor expanded in alpha around 0

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

   15. Applied rewrites 68.4%

       \[\leadsto \frac{\frac{-1 - \mathsf{fma}\left(\beta, \alpha, \beta\right)}{\beta - -2}}{\left(-2 - \beta\right) \cdot \left(\color{blue}{\beta} - -3\right)} \]

   if 9.5e15 < beta

   1. Initial program 94.2%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.2

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

   4. Applied rewrites 29.2%

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

   6. Applied rewrites 29.2%

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

Alternative 9: 89.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.8:\\ \;\;\;\;\frac{\frac{\frac{1 + \alpha}{2 + \alpha}}{2 + \alpha}}{\left(2 + \alpha\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha - -1}{\beta}}{1 - \frac{-3}{\beta}}}{\alpha + \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.8)
   (/ (/ (/ (+ 1.0 alpha) (+ 2.0 alpha)) (+ 2.0 alpha)) (+ (+ 2.0 alpha) 1.0))
   (/ (/ (/ (- alpha -1.0) beta) (- 1.0 (/ -3.0 beta))) (+ alpha beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8) {
		tmp = (((1.0 + alpha) / (2.0 + alpha)) / (2.0 + alpha)) / ((2.0 + alpha) + 1.0);
	} else {
		tmp = (((alpha - -1.0) / beta) / (1.0 - (-3.0 / beta))) / (alpha + 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 <= 3.8d0) then
        tmp = (((1.0d0 + alpha) / (2.0d0 + alpha)) / (2.0d0 + alpha)) / ((2.0d0 + alpha) + 1.0d0)
    else
        tmp = (((alpha - (-1.0d0)) / beta) / (1.0d0 - ((-3.0d0) / beta))) / (alpha + beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8) {
		tmp = (((1.0 + alpha) / (2.0 + alpha)) / (2.0 + alpha)) / ((2.0 + alpha) + 1.0);
	} else {
		tmp = (((alpha - -1.0) / beta) / (1.0 - (-3.0 / beta))) / (alpha + beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 3.8:
		tmp = (((1.0 + alpha) / (2.0 + alpha)) / (2.0 + alpha)) / ((2.0 + alpha) + 1.0)
	else:
		tmp = (((alpha - -1.0) / beta) / (1.0 - (-3.0 / beta))) / (alpha + beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.8)
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + alpha)) / Float64(2.0 + alpha)) / Float64(Float64(2.0 + alpha) + 1.0));
	else
		tmp = Float64(Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(1.0 - Float64(-3.0 / beta))) / Float64(alpha + beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.8)
		tmp = (((1.0 + alpha) / (2.0 + alpha)) / (2.0 + alpha)) / ((2.0 + alpha) + 1.0);
	else
		tmp = (((alpha - -1.0) / beta) / (1.0 - (-3.0 / beta))) / (alpha + beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3.8], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 - N[(-3.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.7999999999999998

   1. Initial program 94.2%

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

   2. Taylor expanded in beta around 0

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

      1. lower-+.f64 70.3

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

   4. Applied rewrites 70.3%

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

   5. Taylor expanded in beta around 0

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

      1. lower-+.f64 69.6

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

   7. Applied rewrites 69.6%

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

   8. Taylor expanded in beta around 0

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

      1. lower-+.f64 69.4

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

   10. Applied rewrites 69.4%

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

   11. Taylor expanded in beta around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-+.f64 70.6

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

   13. Applied rewrites 70.6%

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

   if 3.7999999999999998 < beta

   1. Initial program 94.2%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.2

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

   4. Applied rewrites 29.2%

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

   6. Applied rewrites 29.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\alpha - -1}{\beta}}{1 - \frac{-3}{\alpha + \beta}}}{\alpha + \beta}} \]

   7. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\frac{\alpha - -1}{\beta}}{1 - \color{blue}{\frac{-3}{\beta}}}}{\alpha + \beta} \]
   8. Step-by-step derivation

      1. lower-/.f64 29.7

         \[\leadsto \frac{\frac{\frac{\alpha - -1}{\beta}}{1 - \frac{-3}{\color{blue}{\beta}}}}{\alpha + \beta} \]

