Octave 3.8, jcobi/1

Percentage Accurate: 74.0% → 99.6%

Time: 9.5s

Alternatives: 8

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: 74.0% 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

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.5)
     (/
      (+
       (* (- (- -2.0 beta) beta) (/ (+ beta 2.0) (pow alpha 2.0)))
       (/ (+ beta (- 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 <= -0.5) {
		tmp = ((((-2.0 - beta) - beta) * ((beta + 2.0) / pow(alpha, 2.0))) + ((beta + (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 <= (-0.5d0)) then
        tmp = (((((-2.0d0) - beta) - beta) * ((beta + 2.0d0) / (alpha ** 2.0d0))) + ((beta + (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 <= -0.5) {
		tmp = ((((-2.0 - beta) - beta) * ((beta + 2.0) / Math.pow(alpha, 2.0))) + ((beta + (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 <= -0.5:
		tmp = ((((-2.0 - beta) - beta) * ((beta + 2.0) / math.pow(alpha, 2.0))) + ((beta + (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 <= -0.5)
		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(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 <= -0.5)
		tmp = ((((-2.0 - beta) - beta) * ((beta + 2.0) / (alpha ^ 2.0))) + ((beta + (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, -0.5], 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[(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)) < -0.5

   1. Initial program 6.7%

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

      1. +-commutative 6.7%

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

   3. Simplified 6.7%

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

   5. Taylor expanded in alpha around -inf 91.3%

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

      1. mul-1-neg 91.3%

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

      2. unsub-neg 91.3%

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

   7. Simplified 99.9%

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

   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. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

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

Alternative 2: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\beta + \left(\alpha + 2\right)}, \beta - \alpha, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99999998)
   (/ (+ beta 1.0) alpha)
   (/ (fma (/ 1.0 (+ beta (+ alpha 2.0))) (- beta alpha) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999998) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = fma((1.0 / (beta + (alpha + 2.0))), (beta - alpha), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99999998)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = Float64(fma(Float64(1.0 / Float64(beta + Float64(alpha + 2.0))), Float64(beta - alpha), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99999998], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta - alpha), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999999980000000011

   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 99.7%

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

      1. associate-*r/ 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in beta around 0 99.7%

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

      1. +-commutative 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in beta around 0 99.7%

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

       1. *-rgt-identity 99.7%

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

       2. associate-*r/ 99.6%

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

       3. distribute-rgt1-in 99.6%

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

       4. associate-*r/ 99.7%

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

       5. +-commutative 99.7%

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

       6. *-rgt-identity 99.7%

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

   13. Simplified 99.7%

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

   if -0.999999980000000011 < (/.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. clear-num 99.8%

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

      2. associate-/r/ 99.8%

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

      3. fma-define 99.8%

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

      4. associate-+l+ 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\beta + \left(\alpha + 2\right)}, \beta - \alpha, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.99999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.99999998) (/ (+ 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 <= -0.99999998) {
		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 <= (-0.99999998d0)) 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 <= -0.99999998) {
		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 <= -0.99999998:
		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 <= -0.99999998)
		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 <= -0.99999998)
		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, -0.99999998], 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) 2)) < -0.999999980000000011

   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 99.7%

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

      1. associate-*r/ 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in beta around 0 99.7%

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

      1. +-commutative 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in beta around 0 99.7%

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

       1. *-rgt-identity 99.7%

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

       2. associate-*r/ 99.6%

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

       3. distribute-rgt1-in 99.6%

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

       4. associate-*r/ 99.7%

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

       5. +-commutative 99.7%

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

       6. *-rgt-identity 99.7%

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

   13. Simplified 99.7%

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

   if -0.999999980000000011 < (/.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 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \alpha \cdot -0.25\\ \mathbf{if}\;\alpha \leq -1.15 \cdot 10^{-111}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\alpha \leq -5.4 \cdot 10^{-151}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.95:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* alpha -0.25))))
   (if (<= alpha -1.15e-111)
     t_0
     (if (<= alpha -5.4e-151)
       1.0
       (if (<= alpha 1.95) t_0 (/ (+ beta 1.0) alpha))))))

