Octave 3.8, jcobi/1

Percentage Accurate: 74.2% → 99.9%

Time: 10.8s

Alternatives: 16

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5}{\alpha + \left(\beta + 2\right)}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{2 - \left(\frac{4}{\alpha} + \beta \cdot \left(\frac{6 - \beta \cdot -2}{\alpha} - 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(\alpha \cdot t\_0 - \beta \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ -0.5 (+ alpha (+ beta 2.0)))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9998)
     (/
      (/
       (-
        2.0
        (+ (/ 4.0 alpha) (* beta (- (/ (- 6.0 (* beta -2.0)) alpha) 2.0))))
       alpha)
      2.0)
     (+ 0.5 (- (* alpha t_0) (* beta t_0))))))

C

double code(double alpha, double beta) {
	double t_0 = -0.5 / (alpha + (beta + 2.0));
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9998) {
		tmp = ((2.0 - ((4.0 / alpha) + (beta * (((6.0 - (beta * -2.0)) / alpha) - 2.0)))) / alpha) / 2.0;
	} else {
		tmp = 0.5 + ((alpha * t_0) - (beta * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-0.5d0) / (alpha + (beta + 2.0d0))
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.9998d0)) then
        tmp = ((2.0d0 - ((4.0d0 / alpha) + (beta * (((6.0d0 - (beta * (-2.0d0))) / alpha) - 2.0d0)))) / alpha) / 2.0d0
    else
        tmp = 0.5d0 + ((alpha * t_0) - (beta * t_0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = -0.5 / (alpha + (beta + 2.0));
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9998) {
		tmp = ((2.0 - ((4.0 / alpha) + (beta * (((6.0 - (beta * -2.0)) / alpha) - 2.0)))) / alpha) / 2.0;
	} else {
		tmp = 0.5 + ((alpha * t_0) - (beta * t_0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = -0.5 / (alpha + (beta + 2.0))
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9998:
		tmp = ((2.0 - ((4.0 / alpha) + (beta * (((6.0 - (beta * -2.0)) / alpha) - 2.0)))) / alpha) / 2.0
	else:
		tmp = 0.5 + ((alpha * t_0) - (beta * t_0))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(-0.5 / Float64(alpha + Float64(beta + 2.0)))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9998)
		tmp = Float64(Float64(Float64(2.0 - Float64(Float64(4.0 / alpha) + Float64(beta * Float64(Float64(Float64(6.0 - Float64(beta * -2.0)) / alpha) - 2.0)))) / alpha) / 2.0);
	else
		tmp = Float64(0.5 + Float64(Float64(alpha * t_0) - Float64(beta * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = -0.5 / (alpha + (beta + 2.0));
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9998)
		tmp = ((2.0 - ((4.0 / alpha) + (beta * (((6.0 - (beta * -2.0)) / alpha) - 2.0)))) / alpha) / 2.0;
	else
		tmp = 0.5 + ((alpha * t_0) - (beta * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(-0.5 / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9998], N[(N[(N[(2.0 - N[(N[(4.0 / alpha), $MachinePrecision] + N[(beta * N[(N[(N[(6.0 - N[(beta * -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(0.5 + N[(N[(alpha * t$95$0), $MachinePrecision] - N[(beta * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99980000000000002

   1. Initial program 7.2%

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

      1. +-commutative 7.2%

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

   3. Simplified 7.2%

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

   5. Taylor expanded in alpha around inf 95.0%

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

   7. Simplified 95.0%

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

   8. Taylor expanded in beta around 0 99.8%

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

      1. associate--l+ 99.8%

         \[\leadsto \frac{\frac{\color{blue}{2 + \left(\beta \cdot \left(\left(2 + -2 \cdot \frac{\beta}{\alpha}\right) - 6 \cdot \frac{1}{\alpha}\right) - 4 \cdot \frac{1}{\alpha}\right)}}{\alpha}}{2} \]

      2. associate--l+ 99.8%

         \[\leadsto \frac{\frac{2 + \left(\beta \cdot \color{blue}{\left(2 + \left(-2 \cdot \frac{\beta}{\alpha} - 6 \cdot \frac{1}{\alpha}\right)\right)} - 4 \cdot \frac{1}{\alpha}\right)}{\alpha}}{2} \]