   9. Applied rewrites 29.7%

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

Alternative 10: 72.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.8:\\ \;\;\;\;\frac{\frac{\frac{1 + \alpha}{2 + \alpha}}{2 + \alpha}}{\left(2 + \alpha\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha - -1}{\beta}}{1 - \frac{-3}{\alpha + \beta}}}{\alpha + \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.8)
   (/ (/ (/ (+ 1.0 alpha) (+ 2.0 alpha)) (+ 2.0 alpha)) (+ (+ 2.0 alpha) 1.0))
   (/
    (/ (/ (- alpha -1.0) beta) (- 1.0 (/ -3.0 (+ alpha beta))))
    (+ alpha beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8) {
		tmp = (((1.0 + alpha) / (2.0 + alpha)) / (2.0 + alpha)) / ((2.0 + alpha) + 1.0);
	} else {
		tmp = (((alpha - -1.0) / beta) / (1.0 - (-3.0 / (alpha + beta)))) / (alpha + 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 <= 3.8d0) then
        tmp = (((1.0d0 + alpha) / (2.0d0 + alpha)) / (2.0d0 + alpha)) / ((2.0d0 + alpha) + 1.0d0)
    else
        tmp = (((alpha - (-1.0d0)) / beta) / (1.0d0 - ((-3.0d0) / (alpha + beta)))) / (alpha + beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.8) {
		tmp = (((1.0 + alpha) / (2.0 + alpha)) / (2.0 + alpha)) / ((2.0 + alpha) + 1.0);
	} else {
		tmp = (((alpha - -1.0) / beta) / (1.0 - (-3.0 / (alpha + beta)))) / (alpha + beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 3.8:
		tmp = (((1.0 + alpha) / (2.0 + alpha)) / (2.0 + alpha)) / ((2.0 + alpha) + 1.0)
	else:
		tmp = (((alpha - -1.0) / beta) / (1.0 - (-3.0 / (alpha + beta)))) / (alpha + beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.8)
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + alpha)) / Float64(2.0 + alpha)) / Float64(Float64(2.0 + alpha) + 1.0));
	else
		tmp = Float64(Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(1.0 - Float64(-3.0 / Float64(alpha + beta)))) / Float64(alpha + beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.8)
		tmp = (((1.0 + alpha) / (2.0 + alpha)) / (2.0 + alpha)) / ((2.0 + alpha) + 1.0);
	else
		tmp = (((alpha - -1.0) / beta) / (1.0 - (-3.0 / (alpha + beta)))) / (alpha + beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3.8], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 - N[(-3.0 / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.7999999999999998

