Octave 3.8, jcobi/1

Percentage Accurate: 75.0% → 99.6%

Time: 14.8s

Alternatives: 10

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: 75.0% 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

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

FPCore

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

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.5) {
		tmp = ((((-2.0 - beta) - beta) * ((beta + 2.0) / pow(alpha, 2.0))) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} 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 = beta + (alpha + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.5d0)) then
        tmp = (((((-2.0d0) - beta) - beta) * ((beta + 2.0d0) / (alpha ** 2.0d0))) + ((beta + (beta - (-2.0d0))) / alpha)) / 2.0d0
    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 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.5) {
		tmp = ((((-2.0 - beta) - beta) * ((beta + 2.0) / Math.pow(alpha, 2.0))) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = (((beta / t_0) + 1.0) - (alpha / t_0)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.5:
		tmp = ((((-2.0 - beta) - beta) * ((beta + 2.0) / math.pow(alpha, 2.0))) + ((beta + (beta - -2.0)) / alpha)) / 2.0
	else:
		tmp = (((beta / t_0) + 1.0) - (alpha / t_0)) / 2.0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.5)
		tmp = ((((-2.0 - beta) - beta) * ((beta + 2.0) / (alpha ^ 2.0))) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	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[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / N[Power[alpha, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.5

   1. Initial program 6.9%

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

      1. +-commutative 6.9%

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

   3. Simplified 6.9%

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

   5. Taylor expanded in alpha around -inf 98.0%

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

   7. Simplified 99.6%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   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. +-commutative 100.0%

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

      2. div-sub 100.0%

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

      3. associate-+r- 100.0%

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

      4. associate-+l+ 100.0%

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

      5. associate-+l+ 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \frac{\beta}{t\_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{t\_0 \cdot \left(1 - t\_1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_1 + 1\right) - \frac{\alpha}{t\_0}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))) (t_1 (/ beta t_0)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999996)
     (/ (/ (+ beta (+ beta 2.0)) (* t_0 (- 1.0 t_1))) 2.0)
     (/ (- (+ t_1 1.0) (/ alpha t_0)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double t_1 = beta / t_0;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996) {
		tmp = ((beta + (beta + 2.0)) / (t_0 * (1.0 - t_1))) / 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 = beta + (alpha + 2.0d0)
    t_1 = beta / t_0
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.999996d0)) then
        tmp = ((beta + (beta + 2.0d0)) / (t_0 * (1.0d0 - t_1))) / 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 = beta + (alpha + 2.0);
	double t_1 = beta / t_0;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996) {
		tmp = ((beta + (beta + 2.0)) / (t_0 * (1.0 - t_1))) / 2.0;
	} else {
		tmp = ((t_1 + 1.0) - (alpha / t_0)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	t_1 = beta / t_0
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996:
		tmp = ((beta + (beta + 2.0)) / (t_0 * (1.0 - t_1))) / 2.0
	else:
		tmp = ((t_1 + 1.0) - (alpha / t_0)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	t_1 = Float64(beta / t_0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999996)
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / Float64(t_0 * Float64(1.0 - t_1))) / 2.0);
	else
		tmp = Float64(Float64(Float64(t_1 + 1.0) - Float64(alpha / t_0)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	t_1 = beta / t_0;
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996)
		tmp = ((beta + (beta + 2.0)) / (t_0 * (1.0 - t_1))) / 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[(beta + N[(alpha + 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], -0.999996], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(t$95$1 + 1.0), $MachinePrecision] - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999995999999999996