C

double code(double alpha, double beta) {
	double t_0 = 0.5 + (alpha * -0.25);
	double tmp;
	if (alpha <= -1.15e-111) {
		tmp = t_0;
	} else if (alpha <= -5.4e-151) {
		tmp = 1.0;
	} else if (alpha <= 1.95) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 + (alpha * (-0.25d0))
    if (alpha <= (-1.15d-111)) then
        tmp = t_0
    else if (alpha <= (-5.4d-151)) then
        tmp = 1.0d0
    else if (alpha <= 1.95d0) then
        tmp = t_0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 0.5 + (alpha * -0.25);
	double tmp;
	if (alpha <= -1.15e-111) {
		tmp = t_0;
	} else if (alpha <= -5.4e-151) {
		tmp = 1.0;
	} else if (alpha <= 1.95) {
		tmp = t_0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 0.5 + (alpha * -0.25)
	tmp = 0
	if alpha <= -1.15e-111:
		tmp = t_0
	elif alpha <= -5.4e-151:
		tmp = 1.0
	elif alpha <= 1.95:
		tmp = t_0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(0.5 + Float64(alpha * -0.25))
	tmp = 0.0
	if (alpha <= -1.15e-111)
		tmp = t_0;
	elseif (alpha <= -5.4e-151)
		tmp = 1.0;
	elseif (alpha <= 1.95)
		tmp = t_0;
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 0.5 + (alpha * -0.25);
	tmp = 0.0;
	if (alpha <= -1.15e-111)
		tmp = t_0;
	elseif (alpha <= -5.4e-151)
		tmp = 1.0;
	elseif (alpha <= 1.95)
		tmp = t_0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, -1.15e-111], t$95$0, If[LessEqual[alpha, -5.4e-151], 1.0, If[LessEqual[alpha, 1.95], t$95$0, N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -1.15e-111 or -5.40000000000000014e-151 < alpha < 1.94999999999999996

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around 0 72.0%

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

      1. +-commutative 72.0%

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

   7. Simplified 72.0%

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

   8. Taylor expanded in alpha around 0 71.5%

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

      1. *-commutative 71.5%

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

   10. Simplified 71.5%

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

   if -1.15e-111 < alpha < -5.40000000000000014e-151

   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. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. fma-define 100.0%

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

      4. associate-+l+ 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in beta around inf 91.0%

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

   if 1.94999999999999996 < alpha

   1. Initial program 23.2%

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

      1. +-commutative 23.2%

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

   3. Simplified 23.2%

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

   5. Taylor expanded in alpha around inf 82.6%

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

      1. associate-*r/ 82.6%

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

   7. Simplified 82.6%

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

   8. Taylor expanded in beta around 0 82.6%

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

      1. +-commutative 82.6%

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

   10. Simplified 82.6%

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

   11. Taylor expanded in beta around 0 82.6%

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

       1. *-rgt-identity 82.6%

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

       2. associate-*r/ 82.6%

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

       3. distribute-rgt1-in 82.6%

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

       4. associate-*r/ 82.6%

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

       5. +-commutative 82.6%

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

       6. *-rgt-identity 82.6%

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

   13. Simplified 82.6%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -1.15 \cdot 10^{-111}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq -5.4 \cdot 10^{-151}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.95:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.02 \cdot 10^{-41}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 7.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2.05:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.02e-41)
   0.5
   (if (<= beta 7.5e-34) (/ 1.0 alpha) (if (<= beta 2.05) 0.5 1.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.02e-41) {
		tmp = 0.5;
	} else if (beta <= 7.5e-34) {
		tmp = 1.0 / alpha;
	} else if (beta <= 2.05) {
		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 <= 1.02d-41) then
        tmp = 0.5d0
    else if (beta <= 7.5d-34) then
        tmp = 1.0d0 / alpha
    else if (beta <= 2.05d0) 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 <= 1.02e-41) {
		tmp = 0.5;
	} else if (beta <= 7.5e-34) {
		tmp = 1.0 / alpha;
	} else if (beta <= 2.05) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.02e-41:
		tmp = 0.5
	elif beta <= 7.5e-34:
		tmp = 1.0 / alpha
	elif beta <= 2.05:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.02e-41)
		tmp = 0.5;
	elseif (beta <= 7.5e-34)
		tmp = Float64(1.0 / alpha);
	elseif (beta <= 2.05)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.02e-41)
		tmp = 0.5;
	elseif (beta <= 7.5e-34)
		tmp = 1.0 / alpha;
	elseif (beta <= 2.05)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.02e-41], 0.5, If[LessEqual[beta, 7.5e-34], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[beta, 2.05], 0.5, 1.0]]]