   1. Initial program 94.2%

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

   2. Taylor expanded in beta around 0

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

      1. lower-+.f64 70.3

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

   4. Applied rewrites 70.3%

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

   5. Taylor expanded in beta around 0

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

      1. lower-+.f64 69.6

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

   7. Applied rewrites 69.6%

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

   8. Taylor expanded in beta around 0

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

      1. lower-+.f64 69.4

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

   10. Applied rewrites 69.4%

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

   11. Taylor expanded in beta around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-+.f64 70.6

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

   13. Applied rewrites 70.6%

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

   if 3.7999999999999998 < beta

   1. Initial program 94.2%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.2

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

   4. Applied rewrites 29.2%

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

   6. Applied rewrites 29.2%

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

Alternative 11: 56.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.5:\\ \;\;\;\;\frac{\frac{-1 \cdot \left(-1 \cdot \alpha - 1\right)}{2 + \alpha}}{\left(2 + \alpha\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha - -1}{\beta}}{1 - \frac{-3}{\beta}}}{\alpha + \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.5)
   (/ (/ (* -1.0 (- (* -1.0 alpha) 1.0)) (+ 2.0 alpha)) (+ (+ 2.0 alpha) 1.0))
   (/ (/ (/ (- alpha -1.0) beta) (- 1.0 (/ -3.0 beta))) (+ alpha beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5) {
		tmp = ((-1.0 * ((-1.0 * alpha) - 1.0)) / (2.0 + alpha)) / ((2.0 + alpha) + 1.0);
	} else {
		tmp = (((alpha - -1.0) / beta) / (1.0 - (-3.0 / beta))) / (alpha + 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 <= 1.5d0) then
        tmp = (((-1.0d0) * (((-1.0d0) * alpha) - 1.0d0)) / (2.0d0 + alpha)) / ((2.0d0 + alpha) + 1.0d0)
    else
        tmp = (((alpha - (-1.0d0)) / beta) / (1.0d0 - ((-3.0d0) / beta))) / (alpha + beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5) {
		tmp = ((-1.0 * ((-1.0 * alpha) - 1.0)) / (2.0 + alpha)) / ((2.0 + alpha) + 1.0);
	} else {
		tmp = (((alpha - -1.0) / beta) / (1.0 - (-3.0 / beta))) / (alpha + beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.5:
		tmp = ((-1.0 * ((-1.0 * alpha) - 1.0)) / (2.0 + alpha)) / ((2.0 + alpha) + 1.0)
	else:
		tmp = (((alpha - -1.0) / beta) / (1.0 - (-3.0 / beta))) / (alpha + beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.5)
		tmp = Float64(Float64(Float64(-1.0 * Float64(Float64(-1.0 * alpha) - 1.0)) / Float64(2.0 + alpha)) / Float64(Float64(2.0 + alpha) + 1.0));
	else
		tmp = Float64(Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(1.0 - Float64(-3.0 / beta))) / Float64(alpha + beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.5)
		tmp = ((-1.0 * ((-1.0 * alpha) - 1.0)) / (2.0 + alpha)) / ((2.0 + alpha) + 1.0);
	else
		tmp = (((alpha - -1.0) / beta) / (1.0 - (-3.0 / beta))) / (alpha + beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.5], N[(N[(N[(-1.0 * N[(N[(-1.0 * alpha), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 - N[(-3.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5

   1. Initial program 94.2%

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

   2. Taylor expanded in beta around 0

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

      1. lower-+.f64 70.3

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

   4. Applied rewrites 70.3%

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

   5. Taylor expanded in beta around 0

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

      1. lower-+.f64 69.6

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

   7. Applied rewrites 69.6%

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

   8. Taylor expanded in beta around 0

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

      1. lower-+.f64 69.4

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

   10. Applied rewrites 69.4%

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

   11. Taylor expanded in beta around -inf

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lower-*.f64 11.6

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

   13. Applied rewrites 11.6%

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

   if 1.5 < beta

   1. Initial program 94.2%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.2

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

   4. Applied rewrites 29.2%

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

   6. Applied rewrites 29.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\alpha - -1}{\beta}}{1 - \frac{-3}{\alpha + \beta}}}{\alpha + \beta}} \]

   7. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\frac{\alpha - -1}{\beta}}{1 - \color{blue}{\frac{-3}{\beta}}}}{\alpha + \beta} \]
   8. Step-by-step derivation

      1. lower-/.f64 29.7

         \[\leadsto \frac{\frac{\frac{\alpha - -1}{\beta}}{1 - \frac{-3}{\color{blue}{\beta}}}}{\alpha + \beta} \]