   1. Initial program 5.8%

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

      1. +-commutative 5.8%

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

   3. Simplified 5.8%

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

      1. +-commutative 5.8%

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

      2. div-sub 5.8%

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

      3. associate-+r- 5.8%

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

      4. associate-+l+ 5.8%

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

      5. associate-+l+ 5.8%

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

   6. Applied egg-rr 5.8%

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

      1. flip-+ 5.8%

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

      2. frac-sub 7.0%

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

      3. metadata-eval 7.0%

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

      4. pow2 7.0%

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

   8. Applied egg-rr 7.0%

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

   9. Taylor expanded in alpha around -inf 100.0%

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

       1. sub-neg 100.0%

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

       2. mul-1-neg 100.0%

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

       3. remove-double-neg 100.0%

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

   11. Simplified 100.0%

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

   if -0.999995999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

   3. Simplified 99.8%

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

      1. +-commutative 99.8%

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

      2. div-sub 99.9%

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

      3. associate-+r- 99.9%

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

      4. associate-+l+ 99.9%

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

      5. associate-+l+ 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(1 - \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\beta}{\beta + \left(\alpha + 2\right)} + 1\right) - \frac{\alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\beta}{t\_0} + 1\right) - \frac{\alpha}{t\_0}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999996)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/ (- (+ (/ beta t_0) 1.0) (/ alpha t_0)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} 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 = beta + (alpha + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.999996d0)) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    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 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (((beta / t_0) + 1.0) - (alpha / t_0)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = (((beta / t_0) + 1.0) - (alpha / t_0)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999996)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(beta / t_0) + 1.0) - Float64(alpha / t_0)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	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[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999996], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999995999999999996

   1. Initial program 5.8%

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

      1. +-commutative 5.8%

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

   3. Simplified 5.8%

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

   5. Taylor expanded in alpha around inf 99.9%

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

   if -0.999995999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

   3. Simplified 99.8%

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

      1. +-commutative 99.8%

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

      2. div-sub 99.9%

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

      3. associate-+r- 99.9%

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

      4. associate-+l+ 99.9%

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

      5. associate-+l+ 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\beta}{\beta + \left(\alpha + 2\right)} + 1\right) - \frac{\alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.999996:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.999996)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/ (+ 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 <= -0.999996) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} 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 <= (-0.999996d0)) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    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 <= -0.999996) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} 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 <= -0.999996:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	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 <= -0.999996)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	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 <= -0.999996)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	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, -0.999996], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $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) 2)) < -0.999995999999999996

   1. Initial program 5.8%

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

      1. +-commutative 5.8%

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

   3. Simplified 5.8%

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

   5. Taylor expanded in alpha around inf 99.9%

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

   if -0.999995999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta \cdot 0.5 + 1}{2}\\ \mathbf{if}\;\beta \leq -4.5 \cdot 10^{-141}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\beta \leq -5 \cdot 10^{-179}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 102000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\beta \leq 7.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{1}{\frac{\alpha}{\beta}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (* beta 0.5) 1.0) 2.0)))
   (if (<= beta -4.5e-141)
     t_0
     (if (<= beta -5e-179)
       (/ (/ 2.0 alpha) 2.0)
       (if (<= beta 102000.0)
         t_0
         (if (<= beta 7.2e+47) (/ 1.0 (/ alpha beta)) 1.0))))))

C

double code(double alpha, double beta) {
	double t_0 = ((beta * 0.5) + 1.0) / 2.0;
	double tmp;
	if (beta <= -4.5e-141) {
		tmp = t_0;
	} else if (beta <= -5e-179) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 102000.0) {
		tmp = t_0;
	} else if (beta <= 7.2e+47) {
		tmp = 1.0 / (alpha / beta);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = ((beta * 0.5d0) + 1.0d0) / 2.0d0
    if (beta <= (-4.5d-141)) then
        tmp = t_0
    else if (beta <= (-5d-179)) then
        tmp = (2.0d0 / alpha) / 2.0d0
    else if (beta <= 102000.0d0) then
        tmp = t_0
    else if (beta <= 7.2d+47) then
        tmp = 1.0d0 / (alpha / beta)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = ((beta * 0.5) + 1.0) / 2.0;
	double tmp;
	if (beta <= -4.5e-141) {
		tmp = t_0;
	} else if (beta <= -5e-179) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 102000.0) {
		tmp = t_0;
	} else if (beta <= 7.2e+47) {
		tmp = 1.0 / (alpha / beta);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = ((beta * 0.5) + 1.0) / 2.0
	tmp = 0
	if beta <= -4.5e-141:
		tmp = t_0
	elif beta <= -5e-179:
		tmp = (2.0 / alpha) / 2.0
	elif beta <= 102000.0:
		tmp = t_0
	elif beta <= 7.2e+47:
		tmp = 1.0 / (alpha / beta)
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(Float64(beta * 0.5) + 1.0) / 2.0)
	tmp = 0.0
	if (beta <= -4.5e-141)
		tmp = t_0;
	elseif (beta <= -5e-179)
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	elseif (beta <= 102000.0)
		tmp = t_0;
	elseif (beta <= 7.2e+47)
		tmp = Float64(1.0 / Float64(alpha / beta));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = ((beta * 0.5) + 1.0) / 2.0;
	tmp = 0.0;
	if (beta <= -4.5e-141)
		tmp = t_0;
	elseif (beta <= -5e-179)
		tmp = (2.0 / alpha) / 2.0;
	elseif (beta <= 102000.0)
		tmp = t_0;
	elseif (beta <= 7.2e+47)
		tmp = 1.0 / (alpha / beta);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(N[(beta * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[beta, -4.5e-141], t$95$0, If[LessEqual[beta, -5e-179], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 102000.0], t$95$0, If[LessEqual[beta, 7.2e+47], N[(1.0 / N[(alpha / beta), $MachinePrecision]), $MachinePrecision], 1.0]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < -4.5e-141 or -4.9999999999999998e-179 < beta < 102000