      3. *-commutative 99.8%

         \[\leadsto \frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\color{blue}{\frac{\beta}{\alpha} \cdot -2} - 6 \cdot \frac{1}{\alpha}\right)\right) - 4 \cdot \frac{1}{\alpha}\right)}{\alpha}}{2} \]

      4. associate-*r/ 99.8%

         \[\leadsto \frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \color{blue}{\frac{6 \cdot 1}{\alpha}}\right)\right) - 4 \cdot \frac{1}{\alpha}\right)}{\alpha}}{2} \]

      5. metadata-eval 99.8%

         \[\leadsto \frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \frac{\color{blue}{6}}{\alpha}\right)\right) - 4 \cdot \frac{1}{\alpha}\right)}{\alpha}}{2} \]

      6. associate-*r/ 99.8%

         \[\leadsto \frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \frac{6}{\alpha}\right)\right) - \color{blue}{\frac{4 \cdot 1}{\alpha}}\right)}{\alpha}}{2} \]

      7. metadata-eval 99.8%

         \[\leadsto \frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \frac{6}{\alpha}\right)\right) - \frac{\color{blue}{4}}{\alpha}\right)}{\alpha}}{2} \]

   10. Simplified 99.8%

       \[\leadsto \frac{\frac{\color{blue}{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha}\right)}}{\alpha}}{2} \]

   11. Taylor expanded in alpha around 0 99.8%

       \[\leadsto \frac{\frac{2 + \left(\beta \cdot \left(2 + \color{blue}{\frac{-2 \cdot \beta - 6}{\alpha}}\right) - \frac{4}{\alpha}\right)}{\alpha}}{2} \]

   if -0.99980000000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 99.9%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 99.9%

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

      7. distribute-frac-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. sub-neg 99.9%

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

      10. div-sub 99.9%

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

      11. sub-neg 99.9%

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

      12. metadata-eval 99.9%

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

      13. neg-mul-1 99.9%

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

      14. *-commutative 99.9%

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

      15. +-commutative 99.9%

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

      16. associate-/l/ 99.9%

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

      17. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. associate-+r+ 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+l+ 99.9%

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

      7. associate-+r+ 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+l+ 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{2 - \left(\frac{4}{\alpha} + \beta \cdot \left(\frac{6 - \beta \cdot -2}{\alpha} - 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(\alpha \cdot \frac{-0.5}{\alpha + \left(\beta + 2\right)} - \beta \cdot \frac{-0.5}{\alpha + \left(\beta + 2\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{2 - \left(\frac{4}{\alpha} + \beta \cdot \left(\frac{6 - \beta \cdot -2}{\alpha} - 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{t\_0} + \left(1 - \frac{\alpha}{t\_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9998)
     (/
      (/
       (-
        2.0
        (+ (/ 4.0 alpha) (* beta (- (/ (- 6.0 (* beta -2.0)) alpha) 2.0))))
       alpha)
      2.0)
     (/ (+ (* beta (/ 1.0 t_0)) (- 1.0 (/ alpha t_0))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9998) {
		tmp = ((2.0 - ((4.0 / alpha) + (beta * (((6.0 - (beta * -2.0)) / alpha) - 2.0)))) / alpha) / 2.0;
	} else {
		tmp = ((beta * (1.0 / t_0)) + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.9998d0)) then
        tmp = ((2.0d0 - ((4.0d0 / alpha) + (beta * (((6.0d0 - (beta * (-2.0d0))) / alpha) - 2.0d0)))) / alpha) / 2.0d0
    else
        tmp = ((beta * (1.0d0 / t_0)) + (1.0d0 - (alpha / t_0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9998) {
		tmp = ((2.0 - ((4.0 / alpha) + (beta * (((6.0 - (beta * -2.0)) / alpha) - 2.0)))) / alpha) / 2.0;
	} else {
		tmp = ((beta * (1.0 / t_0)) + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9998:
		tmp = ((2.0 - ((4.0 / alpha) + (beta * (((6.0 - (beta * -2.0)) / alpha) - 2.0)))) / alpha) / 2.0
	else:
		tmp = ((beta * (1.0 / t_0)) + (1.0 - (alpha / t_0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9998)
		tmp = Float64(Float64(Float64(2.0 - Float64(Float64(4.0 / alpha) + Float64(beta * Float64(Float64(Float64(6.0 - Float64(beta * -2.0)) / alpha) - 2.0)))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta * Float64(1.0 / t_0)) + Float64(1.0 - Float64(alpha / t_0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9998)
		tmp = ((2.0 - ((4.0 / alpha) + (beta * (((6.0 - (beta * -2.0)) / alpha) - 2.0)))) / alpha) / 2.0;
	else
		tmp = ((beta * (1.0 / t_0)) + (1.0 - (alpha / t_0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9998], N[(N[(N[(2.0 - N[(N[(4.0 / alpha), $MachinePrecision] + N[(beta * N[(N[(N[(6.0 - N[(beta * -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99980000000000002