Derivation

1. Split input into 3 regimes

2. if beta < 1.02e-41 or 7.5000000000000004e-34 < beta < 2.0499999999999998

   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 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 68.3%

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

   if 1.02e-41 < beta < 7.5000000000000004e-34

   1. Initial program 5.9%

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

      1. +-commutative 5.9%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in beta around 0 5.9%

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

      1. +-commutative 5.9%

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

   7. Simplified 5.9%

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

   8. Taylor expanded in alpha around inf 100.0%

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

   9. Taylor expanded in alpha around 0 100.0%

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

   if 2.0499999999999998 < beta

   1. Initial program 83.5%

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

      1. +-commutative 83.5%

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

   3. Simplified 83.5%

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

      1. clear-num 83.5%

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

      2. associate-/r/ 83.6%

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

      3. fma-define 83.5%

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

      4. associate-+l+ 83.5%

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

   6. Applied egg-rr 83.5%

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

   7. Taylor expanded in beta around inf 82.9%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.02 \cdot 10^{-41}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 7.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2.05:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 6.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 6.6e+16)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 6.6e+16) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 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 <= 6.6d+16) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 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 <= 6.6e+16) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 6.6e+16:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 6.6e+16)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 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 <= 6.6e+16)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 6.6e+16], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 6.6e16

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in alpha around 0 97.7%

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

   if 6.6e16 < alpha

   1. Initial program 19.4%

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

      1. +-commutative 19.4%

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

   3. Simplified 19.4%

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

   5. Taylor expanded in alpha around inf 85.9%

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

      1. associate-*r/ 85.9%

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

   7. Simplified 85.9%

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

   8. Taylor expanded in beta around 0 85.9%

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

      1. +-commutative 85.9%

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

   10. Simplified 85.9%

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

   11. Taylor expanded in beta around 0 85.9%

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

       1. *-rgt-identity 85.9%

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

       2. associate-*r/ 85.8%

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

       3. distribute-rgt1-in 85.8%

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

       4. associate-*r/ 85.9%

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

       5. +-commutative 85.9%

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

       6. *-rgt-identity 85.9%

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

   13. Simplified 85.9%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 6.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.05:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 2.05) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.05) {
		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 <= 2.05d0) 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 <= 2.05) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.05:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.05)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.05)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.05], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2.0499999999999998

   1. Initial program 69.3%

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

      1. +-commutative 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in beta around 0 68.4%

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

      1. +-commutative 68.4%

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

   7. Simplified 68.4%

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

   8. Taylor expanded in alpha around 0 66.3%

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

   if 2.0499999999999998 < beta

   1. Initial program 83.5%

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

      1. +-commutative 83.5%

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

   3. Simplified 83.5%

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

      1. clear-num 83.5%

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

      2. associate-/r/ 83.6%

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

      3. fma-define 83.5%

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

      4. associate-+l+ 83.5%

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

   6. Applied egg-rr 83.5%

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

   7. Taylor expanded in beta around inf 82.9%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.05:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.0% 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 74.5%

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

   1. +-commutative 74.5%

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

3. Simplified 74.5%

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

5. Taylor expanded in beta around 0 48.6%

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

   1. +-commutative 48.6%

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

7. Simplified 48.6%

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

8. Taylor expanded in alpha around 0 48.2%

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

9. Final simplification 48.2%

   \[\leadsto 0.5 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024050 
(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))