   1. Initial program 74.7%

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

      1. +-commutative 74.7%

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

   3. Simplified 74.7%

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

   5. Taylor expanded in alpha around 0 72.8%

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

   6. Taylor expanded in beta around 0 72.1%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \beta} + 1}{2} \]
   7. Step-by-step derivation

      1. *-commutative 72.1%

         \[\leadsto \frac{\color{blue}{\beta \cdot 0.5} + 1}{2} \]

   8. Simplified 72.1%

      \[\leadsto \frac{\color{blue}{\beta \cdot 0.5} + 1}{2} \]

   if -4.5e-141 < beta < -4.9999999999999998e-179

   1. Initial program 31.2%

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

      1. +-commutative 31.2%

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

   3. Simplified 31.2%

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

   5. Taylor expanded in alpha around inf 73.4%

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

   6. Taylor expanded in beta around 0 73.4%

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

   if 102000 < beta < 7.20000000000000015e47

   1. Initial program 26.0%

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

      1. +-commutative 26.0%

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

   3. Simplified 26.0%

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

   5. Taylor expanded in alpha around inf 79.7%

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

   6. Taylor expanded in beta around inf 76.7%

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

      1. associate-*r/ 76.7%

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

   8. Simplified 76.7%

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

      1. clear-num 76.7%

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

      2. inv-pow 76.7%

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

      3. *-commutative 76.7%

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

   10. Applied egg-rr 76.7%

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

       1. unpow-1 76.7%

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

       2. associate-/r/ 76.7%

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

       3. *-commutative 76.7%

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

   12. Simplified 76.7%

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

   13. Taylor expanded in beta around 0 76.7%

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

   if 7.20000000000000015e47 < beta

   1. Initial program 91.4%

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

      1. +-commutative 91.4%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in beta around inf 89.9%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -4.5 \cdot 10^{-141}:\\ \;\;\;\;\frac{\beta \cdot 0.5 + 1}{2}\\ \mathbf{elif}\;\beta \leq -5 \cdot 10^{-179}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 102000:\\ \;\;\;\;\frac{\beta \cdot 0.5 + 1}{2}\\ \mathbf{elif}\;\beta \leq 7.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{1}{\frac{\alpha}{\beta}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 7.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{1}{\frac{\alpha}{\beta}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.8e-23)
   (/ (/ 2.0 alpha) 2.0)
   (if (<= beta 7.2e+47) (/ 1.0 (/ alpha beta)) 1.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.8e-23) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 7.2e+47) {
		tmp = 1.0 / (alpha / beta);
	} 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 <= 8.8d-23) then
        tmp = (2.0d0 / alpha) / 2.0d0
    else if (beta <= 7.2d+47) then
        tmp = 1.0d0 / (alpha / beta)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.8e-23) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 7.2e+47) {
		tmp = 1.0 / (alpha / beta);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 8.8e-23:
		tmp = (2.0 / alpha) / 2.0
	elif beta <= 7.2e+47:
		tmp = 1.0 / (alpha / beta)
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.8e-23)
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	elseif (beta <= 7.2e+47)
		tmp = Float64(1.0 / Float64(alpha / beta));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.8e-23)
		tmp = (2.0 / alpha) / 2.0;
	elseif (beta <= 7.2e+47)
		tmp = 1.0 / (alpha / beta);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 8.8e-23], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 7.2e+47], N[(1.0 / N[(alpha / beta), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if beta < 8.7999999999999998e-23