   1. Initial program 7.2%

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

      1. +-commutative 7.2%

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

   3. Simplified 7.2%

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

   5. Taylor expanded in alpha around inf 95.0%

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

   7. Simplified 95.0%

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

   8. Taylor expanded in beta around 0 99.8%

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

      1. associate--l+ 99.8%

         \[\leadsto \frac{\frac{\color{blue}{2 + \left(\beta \cdot \left(\left(2 + -2 \cdot \frac{\beta}{\alpha}\right) - 6 \cdot \frac{1}{\alpha}\right) - 4 \cdot \frac{1}{\alpha}\right)}}{\alpha}}{2} \]

      2. associate--l+ 99.8%

         \[\leadsto \frac{\frac{2 + \left(\beta \cdot \color{blue}{\left(2 + \left(-2 \cdot \frac{\beta}{\alpha} - 6 \cdot \frac{1}{\alpha}\right)\right)} - 4 \cdot \frac{1}{\alpha}\right)}{\alpha}}{2} \]

      3. *-commutative 99.8%

         \[\leadsto \frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\color{blue}{\frac{\beta}{\alpha} \cdot -2} - 6 \cdot \frac{1}{\alpha}\right)\right) - 4 \cdot \frac{1}{\alpha}\right)}{\alpha}}{2} \]

      4. associate-*r/ 99.8%

         \[\leadsto \frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \color{blue}{\frac{6 \cdot 1}{\alpha}}\right)\right) - 4 \cdot \frac{1}{\alpha}\right)}{\alpha}}{2} \]

      5. metadata-eval 99.8%

         \[\leadsto \frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \frac{\color{blue}{6}}{\alpha}\right)\right) - 4 \cdot \frac{1}{\alpha}\right)}{\alpha}}{2} \]

      6. associate-*r/ 99.8%

         \[\leadsto \frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \frac{6}{\alpha}\right)\right) - \color{blue}{\frac{4 \cdot 1}{\alpha}}\right)}{\alpha}}{2} \]

      7. metadata-eval 99.8%

         \[\leadsto \frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \frac{6}{\alpha}\right)\right) - \frac{\color{blue}{4}}{\alpha}\right)}{\alpha}}{2} \]

   10. Simplified 99.8%

       \[\leadsto \frac{\frac{\color{blue}{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha}\right)}}{\alpha}}{2} \]

   11. Taylor expanded in alpha around 0 99.8%

       \[\leadsto \frac{\frac{2 + \left(\beta \cdot \left(2 + \color{blue}{\frac{-2 \cdot \beta - 6}{\alpha}}\right) - \frac{4}{\alpha}\right)}{\alpha}}{2} \]

   if -0.99980000000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. div-sub 99.9%

