Octave 3.8, jcobi/1

Percentage Accurate: 74.4% → 99.6%

Time: 11.7s

Alternatives: 12

Speedup: 0.9×

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: 74.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{\alpha + \left(\beta + 2\right)}, \alpha - \beta, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99998)
   (/ (+ beta 1.0) alpha)
   (fma (/ -0.5 (+ alpha (+ beta 2.0))) (- alpha beta) 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99998) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = fma((-0.5 / (alpha + (beta + 2.0))), (alpha - beta), 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99998)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = fma(Float64(-0.5 / Float64(alpha + Float64(beta + 2.0))), Float64(alpha - beta), 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99998], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(-0.5 / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(alpha - beta), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99997999999999998

   1. Initial program 7.9%

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

      1. +-commutative 7.9%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in alpha around inf 98.8%

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

      1. *-commutative 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in beta around 0 98.8%

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

   9. Taylor expanded in alpha around 0 98.8%

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

   if -0.99997999999999998 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   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%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. neg-sub0 99.6%

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

      5. associate-+l- 99.6%

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

      6. sub0-neg 99.6%

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

      7. distribute-frac-neg 99.6%

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

      8. +-commutative 99.6%

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

      9. sub-neg 99.6%

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

      10. div-sub 99.6%

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

      11. sub-neg 99.6%

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

      12. metadata-eval 99.6%

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

      13. neg-mul-1 99.6%

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

      14. *-commutative 99.6%

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

      15. +-commutative 99.6%

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

      16. associate-/l/ 99.6%

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

      17. associate-*l/ 99.6%

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

   3. Simplified 99.6%

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

      1. +-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. fma-define 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+l+ 99.7%

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

   6. Applied egg-rr 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{\alpha + \left(\beta + 2\right)}, \alpha - \beta, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ t_1 := \frac{\alpha}{t\_0}\\ t_2 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t\_2} \leq -0.99998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t\_0} + \frac{t\_1 \cdot t\_1 + -1}{-1 - \frac{\alpha}{t\_2}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0)))
        (t_1 (/ alpha t_0))
        (t_2 (+ (+ beta alpha) 2.0)))
   (if (<= (/ (- beta alpha) t_2) -0.99998)
     (/ (+ beta 1.0) alpha)
     (/
      (+ (/ beta t_0) (/ (+ (* t_1 t_1) -1.0) (- -1.0 (/ alpha t_2))))
      2.0))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double t_1 = alpha / t_0;
	double t_2 = (beta + alpha) + 2.0;
	double tmp;
	if (((beta - alpha) / t_2) <= -0.99998) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = ((beta / t_0) + (((t_1 * t_1) + -1.0) / (-1.0 - (alpha / t_2)))) / 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) :: t_2
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    t_1 = alpha / t_0
    t_2 = (beta + alpha) + 2.0d0
    if (((beta - alpha) / t_2) <= (-0.99998d0)) then
        tmp = (beta + 1.0d0) / alpha
    else
        tmp = ((beta / t_0) + (((t_1 * t_1) + (-1.0d0)) / ((-1.0d0) - (alpha / t_2)))) / 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 = alpha / t_0;
	double t_2 = (beta + alpha) + 2.0;
	double tmp;
	if (((beta - alpha) / t_2) <= -0.99998) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = ((beta / t_0) + (((t_1 * t_1) + -1.0) / (-1.0 - (alpha / t_2)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	t_1 = alpha / t_0
	t_2 = (beta + alpha) + 2.0
	tmp = 0
	if ((beta - alpha) / t_2) <= -0.99998:
		tmp = (beta + 1.0) / alpha
	else:
		tmp = ((beta / t_0) + (((t_1 * t_1) + -1.0) / (-1.0 - (alpha / t_2)))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	t_1 = Float64(alpha / t_0)
	t_2 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / t_2) <= -0.99998)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = Float64(Float64(Float64(beta / t_0) + Float64(Float64(Float64(t_1 * t_1) + -1.0) / Float64(-1.0 - Float64(alpha / t_2)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	t_1 = alpha / t_0;
	t_2 = (beta + alpha) + 2.0;
	tmp = 0.0;
	if (((beta - alpha) / t_2) <= -0.99998)
		tmp = (beta + 1.0) / alpha;
	else
		tmp = ((beta / t_0) + (((t_1 * t_1) + -1.0) / (-1.0 - (alpha / t_2)))) / 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[(alpha / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / t$95$2), $MachinePrecision], -0.99998], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] + N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] + -1.0), $MachinePrecision] / N[(-1.0 - N[(alpha / t$95$2), $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))) < -0.99997999999999998