   9. Applied rewrites 29.7%

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

Alternative 12: 37.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.5:\\ \;\;\;\;\frac{\beta - -1}{\left(-2 - \alpha\right) - \beta} \cdot \frac{1}{-3 - \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha - -1}{\beta}}{1 - \frac{-3}{\beta}}}{\alpha + \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.5)
   (* (/ (- beta -1.0) (- (- -2.0 alpha) beta)) (/ 1.0 (- -3.0 beta)))
   (/ (/ (/ (- alpha -1.0) beta) (- 1.0 (/ -3.0 beta))) (+ alpha beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5) {
		tmp = ((beta - -1.0) / ((-2.0 - alpha) - beta)) * (1.0 / (-3.0 - beta));
	} else {
		tmp = (((alpha - -1.0) / beta) / (1.0 - (-3.0 / beta))) / (alpha + 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 <= 1.5d0) then
        tmp = ((beta - (-1.0d0)) / (((-2.0d0) - alpha) - beta)) * (1.0d0 / ((-3.0d0) - beta))
    else
        tmp = (((alpha - (-1.0d0)) / beta) / (1.0d0 - ((-3.0d0) / beta))) / (alpha + beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5) {
		tmp = ((beta - -1.0) / ((-2.0 - alpha) - beta)) * (1.0 / (-3.0 - beta));
	} else {
		tmp = (((alpha - -1.0) / beta) / (1.0 - (-3.0 / beta))) / (alpha + beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.5:
		tmp = ((beta - -1.0) / ((-2.0 - alpha) - beta)) * (1.0 / (-3.0 - beta))
	else:
		tmp = (((alpha - -1.0) / beta) / (1.0 - (-3.0 / beta))) / (alpha + beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.5)
		tmp = Float64(Float64(Float64(beta - -1.0) / Float64(Float64(-2.0 - alpha) - beta)) * Float64(1.0 / Float64(-3.0 - beta)));
	else
		tmp = Float64(Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(1.0 - Float64(-3.0 / beta))) / Float64(alpha + beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.5)
		tmp = ((beta - -1.0) / ((-2.0 - alpha) - beta)) * (1.0 / (-3.0 - beta));
	else
		tmp = (((alpha - -1.0) / beta) / (1.0 - (-3.0 / beta))) / (alpha + beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.5], N[(N[(N[(beta - -1.0), $MachinePrecision] / N[(N[(-2.0 - alpha), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(-3.0 - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 - N[(-3.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5

   1. Initial program 94.2%

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

   2. Taylor expanded in alpha around -inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 37.2

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

   4. Applied rewrites 37.2%

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

   6. Applied rewrites 37.2%

      \[\leadsto \color{blue}{\frac{\beta - -1}{\left(-2 - \alpha\right) - \beta} \cdot \frac{1}{\left(-3 - \alpha\right) - \beta}} \]

   7. Taylor expanded in alpha around 0

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

   9. Applied rewrites 12.4%

      \[\leadsto \frac{\beta - -1}{\left(-2 - \alpha\right) - \beta} \cdot \frac{1}{\color{blue}{-3} - \beta} \]

   if 1.5 < beta

   1. Initial program 94.2%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.2

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

   4. Applied rewrites 29.2%

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

   6. Applied rewrites 29.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\alpha - -1}{\beta}}{1 - \frac{-3}{\alpha + \beta}}}{\alpha + \beta}} \]

   7. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\frac{\alpha - -1}{\beta}}{1 - \color{blue}{\frac{-3}{\beta}}}}{\alpha + \beta} \]
   8. Step-by-step derivation

      1. lower-/.f64 29.7

         \[\leadsto \frac{\frac{\frac{\alpha - -1}{\beta}}{1 - \frac{-3}{\color{blue}{\beta}}}}{\alpha + \beta} \]

   9. Applied rewrites 29.7%

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

Alternative 13: 37.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.5:\\ \;\;\;\;\frac{\beta - -1}{\left(-2 - \alpha\right) - \beta} \cdot \frac{1}{-3 - \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.5)
   (* (/ (- beta -1.0) (- (- -2.0 alpha) beta)) (/ 1.0 (- -3.0 beta)))
   (/ (/ (+ 1.0 alpha) beta) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5) {
		tmp = ((beta - -1.0) / ((-2.0 - alpha) - beta)) * (1.0 / (-3.0 - beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / (((alpha + beta) + (2.0 * 1.0)) + 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 <= 1.5d0) then
        tmp = ((beta - (-1.0d0)) / (((-2.0d0) - alpha) - beta)) * (1.0d0 / ((-3.0d0) - beta))
    else
        tmp = ((1.0d0 + alpha) / beta) / (((alpha + beta) + (2.0d0 * 1.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5) {
		tmp = ((beta - -1.0) / ((-2.0 - alpha) - beta)) * (1.0 / (-3.0 - beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.5:
		tmp = ((beta - -1.0) / ((-2.0 - alpha) - beta)) * (1.0 / (-3.0 - beta))
	else:
		tmp = ((1.0 + alpha) / beta) / (((alpha + beta) + (2.0 * 1.0)) + 1.0)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.5)
		tmp = Float64(Float64(Float64(beta - -1.0) / Float64(Float64(-2.0 - alpha) - beta)) * Float64(1.0 / Float64(-3.0 - beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * 1.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.5)
		tmp = ((beta - -1.0) / ((-2.0 - alpha) - beta)) * (1.0 / (-3.0 - beta));
	else
		tmp = ((1.0 + alpha) / beta) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.5], N[(N[(N[(beta - -1.0), $MachinePrecision] / N[(N[(-2.0 - alpha), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(-3.0 - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5