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

      2. associate-+l- 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+l+ 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. div-inv 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{2 - \left(\frac{4}{\alpha} + \beta \cdot \left(\frac{6 - \beta \cdot -2}{\alpha} - 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{\alpha + \left(\beta + 2\right)} + \left(1 - \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{\beta}{t\_0} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{t\_0} + \left(1 - \frac{\alpha}{t\_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9999999)
     (/ (+ (/ beta t_0) (/ (- beta -2.0) alpha)) 2.0)
     (/ (+ (* beta (/ 1.0 t_0)) (- 1.0 (/ alpha t_0))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999) {
		tmp = ((beta / t_0) + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = ((beta * (1.0 / t_0)) + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.9999999d0)) then
        tmp = ((beta / t_0) + ((beta - (-2.0d0)) / alpha)) / 2.0d0
    else
        tmp = ((beta * (1.0d0 / t_0)) + (1.0d0 - (alpha / t_0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999) {
		tmp = ((beta / t_0) + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = ((beta * (1.0 / t_0)) + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999:
		tmp = ((beta / t_0) + ((beta - -2.0) / alpha)) / 2.0
	else:
		tmp = ((beta * (1.0 / t_0)) + (1.0 - (alpha / t_0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9999999)
		tmp = Float64(Float64(Float64(beta / t_0) + Float64(Float64(beta - -2.0) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta * Float64(1.0 / t_0)) + Float64(1.0 - Float64(alpha / t_0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999)
		tmp = ((beta / t_0) + ((beta - -2.0) / alpha)) / 2.0;
	else
		tmp = ((beta * (1.0 / t_0)) + (1.0 - (alpha / t_0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999999], N[(N[(N[(beta / t$95$0), $MachinePrecision] + N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999999900000000053

   1. Initial program 6.5%

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

      1. +-commutative 6.5%

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

   3. Simplified 6.5%

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

      1. div-sub 6.5%

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

      2. associate-+l- 9.4%

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

      3. +-commutative 9.4%

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

      4. associate-+l+ 9.4%

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

      5. +-commutative 9.4%

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

      6. associate-+l+ 9.4%

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

   6. Applied egg-rr 9.4%

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

   7. Taylor expanded in alpha around inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. distribute-lft-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. neg-mul-1 99.6%

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

      5. sub-neg 99.6%

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

   9. Simplified 99.6%

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

   if -0.999999900000000053 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. div-sub 99.7%

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

      2. associate-+l- 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+l+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+l+ 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. div-inv 99.7%

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

   8. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{\beta}{\alpha + \left(\beta + 2\right)} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{\alpha + \left(\beta + 2\right)} + \left(1 - \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ t_1 := \frac{\beta}{t\_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;\frac{t\_1 + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 + \left(1 - \frac{\alpha}{t\_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))) (t_1 (/ beta t_0)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9999999)
     (/ (+ t_1 (/ (- beta -2.0) alpha)) 2.0)
     (/ (+ t_1 (- 1.0 (/ alpha t_0))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double t_1 = beta / t_0;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999) {
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (t_1 + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    t_1 = beta / t_0
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.9999999d0)) then
        tmp = (t_1 + ((beta - (-2.0d0)) / alpha)) / 2.0d0
    else
        tmp = (t_1 + (1.0d0 - (alpha / t_0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double t_1 = beta / t_0;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999) {
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (t_1 + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	t_1 = beta / t_0
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999:
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0
	else:
		tmp = (t_1 + (1.0 - (alpha / t_0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	t_1 = Float64(beta / t_0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9999999)
		tmp = Float64(Float64(t_1 + Float64(Float64(beta - -2.0) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(t_1 + Float64(1.0 - Float64(alpha / t_0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	t_1 = beta / t_0;
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999)
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0;
	else
		tmp = (t_1 + (1.0 - (alpha / t_0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta / t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999999], N[(N[(t$95$1 + N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$1 + N[(1.0 - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999999900000000053

   1. Initial program 6.5%

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

      1. +-commutative 6.5%

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

   3. Simplified 6.5%

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

      1. div-sub 6.5%

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

      2. associate-+l- 9.4%

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

      3. +-commutative 9.4%

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

      4. associate-+l+ 9.4%

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

      5. +-commutative 9.4%

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

      6. associate-+l+ 9.4%

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

   6. Applied egg-rr 9.4%

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

   7. Taylor expanded in alpha around inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. distribute-lft-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. neg-mul-1 99.6%

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

      5. sub-neg 99.6%

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

   9. Simplified 99.6%

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

   if -0.999999900000000053 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. div-sub 99.7%

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

      2. associate-+l- 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+l+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+l+ 99.7%