   1. Initial program 71.0%

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

      1. +-commutative 71.0%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in alpha around inf 33.2%

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

   6. Taylor expanded in beta around 0 32.6%

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

   if 8.7999999999999998e-23 < beta < 7.20000000000000015e47

   1. Initial program 40.8%

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

      1. +-commutative 40.8%

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

   3. Simplified 40.8%

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

   5. Taylor expanded in alpha around inf 64.3%

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

   6. Taylor expanded in beta around inf 61.9%

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

      1. associate-*r/ 61.9%

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

   8. Simplified 61.9%

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

      1. clear-num 61.9%

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

      2. inv-pow 61.9%

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

      3. *-commutative 61.9%

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

   10. Applied egg-rr 61.9%

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

       1. unpow-1 61.9%

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

       2. associate-/r/ 61.9%

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

       3. *-commutative 61.9%

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

   12. Simplified 61.9%

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

   13. Taylor expanded in beta around 0 61.9%

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

   if 7.20000000000000015e47 < beta

   1. Initial program 91.4%

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

      1. +-commutative 91.4%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in beta around inf 89.9%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 7.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{1}{\frac{\alpha}{\beta}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1600000000000:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1600000000000.0)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1600000000000.0) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = (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 <= 1600000000000.0d0) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1600000000000.0) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1600000000000.0:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1600000000000.0)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1600000000000.0)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1600000000000.0], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.6e12

   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 98.8%

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

   if 1.6e12 < alpha

   1. Initial program 27.9%

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

      1. +-commutative 27.9%

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

   3. Simplified 27.9%

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

   5. Taylor expanded in alpha around inf 78.3%

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

   6. Taylor expanded in beta around 0 57.5%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1600000000000:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.1 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 4.1e+15)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 4.1e+15) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 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 <= 4.1d+15) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 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 <= 4.1e+15) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 4.1e+15:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 4.1e+15)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 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 <= 4.1e+15)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 4.1e+15], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $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 < 4.1e15

   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 98.8%

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

   if 4.1e15 < alpha

   1. Initial program 27.9%

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

      1. +-commutative 27.9%

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

   3. Simplified 27.9%

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

   5. Taylor expanded in alpha around inf 78.3%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.1 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.4 \cdot 10^{+205}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.4e+205) 1.0 (/ beta alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.4e+205) {
		tmp = 1.0;
	} else {
		tmp = beta / 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 <= 2.4d+205) then
        tmp = 1.0d0
    else
        tmp = beta / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.4e+205) {
		tmp = 1.0;
	} else {
		tmp = beta / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2.4e+205:
		tmp = 1.0
	else:
		tmp = beta / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.4e+205)
		tmp = 1.0;
	else
		tmp = Float64(beta / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2.4e+205)
		tmp = 1.0;
	else
		tmp = beta / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.4e+205], 1.0, N[(beta / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.39999999999999986e205

   1. Initial program 85.8%

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

      1. +-commutative 85.8%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in beta around inf 46.0%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 2.39999999999999986e205 < alpha

   1. Initial program 9.9%

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

      1. +-commutative 9.9%

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

   3. Simplified 9.9%

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

   5. Taylor expanded in alpha around inf 97.1%

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

   6. Taylor expanded in beta around inf 30.8%

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

      1. associate-*r/ 30.8%

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

   8. Simplified 30.8%

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

   9. Taylor expanded in beta around 0 30.8%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.4 \cdot 10^{+205}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 37.1% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 1.0

Julia

function code(alpha, beta)
	return 1.0
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 1.0

Derivation

1. Initial program 76.3%

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

   1. +-commutative 76.3%

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

3. Simplified 76.3%

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

5. Taylor expanded in beta around inf 41.1%

   \[\leadsto \frac{\color{blue}{2}}{2} \]

6. Final simplification 41.1%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024044 
(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))