   1. Initial program 94.2%

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

   2. Taylor expanded in alpha around -inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 37.2

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

   4. Applied rewrites 37.2%

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

   6. Applied rewrites 37.2%

      \[\leadsto \color{blue}{\frac{\beta - -1}{\left(-2 - \alpha\right) - \beta} \cdot \frac{1}{\left(-3 - \alpha\right) - \beta}} \]

   7. Taylor expanded in alpha around 0

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

   9. Applied rewrites 12.4%

      \[\leadsto \frac{\beta - -1}{\left(-2 - \alpha\right) - \beta} \cdot \frac{1}{\color{blue}{-3} - \beta} \]

   if 1.5 < beta

   1. Initial program 94.2%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.2

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

   4. Applied rewrites 29.2%

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

Alternative 14: 37.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.55:\\ \;\;\;\;\frac{\beta - -1}{\left(-2 - \alpha\right) - \beta} \cdot \frac{1}{-3 - \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.55)
   (* (/ (- beta -1.0) (- (- -2.0 alpha) beta)) (/ 1.0 (- -3.0 beta)))
   (/ (/ (+ 1.0 alpha) beta) (+ 3.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.55) {
		tmp = ((beta - -1.0) / ((-2.0 - alpha) - beta)) * (1.0 / (-3.0 - beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + 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 <= 1.55d0) then
        tmp = ((beta - (-1.0d0)) / (((-2.0d0) - alpha) - beta)) * (1.0d0 / ((-3.0d0) - beta))
    else
        tmp = ((1.0d0 + alpha) / beta) / (3.0d0 + beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.55) {
		tmp = ((beta - -1.0) / ((-2.0 - alpha) - beta)) * (1.0 / (-3.0 - beta));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.55:
		tmp = ((beta - -1.0) / ((-2.0 - alpha) - beta)) * (1.0 / (-3.0 - beta))
	else:
		tmp = ((1.0 + alpha) / beta) / (3.0 + beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.55)
		tmp = Float64(Float64(Float64(beta - -1.0) / Float64(Float64(-2.0 - alpha) - beta)) * Float64(1.0 / Float64(-3.0 - beta)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(3.0 + beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.55)
		tmp = ((beta - -1.0) / ((-2.0 - alpha) - beta)) * (1.0 / (-3.0 - beta));
	else
		tmp = ((1.0 + alpha) / beta) / (3.0 + beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.55], N[(N[(N[(beta - -1.0), $MachinePrecision] / N[(N[(-2.0 - alpha), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(-3.0 - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.55000000000000004

   1. Initial program 94.2%

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

   2. Taylor expanded in alpha around -inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 37.2

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

   4. Applied rewrites 37.2%

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

   6. Applied rewrites 37.2%

      \[\leadsto \color{blue}{\frac{\beta - -1}{\left(-2 - \alpha\right) - \beta} \cdot \frac{1}{\left(-3 - \alpha\right) - \beta}} \]

   7. Taylor expanded in alpha around 0

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

   9. Applied rewrites 12.4%

      \[\leadsto \frac{\beta - -1}{\left(-2 - \alpha\right) - \beta} \cdot \frac{1}{\color{blue}{-3} - \beta} \]

   if 1.55000000000000004 < beta

   1. Initial program 94.2%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.2

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

   4. Applied rewrites 29.2%

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

   5. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{3 + \beta}} \]
   6. Step-by-step derivation

      1. lower-+.f64 29.0

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{3 + \color{blue}{\beta}} \]