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

   6. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{\beta}{\alpha + \left(\beta + 2\right)} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\alpha + \left(\beta + 2\right)} + \left(1 - \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{\beta}{t\_0} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{t\_0 \cdot \frac{-1}{\alpha - \beta}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9999999)
     (/ (+ (/ beta t_0) (/ (- beta -2.0) alpha)) 2.0)
     (/ (+ 1.0 (/ 1.0 (* t_0 (/ -1.0 (- alpha beta))))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999) {
		tmp = ((beta / t_0) + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (1.0 + (1.0 / (t_0 * (-1.0 / (alpha - beta))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.9999999d0)) then
        tmp = ((beta / t_0) + ((beta - (-2.0d0)) / alpha)) / 2.0d0
    else
        tmp = (1.0d0 + (1.0d0 / (t_0 * ((-1.0d0) / (alpha - beta))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999) {
		tmp = ((beta / t_0) + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (1.0 + (1.0 / (t_0 * (-1.0 / (alpha - beta))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999:
		tmp = ((beta / t_0) + ((beta - -2.0) / alpha)) / 2.0
	else:
		tmp = (1.0 + (1.0 / (t_0 * (-1.0 / (alpha - beta))))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9999999)
		tmp = Float64(Float64(Float64(beta / t_0) + Float64(Float64(beta - -2.0) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(1.0 / Float64(t_0 * Float64(-1.0 / Float64(alpha - beta))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999)
		tmp = ((beta / t_0) + ((beta - -2.0) / alpha)) / 2.0;
	else
		tmp = (1.0 + (1.0 / (t_0 * (-1.0 / (alpha - beta))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999999], N[(N[(N[(beta / t$95$0), $MachinePrecision] + N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(1.0 / N[(t$95$0 * N[(-1.0 / N[(alpha - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999999900000000053

   1. Initial program 6.5%

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

      1. +-commutative 6.5%

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

   3. Simplified 6.5%

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

      1. div-sub 6.5%

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

      2. associate-+l- 9.4%

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

      3. +-commutative 9.4%

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

      4. associate-+l+ 9.4%

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

      5. +-commutative 9.4%

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

      6. associate-+l+ 9.4%

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

   6. Applied egg-rr 9.4%

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

   7. Taylor expanded in alpha around inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. distribute-lft-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. neg-mul-1 99.6%

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

      5. sub-neg 99.6%

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

   9. Simplified 99.6%

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

   if -0.999999900000000053 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+l+ 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

   8. Simplified 99.7%

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

      1. div-inv 99.7%

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

   10. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{\beta}{\alpha + \left(\beta + 2\right)} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{-1}{\alpha - \beta}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{\beta}{t\_0} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1}{\frac{t\_0}{\alpha - \beta}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9999999)
     (/ (+ (/ beta t_0) (/ (- beta -2.0) alpha)) 2.0)
     (/ (+ 1.0 (/ -1.0 (/ t_0 (- alpha beta)))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999) {
		tmp = ((beta / t_0) + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (1.0 + (-1.0 / (t_0 / (alpha - beta)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.9999999d0)) then
        tmp = ((beta / t_0) + ((beta - (-2.0d0)) / alpha)) / 2.0d0
    else
        tmp = (1.0d0 + ((-1.0d0) / (t_0 / (alpha - beta)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999) {
		tmp = ((beta / t_0) + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (1.0 + (-1.0 / (t_0 / (alpha - beta)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999:
		tmp = ((beta / t_0) + ((beta - -2.0) / alpha)) / 2.0
	else:
		tmp = (1.0 + (-1.0 / (t_0 / (alpha - beta)))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9999999)
		tmp = Float64(Float64(Float64(beta / t_0) + Float64(Float64(beta - -2.0) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(-1.0 / Float64(t_0 / Float64(alpha - beta)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999)
		tmp = ((beta / t_0) + ((beta - -2.0) / alpha)) / 2.0;
	else
		tmp = (1.0 + (-1.0 / (t_0 / (alpha - beta)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999999], N[(N[(N[(beta / t$95$0), $MachinePrecision] + N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(-1.0 / N[(t$95$0 / N[(alpha - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999999900000000053

