Octave 3.8, jcobi/1

Percentage Accurate: 74.1% → 99.8%

Time: 10.0s

Alternatives: 12

Speedup: 1.3×

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: 74.1% 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: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{0.5 \cdot \left(\left(2 + \beta \cdot 2\right) + \left(\beta + 2\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)} + 1}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9995)
   (/
    (*
     0.5
     (+
      (+ 2.0 (* beta 2.0))
      (* (+ beta 2.0) (/ (- (- -2.0 beta) beta) alpha))))
    alpha)
   (/ (log (exp (+ (/ (- beta alpha) (+ alpha (+ beta 2.0))) 1.0))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995) {
		tmp = (0.5 * ((2.0 + (beta * 2.0)) + ((beta + 2.0) * (((-2.0 - beta) - beta) / alpha)))) / alpha;
	} else {
		tmp = log(exp((((beta - alpha) / (alpha + (beta + 2.0))) + 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) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.9995d0)) then
        tmp = (0.5d0 * ((2.0d0 + (beta * 2.0d0)) + ((beta + 2.0d0) * ((((-2.0d0) - beta) - beta) / alpha)))) / alpha
    else
        tmp = log(exp((((beta - alpha) / (alpha + (beta + 2.0d0))) + 1.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995) {
		tmp = (0.5 * ((2.0 + (beta * 2.0)) + ((beta + 2.0) * (((-2.0 - beta) - beta) / alpha)))) / alpha;
	} else {
		tmp = Math.log(Math.exp((((beta - alpha) / (alpha + (beta + 2.0))) + 1.0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995:
		tmp = (0.5 * ((2.0 + (beta * 2.0)) + ((beta + 2.0) * (((-2.0 - beta) - beta) / alpha)))) / alpha
	else:
		tmp = math.log(math.exp((((beta - alpha) / (alpha + (beta + 2.0))) + 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)) <= -0.9995)
		tmp = Float64(Float64(0.5 * Float64(Float64(2.0 + Float64(beta * 2.0)) + Float64(Float64(beta + 2.0) * Float64(Float64(Float64(-2.0 - beta) - beta) / alpha)))) / alpha);
	else
		tmp = Float64(log(exp(Float64(Float64(Float64(beta - alpha) / Float64(alpha + Float64(beta + 2.0))) + 1.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995)
		tmp = (0.5 * ((2.0 + (beta * 2.0)) + ((beta + 2.0) * (((-2.0 - beta) - beta) / alpha)))) / alpha;
	else
		tmp = log(exp((((beta - alpha) / (alpha + (beta + 2.0))) + 1.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9995], N[(N[(0.5 * N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[Log[N[Exp[N[(N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99950000000000006

   1. Initial program 6.3%

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

      1. +-commutative 6.3%

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

   3. Simplified 6.3%

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

   5. Taylor expanded in alpha around inf 94.8%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(2 + 2 \cdot \beta\right) + 0.5 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha}}{\alpha}} \]
   6. Step-by-step derivation

   7. Simplified 99.9%

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

   if -0.99950000000000006 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. add-log-exp 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{0.5 \cdot \left(\left(2 + \beta \cdot 2\right) + \left(\beta + 2\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)} + 1}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.9995:\\ \;\;\;\;\frac{0.5 \cdot \left(\left(2 + \beta \cdot 2\right) + \left(\beta + 2\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9995)
     (/
      (*
       0.5
       (+
        (+ 2.0 (* beta 2.0))
        (* (+ beta 2.0) (/ (- (- -2.0 beta) beta) alpha))))
      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 <= -0.9995) {
		tmp = (0.5 * ((2.0 + (beta * 2.0)) + ((beta + 2.0) * (((-2.0 - beta) - beta) / alpha)))) / 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 <= (-0.9995d0)) then
        tmp = (0.5d0 * ((2.0d0 + (beta * 2.0d0)) + ((beta + 2.0d0) * ((((-2.0d0) - beta) - beta) / alpha)))) / 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 <= -0.9995) {
		tmp = (0.5 * ((2.0 + (beta * 2.0)) + ((beta + 2.0) * (((-2.0 - beta) - beta) / alpha)))) / 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 <= -0.9995:
		tmp = (0.5 * ((2.0 + (beta * 2.0)) + ((beta + 2.0) * (((-2.0 - beta) - beta) / alpha)))) / 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 <= -0.9995)
		tmp = Float64(Float64(0.5 * Float64(Float64(2.0 + Float64(beta * 2.0)) + Float64(Float64(beta + 2.0) * Float64(Float64(Float64(-2.0 - beta) - beta) / alpha)))) / 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 <= -0.9995)
		tmp = (0.5 * ((2.0 + (beta * 2.0)) + ((beta + 2.0) * (((-2.0 - beta) - beta) / alpha)))) / 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, -0.9995], N[(N[(0.5 * N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99950000000000006