   1. Initial program 7.9%

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

      1. +-commutative 7.9%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in alpha around inf 98.8%

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

      1. *-commutative 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in beta around 0 98.8%

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

   9. Taylor expanded in alpha around 0 98.8%

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

   if -0.99997999999999998 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   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. Step-by-step derivation

      1. div-sub 99.6%

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

      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. flip-- 99.7%

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

      2. pow2 99.7%

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

      3. associate-+r+ 99.7%

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

      4. metadata-eval 99.7%

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

      5. associate-+r+ 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. unpow2 99.7%

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

      2. associate-+r+ 99.7%

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

      3. associate-+r+ 99.7%

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

   10. Applied egg-rr 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\alpha + \left(\beta + 2\right)} + \frac{\frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha}{\alpha + \left(\beta + 2\right)} + -1}{-1 - \frac{\alpha}{\left(\beta + \alpha\right) + 2}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t\_0} + \left(1 - \frac{\alpha}{t\_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99998)
     (/ (+ beta 1.0) alpha)
     (/ (+ (/ beta 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)) <= -0.99998) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = ((beta / 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)) <= (-0.99998d0)) then
        tmp = (beta + 1.0d0) / alpha
    else
        tmp = ((beta / 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)) <= -0.99998) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = ((beta / 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)) <= -0.99998:
		tmp = (beta + 1.0) / alpha
	else:
		tmp = ((beta / 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)) <= -0.99998)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = Float64(Float64(Float64(beta / 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)) <= -0.99998)
		tmp = (beta + 1.0) / alpha;
	else
		tmp = ((beta / 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], -0.99998], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta / t$95$0), $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))) < -0.99997999999999998

   1. Initial program 7.9%

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

      1. +-commutative 7.9%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in alpha around inf 98.8%

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

      1. *-commutative 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in beta around 0 98.8%

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

   9. Taylor expanded in alpha around 0 98.8%

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

   if -0.99997999999999998 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   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. Step-by-step derivation

      1. div-sub 99.6%

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

      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 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\alpha + \left(\beta + 2\right)} + \left(1 - \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.99998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.99998) (/ (+ beta 1.0) alpha) (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.99998) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 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 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.99998d0)) then
        tmp = (beta + 1.0d0) / alpha
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.99998) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.99998:
		tmp = (beta + 1.0) / alpha
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.99998)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.99998)
		tmp = (beta + 1.0) / alpha;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.99998], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(t$95$0 + 1.0), $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))) < -0.99997999999999998

   1. Initial program 7.9%

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

      1. +-commutative 7.9%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in alpha around inf 98.8%

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

      1. *-commutative 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in beta around 0 98.8%

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

   9. Taylor expanded in alpha around 0 98.8%

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

   if -0.99997999999999998 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2600000000:\\ \;\;\;\;0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2600000000.0)
   (+ 0.5 (* (- alpha beta) (/ -0.5 (+ beta (+ alpha 2.0)))))
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2600000000.0) {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	} else {
		tmp = (beta + 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 <= 2600000000.0d0) then
        tmp = 0.5d0 + ((alpha - beta) * ((-0.5d0) / (beta + (alpha + 2.0d0))))
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2600000000.0) {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2600000000.0:
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))))
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2600000000.0)
		tmp = Float64(0.5 + Float64(Float64(alpha - beta) * Float64(-0.5 / Float64(beta + Float64(alpha + 2.0)))));
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2600000000.0)
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2600000000.0], N[(0.5 + N[(N[(alpha - beta), $MachinePrecision] * N[(-0.5 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.6e9