   1. Initial program 6.5%

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

      1. +-commutative 6.5%

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

   3. Simplified 6.5%

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

      1. div-sub 6.5%

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

      2. associate-+l- 9.4%

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

      3. +-commutative 9.4%

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

      4. associate-+l+ 9.4%

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

      5. +-commutative 9.4%

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

      6. associate-+l+ 9.4%

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

   6. Applied egg-rr 9.4%

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

   7. Taylor expanded in alpha around inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. distribute-lft-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. neg-mul-1 99.6%

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

      5. sub-neg 99.6%

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

   9. Simplified 99.6%

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

   if -0.999999900000000053 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+l+ 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{\beta}{\alpha + \left(\beta + 2\right)} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1}{\frac{\alpha + \left(\beta + 2\right)}{\alpha - \beta}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1}{\frac{\alpha + \left(\beta + 2\right)}{\alpha - \beta}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9999999)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
   (/ (+ 1.0 (/ -1.0 (/ (+ alpha (+ beta 2.0)) (- alpha beta)))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (-1.0 / ((alpha + (beta + 2.0)) / (alpha - beta)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.9999999d0)) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + ((-1.0d0) / ((alpha + (beta + 2.0d0)) / (alpha - beta)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (-1.0 / ((alpha + (beta + 2.0)) / (alpha - beta)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = (1.0 + (-1.0 / ((alpha + (beta + 2.0)) / (alpha - beta)))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9999999)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(-1.0 / Float64(Float64(alpha + Float64(beta + 2.0)) / Float64(alpha - beta)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = (1.0 + (-1.0 / ((alpha + (beta + 2.0)) / (alpha - beta)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999999], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(-1.0 / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(alpha - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999999900000000053

   1. Initial program 6.5%

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

      1. +-commutative 6.5%

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

   3. Simplified 6.5%

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

   5. Taylor expanded in alpha around inf 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

   if -0.999999900000000053 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+l+ 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1}{\frac{\alpha + \left(\beta + 2\right)}{\alpha - \beta}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 - \left(\beta - \alpha\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9999999)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
   (- 0.5 (* (- beta alpha) (/ -0.5 (+ beta (+ alpha 2.0)))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = 0.5 - ((beta - alpha) * (-0.5 / (beta + (alpha + 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.9999999d0)) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = 0.5d0 - ((beta - alpha) * ((-0.5d0) / (beta + (alpha + 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = 0.5 - ((beta - alpha) * (-0.5 / (beta + (alpha + 2.0))));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = 0.5 - ((beta - alpha) * (-0.5 / (beta + (alpha + 2.0))))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9999999)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(0.5 - Float64(Float64(beta - alpha) * Float64(-0.5 / Float64(beta + Float64(alpha + 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = 0.5 - ((beta - alpha) * (-0.5 / (beta + (alpha + 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999999], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(0.5 - N[(N[(beta - alpha), $MachinePrecision] * N[(-0.5 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999999900000000053

   1. Initial program 6.5%

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

      1. +-commutative 6.5%

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

   3. Simplified 6.5%

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

   5. Taylor expanded in alpha around inf 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

   if -0.999999900000000053 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. neg-sub0 99.7%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 99.7%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 99.7%

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

      7. distribute-frac-neg 99.7%

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

      8. +-commutative 99.7%

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

      9. sub-neg 99.7%

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

      10. div-sub 99.7%

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

      11. sub-neg 99.7%

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

      12. metadata-eval 99.7%

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

      13. neg-mul-1 99.7%

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

      14. *-commutative 99.7%

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

      15. +-commutative 99.7%

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

      16. associate-/l/ 99.7%

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

      17. associate-*l/ 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 - \left(\beta - \alpha\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3300000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 3300000000000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3300000000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 3300000000000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3300000000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 3300000000000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 3300000000000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 3300000000000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 3300000000000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3.3e12