   1. Initial program 6.3%

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

      1. +-commutative 6.3%

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

   3. Simplified 6.3%

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

   5. Taylor expanded in alpha around inf 94.8%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(2 + 2 \cdot \beta\right) + 0.5 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha}}{\alpha}} \]
   6. Step-by-step derivation

   7. Simplified 99.9%

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

   if -0.99950000000000006 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{0.5 \cdot \left(\left(2 + \beta \cdot 2\right) + \left(\beta + 2\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.9995:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9995) (+ (/ beta alpha) (/ 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 <= -0.9995) {
		tmp = (beta / alpha) + (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 <= (-0.9995d0)) then
        tmp = (beta / alpha) + (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 <= -0.9995) {
		tmp = (beta / alpha) + (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 <= -0.9995:
		tmp = (beta / alpha) + (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 <= -0.9995)
		tmp = Float64(Float64(beta / alpha) + Float64(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 <= -0.9995)
		tmp = (beta / alpha) + (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, -0.9995], N[(N[(beta / alpha), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99950000000000006

   1. Initial program 6.3%

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

      1. +-commutative 6.3%

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

   3. Simplified 6.3%

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

   5. Taylor expanded in alpha around inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. distribute-lft-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in beta around 0 99.6%

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

      1. +-commutative 99.6%

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

   10. Simplified 99.6%

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

   if -0.99950000000000006 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 52000000000:\\ \;\;\;\;0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 52000000000.0)
   (+ 0.5 (* (- alpha beta) (/ -0.5 (+ beta (+ alpha 2.0)))))
   (+ (/ beta alpha) (/ 1.0 alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 52000000000.0) {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	} else {
		tmp = (beta / alpha) + (1.0 / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 52000000000.0d0) then
        tmp = 0.5d0 + ((alpha - beta) * ((-0.5d0) / (beta + (alpha + 2.0d0))))
    else
        tmp = (beta / alpha) + (1.0d0 / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 52000000000.0) {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	} else {
		tmp = (beta / alpha) + (1.0 / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 52000000000.0:
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))))
	else:
		tmp = (beta / alpha) + (1.0 / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 52000000000.0)
		tmp = Float64(0.5 + Float64(Float64(alpha - beta) * Float64(-0.5 / Float64(beta + Float64(alpha + 2.0)))));
	else
		tmp = Float64(Float64(beta / alpha) + Float64(1.0 / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 52000000000.0)
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	else
		tmp = (beta / alpha) + (1.0 / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 52000000000.0], N[(0.5 + N[(N[(alpha - beta), $MachinePrecision] * N[(-0.5 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(beta / alpha), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 5.2e10

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

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

      5. associate-+l- 99.9%

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

      6. sub0-neg 99.9%

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

      7. distribute-frac-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. sub-neg 99.9%

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

      10. div-sub 99.9%

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

      11. sub-neg 99.9%

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

      12. metadata-eval 99.9%

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

      13. neg-mul-1 99.9%

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

      14. *-commutative 99.9%

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

      15. +-commutative 99.9%

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

      16. associate-/l/ 99.9%

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

      17. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   if 5.2e10 < alpha

   1. Initial program 20.7%

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

      1. +-commutative 20.7%

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

   3. Simplified 20.7%

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

   5. Taylor expanded in alpha around inf 85.5%

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

      1. associate-*r/ 85.5%

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

      2. distribute-lft-in 85.5%

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

      3. metadata-eval 85.5%

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

      4. *-commutative 85.5%

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

   7. Simplified 85.5%

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

   8. Taylor expanded in beta around 0 85.5%

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

      1. +-commutative 85.5%

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

   10. Simplified 85.5%

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

Alternative 5: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3200:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 3200.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (+ (/ beta alpha) (/ 1.0 alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3200.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta / alpha) + (1.0 / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 3200.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (beta / alpha) + (1.0d0 / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3200.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta / alpha) + (1.0 / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 3200.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (beta / alpha) + (1.0 / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 3200.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(beta / alpha) + Float64(1.0 / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 3200.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (beta / alpha) + (1.0 / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 3200.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta / alpha), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3200