   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

   if 2.6e9 < alpha

   1. Initial program 21.3%

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

      1. +-commutative 21.3%

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

   3. Simplified 21.3%

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

   5. Taylor expanded in alpha around inf 85.8%

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

      1. *-commutative 85.8%

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

   7. Simplified 85.8%

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

   8. Taylor expanded in beta around 0 85.8%

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

   9. Taylor expanded in alpha around 0 85.8%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2600000000:\\ \;\;\;\;0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 8.6 \cdot 10^{-45}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq 45000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 8.6e-45)
   (+ 0.5 (* alpha -0.25))
   (if (<= alpha 45000.0) 1.0 (/ (+ beta 1.0) alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 8.6e-45) {
		tmp = 0.5 + (alpha * -0.25);
	} else if (alpha <= 45000.0) {
		tmp = 1.0;
	} else {
		tmp = (beta + 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 <= 8.6d-45) then
        tmp = 0.5d0 + (alpha * (-0.25d0))
    else if (alpha <= 45000.0d0) then
        tmp = 1.0d0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 8.6e-45) {
		tmp = 0.5 + (alpha * -0.25);
	} else if (alpha <= 45000.0) {
		tmp = 1.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 8.6e-45:
		tmp = 0.5 + (alpha * -0.25)
	elif alpha <= 45000.0:
		tmp = 1.0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 8.6e-45)
		tmp = Float64(0.5 + Float64(alpha * -0.25));
	elseif (alpha <= 45000.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 8.6e-45)
		tmp = 0.5 + (alpha * -0.25);
	elseif (alpha <= 45000.0)
		tmp = 1.0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 8.6e-45], N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[alpha, 45000.0], 1.0, N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 8.5999999999999998e-45

   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 70.3%

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

      1. +-commutative 70.3%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in alpha around 0 69.4%

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

      1. *-commutative 69.4%

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

   10. Simplified 69.4%

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

   if 8.5999999999999998e-45 < alpha < 45000

   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. Step-by-step derivation

      1. div-sub 100.0%

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

      2. associate-+l- 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-+l+ 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. flip-- 99.9%

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

      2. pow2 99.9%

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

      3. associate-+r+ 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-+r+ 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. add-log-exp 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+r+ 99.9%

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

      4. metadata-eval 99.9%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in beta around inf 81.2%

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

   if 45000 < alpha

   1. Initial program 21.9%

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

      1. +-commutative 21.9%

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

   3. Simplified 21.9%

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

   5. Taylor expanded in alpha around inf 85.3%

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

      1. *-commutative 85.3%

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

   7. Simplified 85.3%

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

   8. Taylor expanded in beta around 0 85.3%

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

   9. Taylor expanded in alpha around 0 85.3%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 8.6 \cdot 10^{-45}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq 45000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 66000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 66000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 66000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 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 <= 66000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 66000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 66000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 66000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 66000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 66000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 66000

   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 alpha around 0 98.6%

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

      1. +-commutative 98.6%

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

   7. Simplified 98.6%

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

   if 66000 < alpha

   1. Initial program 21.9%

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

      1. +-commutative 21.9%

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

   3. Simplified 21.9%

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

   5. Taylor expanded in alpha around inf 85.3%

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

      1. *-commutative 85.3%

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

   7. Simplified 85.3%

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

   8. Taylor expanded in beta around 0 85.3%

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

   9. Taylor expanded in alpha around 0 85.3%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 66000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 8.6 \cdot 10^{-45}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq 45000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 8.6e-45)
   (+ 0.5 (* alpha -0.25))
   (if (<= alpha 45000.0) 1.0 (/ 1.0 alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 8.6e-45) {
		tmp = 0.5 + (alpha * -0.25);
	} else if (alpha <= 45000.0) {
		tmp = 1.0;
	} 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 <= 8.6d-45) then
        tmp = 0.5d0 + (alpha * (-0.25d0))
    else if (alpha <= 45000.0d0) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 8.6e-45) {
		tmp = 0.5 + (alpha * -0.25);
	} else if (alpha <= 45000.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 8.6e-45:
		tmp = 0.5 + (alpha * -0.25)
	elif alpha <= 45000.0:
		tmp = 1.0
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 8.6e-45)
		tmp = Float64(0.5 + Float64(alpha * -0.25));
	elseif (alpha <= 45000.0)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 8.6e-45)
		tmp = 0.5 + (alpha * -0.25);
	elseif (alpha <= 45000.0)
		tmp = 1.0;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 8.6e-45], N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[alpha, 45000.0], 1.0, N[(1.0 / alpha), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 8.5999999999999998e-45

   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 70.3%

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

      1. +-commutative 70.3%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in alpha around 0 69.4%

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

      1. *-commutative 69.4%

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

   10. Simplified 69.4%

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

   if 8.5999999999999998e-45 < alpha < 45000

   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. Step-by-step derivation

      1. div-sub 100.0%

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

      2. associate-+l- 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-+l+ 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. flip-- 99.9%

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

      2. pow2 99.9%

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

      3. associate-+r+ 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-+r+ 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. add-log-exp 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+r+ 99.9%

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

      4. metadata-eval 99.9%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in beta around inf 81.2%