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in alpha around 0 96.8%

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

      1. +-commutative 96.8%

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

   7. Simplified 96.8%

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

   if 3.3e12 < alpha

   1. Initial program 22.9%

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

      1. +-commutative 22.9%

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

   3. Simplified 22.9%

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

   5. Taylor expanded in alpha around inf 83.4%

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

      1. *-commutative 83.4%

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

   7. Simplified 83.4%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3300000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2200000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} - \frac{-1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2200000000000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (- (/ beta alpha) (/ -1.0 alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2200000000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta / alpha) - (-1.0 / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 2200000000000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (beta / alpha) - ((-1.0d0) / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2200000000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta / alpha) - (-1.0 / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2200000000000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (beta / alpha) - (-1.0 / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2200000000000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(beta / alpha) - Float64(-1.0 / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2200000000000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (beta / alpha) - (-1.0 / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2200000000000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta / alpha), $MachinePrecision] - N[(-1.0 / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.2e12

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in alpha around 0 96.8%

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

      1. +-commutative 96.8%

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

   7. Simplified 96.8%

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

   if 2.2e12 < alpha

   1. Initial program 22.9%

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

      1. +-commutative 22.9%

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

   3. Simplified 22.9%

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

   5. Taylor expanded in alpha around inf 83.4%

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

      1. *-commutative 83.4%

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

   7. Simplified 83.4%

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

   8. Taylor expanded in beta around 0 83.4%

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

      1. +-commutative 83.4%

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

   10. Simplified 83.4%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2200000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} - \frac{-1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.9:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} - \frac{-1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.9) (+ 0.5 (* alpha -0.25)) (- (/ beta alpha) (/ -1.0 alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.9) {
		tmp = 0.5 + (alpha * -0.25);
	} else {
		tmp = (beta / alpha) - (-1.0 / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.9d0) then
        tmp = 0.5d0 + (alpha * (-0.25d0))
    else
        tmp = (beta / alpha) - ((-1.0d0) / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.9) {
		tmp = 0.5 + (alpha * -0.25);
	} else {
		tmp = (beta / alpha) - (-1.0 / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.9:
		tmp = 0.5 + (alpha * -0.25)
	else:
		tmp = (beta / alpha) - (-1.0 / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.9)
		tmp = Float64(0.5 + Float64(alpha * -0.25));
	else
		tmp = Float64(Float64(beta / alpha) - Float64(-1.0 / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.9)
		tmp = 0.5 + (alpha * -0.25);
	else
		tmp = (beta / alpha) - (-1.0 / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.9], N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision], N[(N[(beta / alpha), $MachinePrecision] - N[(-1.0 / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.8999999999999999

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around 0 72.5%

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

      1. +-commutative 72.5%

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

   7. Simplified 72.5%

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

   8. Taylor expanded in alpha around 0 71.1%

      \[\leadsto \color{blue}{0.5 + -0.25 \cdot \alpha} \]
   9. Step-by-step derivation

      1. *-commutative 71.1%

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

   10. Simplified 71.1%

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

   if 1.8999999999999999 < alpha

   1. Initial program 24.5%

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

      1. +-commutative 24.5%

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

   3. Simplified 24.5%

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

   5. Taylor expanded in alpha around inf 82.1%

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

      1. *-commutative 82.1%

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

   7. Simplified 82.1%

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

   8. Taylor expanded in beta around 0 82.1%

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

      1. +-commutative 82.1%

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

   10. Simplified 82.1%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.9:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} - \frac{-1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 0.95:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 0.95) (+ 0.5 (* alpha -0.25)) (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.95) {
		tmp = 0.5 + (alpha * -0.25);
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 0.95d0) then
        tmp = 0.5d0 + (alpha * (-0.25d0))
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.95) {
		tmp = 0.5 + (alpha * -0.25);
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 0.95:
		tmp = 0.5 + (alpha * -0.25)
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 0.95)
		tmp = Float64(0.5 + Float64(alpha * -0.25));
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 0.95)
		tmp = 0.5 + (alpha * -0.25);
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 0.95], N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision], N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 0.94999999999999996

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around 0 72.5%

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

      1. +-commutative 72.5%

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

   7. Simplified 72.5%

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

   8. Taylor expanded in alpha around 0 71.1%

      \[\leadsto \color{blue}{0.5 + -0.25 \cdot \alpha} \]
   9. Step-by-step derivation