   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 97.7%

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

      1. +-commutative 97.7%

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

   7. Simplified 97.7%

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

   if 3200 < alpha

   1. Initial program 21.4%

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

      1. +-commutative 21.4%

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

   3. Simplified 21.4%

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

   5. Taylor expanded in alpha around inf 85.0%

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

      1. associate-*r/ 85.0%

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

      2. distribute-lft-in 85.0%

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

      3. metadata-eval 85.0%

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

      4. *-commutative 85.0%

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

   7. Simplified 85.0%

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

   8. Taylor expanded in beta around 0 85.0%

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

      1. +-commutative 85.0%

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

   10. Simplified 85.0%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3200:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.0) (+ 0.5 (* alpha -0.25)) (+ (/ beta alpha) (/ 1.0 alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.0) {
		tmp = 0.5 + (alpha * -0.25);
	} else {
		tmp = (beta / alpha) + (1.0 / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 2.0d0) then
        tmp = 0.5d0 + (alpha * (-0.25d0))
    else
        tmp = (beta / alpha) + (1.0d0 / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.0) {
		tmp = 0.5 + (alpha * -0.25);
	} else {
		tmp = (beta / alpha) + (1.0 / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2.0:
		tmp = 0.5 + (alpha * -0.25)
	else:
		tmp = (beta / alpha) + (1.0 / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.0)
		tmp = Float64(0.5 + Float64(alpha * -0.25));
	else
		tmp = Float64(Float64(beta / alpha) + Float64(1.0 / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2.0)
		tmp = 0.5 + (alpha * -0.25);
	else
		tmp = (beta / alpha) + (1.0 / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.0], N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision], N[(N[(beta / alpha), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 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. Taylor expanded in beta around 0 73.1%

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

      1. +-commutative 73.1%

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

   7. Simplified 73.1%

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

   8. Taylor expanded in alpha around 0 72.3%

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

      1. *-commutative 72.3%

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

   10. Simplified 72.3%

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

   if 2 < alpha

   1. Initial program 21.4%

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

      1. +-commutative 21.4%

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

   3. Simplified 21.4%

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

   5. Taylor expanded in alpha around inf 85.0%

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

      1. associate-*r/ 85.0%

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

      2. distribute-lft-in 85.0%

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

      3. metadata-eval 85.0%

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

      4. *-commutative 85.0%

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

   7. Simplified 85.0%

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

   8. Taylor expanded in beta around 0 85.0%

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

      1. +-commutative 85.0%

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

   10. Simplified 85.0%

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

Alternative 7: 75.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if alpha < 1.8999999999999999

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around 0 73.1%

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

      1. +-commutative 73.1%

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

   7. Simplified 73.1%

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

   8. Taylor expanded in alpha around 0 72.3%

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

      1. *-commutative 72.3%

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

   10. Simplified 72.3%

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

   if 1.8999999999999999 < alpha

   1. Initial program 21.4%

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

      1. +-commutative 21.4%

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

   3. Simplified 21.4%

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

   5. Taylor expanded in alpha around inf 85.0%

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

      1. associate-*r/ 85.0%

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

      2. distribute-lft-in 85.0%

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

      3. metadata-eval 85.0%

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

      4. *-commutative 85.0%

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

   7. Simplified 85.0%

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

   8. Taylor expanded in alpha around 0 85.0%

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

4. Final simplification 76.8%

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

Alternative 8: 69.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if alpha < 0.94999999999999996

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around 0 73.1%

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

      1. +-commutative 73.1%

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

   7. Simplified 73.1%

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

   8. Taylor expanded in alpha around 0 72.3%

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

      1. *-commutative 72.3%

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

   10. Simplified 72.3%

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

   if 0.94999999999999996 < alpha

   1. Initial program 21.4%

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

      1. +-commutative 21.4%

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

   3. Simplified 21.4%

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

   5. Taylor expanded in alpha around inf 85.0%

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

      1. associate-*r/ 85.0%

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

      2. distribute-lft-in 85.0%

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

      3. metadata-eval 85.0%

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

      4. *-commutative 85.0%

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

   7. Simplified 85.0%

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

   8. Taylor expanded in beta around 0 67.8%

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

Alternative 9: 69.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= alpha 2.0) 0.5 (/ 1.0 alpha)))