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

   if 45000 < alpha

   1. Initial program 21.9%

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

      1. +-commutative 21.9%

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

   3. Simplified 21.9%

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

   5. Taylor expanded in beta around 0 7.9%

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

      1. +-commutative 7.9%

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

   7. Simplified 7.9%

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

   8. Taylor expanded in alpha around inf 70.5%

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

Alternative 9: 67.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 10^{-47}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 37000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1e-47) 0.5 (if (<= alpha 37000.0) 1.0 (/ 1.0 alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1e-47) {
		tmp = 0.5;
	} else if (alpha <= 37000.0) {
		tmp = 1.0;
	} 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 <= 1d-47) then
        tmp = 0.5d0
    else if (alpha <= 37000.0d0) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1e-47) {
		tmp = 0.5;
	} else if (alpha <= 37000.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1e-47:
		tmp = 0.5
	elif alpha <= 37000.0:
		tmp = 1.0
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1e-47)
		tmp = 0.5;
	elseif (alpha <= 37000.0)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1e-47)
		tmp = 0.5;
	elseif (alpha <= 37000.0)
		tmp = 1.0;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1e-47], 0.5, If[LessEqual[alpha, 37000.0], 1.0, N[(1.0 / alpha), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 9.9999999999999997e-48

   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 70.3%

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

      1. +-commutative 70.3%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in alpha around 0 69.0%

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

   if 9.9999999999999997e-48 < alpha < 37000

   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. Step-by-step derivation

      1. div-sub 100.0%

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

      2. associate-+l- 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-+l+ 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. flip-- 99.9%

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

      2. pow2 99.9%

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

      3. associate-+r+ 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-+r+ 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. add-log-exp 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+r+ 99.9%

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

      4. metadata-eval 99.9%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in beta around inf 81.2%

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

   if 37000 < alpha

   1. Initial program 21.9%

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

      1. +-commutative 21.9%

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

   3. Simplified 21.9%

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

   5. Taylor expanded in beta around 0 7.9%

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

      1. +-commutative 7.9%

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

   7. Simplified 7.9%

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

   8. Taylor expanded in alpha around inf 70.5%

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

Alternative 10: 69.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.6 \cdot 10^{+27}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 4.6e+27) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6e+27) {
		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 <= 4.6d+27) 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 <= 4.6e+27) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.6e+27:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.6e+27)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.6e+27)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.6e+27], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 4.6000000000000001e27

   1. Initial program 62.1%

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

      1. +-commutative 62.1%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in beta around 0 60.1%

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

      1. +-commutative 60.1%

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

   7. Simplified 60.1%

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

   8. Taylor expanded in alpha around 0 57.3%

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

   if 4.6000000000000001e27 < beta

   1. Initial program 91.2%

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

      1. +-commutative 91.2%

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

   3. Simplified 91.2%

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

      1. div-sub 91.2%

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

      2. associate-+l- 92.5%

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

      3. +-commutative 92.5%

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

      4. associate-+l+ 92.5%

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

      5. +-commutative 92.5%

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

      6. associate-+l+ 92.5%

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

   6. Applied egg-rr 92.5%

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

      1. flip-- 92.5%

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

      2. pow2 92.5%

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

      3. associate-+r+ 92.5%

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

      4. metadata-eval 92.5%

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

      5. associate-+r+ 92.5%

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

   8. Applied egg-rr 92.5%

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

      1. add-log-exp 92.5%

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

      2. sub-neg 92.5%

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

      3. associate-+r+ 92.5%

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

      4. metadata-eval 92.5%

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

   10. Applied egg-rr 92.5%

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

   11. Taylor expanded in beta around inf 88.8%

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

Alternative 11: 48.9% 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 71.3%

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

   1. +-commutative 71.3%

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

3. Simplified 71.3%

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

5. Taylor expanded in beta around 0 45.8%

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

   1. +-commutative 45.8%

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

7. Simplified 45.8%

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

8. Taylor expanded in alpha around 0 44.7%

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

Alternative 12: 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 71.3%

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

   1. +-commutative 71.3%

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

   2. sub-neg 71.3%

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

   3. +-commutative 71.3%

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

   4. neg-sub0 71.3%

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

   5. associate-+l- 71.3%

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

   6. sub0-neg 71.3%

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

   7. distribute-frac-neg 71.3%

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

   8. +-commutative 71.3%

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

   9. sub-neg 71.3%

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

   10. div-sub 71.3%

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

   11. sub-neg 71.3%

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

   12. metadata-eval 71.3%

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

   13. neg-mul-1 71.3%

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

   14. *-commutative 71.3%

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

   15. +-commutative 71.3%

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

   16. associate-/l/ 71.3%

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

   17. associate-*l/ 71.3%

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

3. Simplified 71.4%

   \[\leadsto \color{blue}{0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}} \]
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 2024155 
(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))