   7. Applied rewrites 29.0%

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

Alternative 15: 37.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.8:\\ \;\;\;\;\frac{\beta - -1}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\beta - \left(-3 - \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.8)
   (/ (- beta -1.0) (* (- alpha (- -2.0 beta)) (- beta (- -3.0 alpha))))
   (/ (/ (+ 1.0 alpha) beta) (+ 3.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.8) {
		tmp = (beta - -1.0) / ((alpha - (-2.0 - beta)) * (beta - (-3.0 - alpha)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + 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 <= 5.8d0) then
        tmp = (beta - (-1.0d0)) / ((alpha - ((-2.0d0) - beta)) * (beta - ((-3.0d0) - alpha)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (3.0d0 + beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.8) {
		tmp = (beta - -1.0) / ((alpha - (-2.0 - beta)) * (beta - (-3.0 - alpha)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (3.0 + beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 5.8:
		tmp = (beta - -1.0) / ((alpha - (-2.0 - beta)) * (beta - (-3.0 - alpha)))
	else:
		tmp = ((1.0 + alpha) / beta) / (3.0 + beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.8)
		tmp = Float64(Float64(beta - -1.0) / Float64(Float64(alpha - Float64(-2.0 - beta)) * Float64(beta - Float64(-3.0 - alpha))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(3.0 + beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.8)
		tmp = (beta - -1.0) / ((alpha - (-2.0 - beta)) * (beta - (-3.0 - alpha)));
	else
		tmp = ((1.0 + alpha) / beta) / (3.0 + beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 5.8], N[(N[(beta - -1.0), $MachinePrecision] / N[(N[(alpha - N[(-2.0 - beta), $MachinePrecision]), $MachinePrecision] * N[(beta - N[(-3.0 - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.79999999999999982

   1. Initial program 94.2%

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

   2. Taylor expanded in alpha around -inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 37.2

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

   4. Applied rewrites 37.2%

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

   6. Applied rewrites 37.2%

      \[\leadsto \color{blue}{\frac{\beta - -1}{\left(-2 - \alpha\right) - \beta} \cdot \frac{1}{\left(-3 - \alpha\right) - \beta}} \]

   8. Applied rewrites 48.2%

      \[\leadsto \color{blue}{\frac{\beta - -1}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\beta - \left(-3 - \alpha\right)\right)}} \]

   if 5.79999999999999982 < beta

   1. Initial program 94.2%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.2

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

   4. Applied rewrites 29.2%

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

   5. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{3 + \beta}} \]
   6. Step-by-step derivation

      1. lower-+.f64 29.0

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{3 + \color{blue}{\beta}} \]

   7. Applied rewrites 29.0%

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

Alternative 16: 29.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1 + \alpha}{\beta}}{3 + \beta} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (/ (/ (+ 1.0 alpha) beta) (+ 3.0 beta)))

C

double code(double alpha, double beta) {
	return ((1.0 + alpha) / beta) / (3.0 + beta);
}

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 + alpha) / beta) / (3.0d0 + beta)
end function

Java

public static double code(double alpha, double beta) {
	return ((1.0 + alpha) / beta) / (3.0 + beta);
}

Python

def code(alpha, beta):
	return ((1.0 + alpha) / beta) / (3.0 + beta)

Julia

function code(alpha, beta)
	return Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(3.0 + beta))
end

MATLAB

function tmp = code(alpha, beta)
	tmp = ((1.0 + alpha) / beta) / (3.0 + beta);
end

Wolfram

code[alpha_, beta_] := N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.2%

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

2. Taylor expanded in beta around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 29.2

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

4. Applied rewrites 29.2%

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

5. Taylor expanded in alpha around 0

   \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{3 + \beta}} \]
6. Step-by-step derivation

   1. lower-+.f64 29.0

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{3 + \color{blue}{\beta}} \]

7. Applied rewrites 29.0%

   \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{3 + \beta}} \]
8. Add Preprocessing

Alternative 17: 28.8% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \frac{\beta - -1}{\alpha \cdot \alpha} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (/ (- beta -1.0) (* alpha alpha)))

C

double code(double alpha, double beta) {
	return (beta - -1.0) / (alpha * alpha);
}

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 - (-1.0d0)) / (alpha * alpha)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

2. Taylor expanded in alpha around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-pow.f64 28.8

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

4. Applied rewrites 28.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. metadata-eval N/A

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

   5. lower--.f64 28.8

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 28.8

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

6. Applied rewrites 28.8%

   \[\leadsto \frac{\beta - -1}{\color{blue}{\alpha \cdot \alpha}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))