      1. *-commutative 71.1%

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

   10. Simplified 71.1%

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

   if 0.94999999999999996 < alpha

   1. Initial program 24.5%

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

      1. +-commutative 24.5%

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

   3. Simplified 24.5%

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

   5. Taylor expanded in beta around 0 6.5%

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

      1. +-commutative 6.5%

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

   7. Simplified 6.5%

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

   8. Taylor expanded in alpha around inf 64.1%

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

Alternative 13: 68.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.8:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= alpha 2.8) 0.5 (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.8) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 2.8d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.8) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2.8:
		tmp = 0.5
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.8)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2.8)
		tmp = 0.5;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.8], 0.5, N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.7999999999999998

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around 0 72.5%

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

      1. +-commutative 72.5%

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

   7. Simplified 72.5%

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

   8. Taylor expanded in alpha around 0 70.3%

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

   if 2.7999999999999998 < alpha

   1. Initial program 24.5%

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

      1. +-commutative 24.5%

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

   3. Simplified 24.5%

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

   5. Taylor expanded in beta around 0 6.5%

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

      1. +-commutative 6.5%

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

   7. Simplified 6.5%

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

   8. Taylor expanded in alpha around inf 64.1%

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

Alternative 14: 70.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 60000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 60000.0) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 60000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 60000.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 60000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 60000.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 60000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 60000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 60000.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 6e4

   1. Initial program 63.9%

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

      1. +-commutative 63.9%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in beta around 0 63.1%

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

      1. +-commutative 63.1%

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

   7. Simplified 63.1%

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

   8. Taylor expanded in alpha around 0 60.2%

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

   if 6e4 < beta

   1. Initial program 83.0%

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

      1. +-commutative 83.0%

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

   3. Simplified 83.0%

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

      1. clear-num 83.0%

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

      2. inv-pow 83.0%

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

      3. +-commutative 83.0%

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

      4. associate-+l+ 83.0%

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

   6. Applied egg-rr 83.0%

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

      1. unpow-1 83.0%

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

   8. Simplified 83.0%

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

   9. Taylor expanded in beta around inf 79.5%

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

Alternative 15: 49.4% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 0.5)

C

double code(double alpha, double beta) {
	return 0.5;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.5d0
end function

Java

public static double code(double alpha, double beta) {
	return 0.5;
}

Python

def code(alpha, beta):
	return 0.5

Julia

function code(alpha, beta)
	return 0.5
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.5;
end

Wolfram

code[alpha_, beta_] := 0.5

Derivation

1. Initial program 70.2%

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

   1. +-commutative 70.2%

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

3. Simplified 70.2%

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

5. Taylor expanded in beta around 0 46.5%

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

   1. +-commutative 46.5%

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

7. Simplified 46.5%

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

8. Taylor expanded in alpha around 0 45.7%

   \[\leadsto \color{blue}{0.5} \]
9. Add Preprocessing

Alternative 16: 3.7% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 0.0)

C

double code(double alpha, double beta) {
	return 0.0;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.0d0
end function

Java

public static double code(double alpha, double beta) {
	return 0.0;
}

Python

def code(alpha, beta):
	return 0.0

Julia

function code(alpha, beta)
	return 0.0
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.0;
end

Wolfram

code[alpha_, beta_] := 0.0

Derivation

1. Initial program 70.2%

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

   1. +-commutative 70.2%

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

   2. sub-neg 70.2%

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

   3. +-commutative 70.2%

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

   4. neg-sub0 70.2%

      \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

   5. associate-+l- 70.2%

      \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

   6. sub0-neg 70.2%

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

   7. distribute-frac-neg 70.2%

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

   8. +-commutative 70.2%

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

   9. sub-neg 70.2%

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

   10. div-sub 70.2%

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

   11. sub-neg 70.2%

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

   12. metadata-eval 70.2%

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

   13. neg-mul-1 70.2%

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

   14. *-commutative 70.2%

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

   15. +-commutative 70.2%

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

   16. associate-/l/ 70.2%

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

   17. associate-*l/ 70.2%

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

3. Simplified 70.2%

   \[\leadsto \color{blue}{0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in alpha around inf 3.8%

   \[\leadsto 0.5 + \color{blue}{-0.5} \]
6. Step-by-step derivation

   1. metadata-eval 3.8%

      \[\leadsto \color{blue}{0} \]

7. Applied egg-rr 3.8%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024169 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))