C

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

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 2.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if alpha < 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%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-mul-1 100.0%

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

      14. *-commutative 100.0%

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

      15. +-commutative 100.0%

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

      16. associate-/l/ 100.0%

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

      17. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around 0 69.1%

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

      1. metadata-eval 69.1%

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

      2. distribute-neg-frac 69.1%

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

      3. distribute-neg-frac2 69.1%

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

      4. distribute-neg-in 69.1%

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

      5. metadata-eval 69.1%

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

      6. unsub-neg 69.1%

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

   7. Simplified 69.1%

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

   8. Taylor expanded in alpha around 0 67.1%

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

      1. *-commutative 67.1%

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

   10. Simplified 67.1%

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

   11. Taylor expanded in beta around 0 71.2%

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

   if 2 < alpha

   1. Initial program 21.4%

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

      1. +-commutative 21.4%

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

   3. Simplified 21.4%

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

   5. Taylor expanded in alpha around inf 85.0%

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

      1. associate-*r/ 85.0%

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

      2. distribute-lft-in 85.0%

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

      3. metadata-eval 85.0%

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

      4. *-commutative 85.0%

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

   7. Simplified 85.0%

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

   8. Taylor expanded in beta around 0 67.8%

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

Alternative 10: 70.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 2.0) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 66.2%

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

      1. +-commutative 66.2%

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

      2. sub-neg 66.2%

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

      3. +-commutative 66.2%

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

      4. neg-sub0 66.2%

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

      5. associate-+l- 66.2%

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

      6. sub0-neg 66.2%

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

      7. distribute-frac-neg 66.2%

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

      8. +-commutative 66.2%

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

      9. sub-neg 66.2%

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

      10. div-sub 66.2%

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

      11. sub-neg 66.2%

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

      12. metadata-eval 66.2%

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

      13. neg-mul-1 66.2%

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

      14. *-commutative 66.2%

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

      15. +-commutative 66.2%

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

      16. associate-/l/ 66.2%

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

      17. associate-*l/ 66.2%

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

   3. Simplified 66.1%

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

   5. Taylor expanded in beta around 0 65.3%

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

      1. metadata-eval 65.3%

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

      2. distribute-neg-frac 65.3%

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

      3. distribute-neg-frac2 65.3%

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

      4. distribute-neg-in 65.3%

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

      5. metadata-eval 65.3%

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

      6. unsub-neg 65.3%

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

   7. Simplified 65.3%

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

   8. Taylor expanded in alpha around 0 62.6%

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

      1. *-commutative 62.6%

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

   10. Simplified 62.6%

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

   11. Taylor expanded in beta around 0 62.4%

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

   if 2 < beta

   1. Initial program 84.1%

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

      1. +-commutative 84.1%

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

   3. Simplified 84.1%

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

      1. add-log-exp 84.1%

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

      2. +-commutative 84.1%

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

      3. associate-+l+ 84.1%

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

   6. Applied egg-rr 84.1%

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

   7. Taylor expanded in beta around inf 80.7%

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

Alternative 11: 49.1% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.5

Julia

function code(alpha, beta)
	return 0.5
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.5

Derivation

1. Initial program 71.8%

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

   1. +-commutative 71.8%

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

   2. sub-neg 71.8%

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

   3. +-commutative 71.8%

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

   4. neg-sub0 71.8%

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

   5. associate-+l- 71.8%

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

   6. sub0-neg 71.8%

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

   7. distribute-frac-neg 71.8%

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

   8. +-commutative 71.8%

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

   9. sub-neg 71.8%

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

   10. div-sub 71.8%

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

   11. sub-neg 71.8%

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

   12. metadata-eval 71.8%

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

   13. neg-mul-1 71.8%

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

   14. *-commutative 71.8%

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

   15. +-commutative 71.8%

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

   16. associate-/l/ 71.8%

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

   17. associate-*l/ 71.8%

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

3. Simplified 71.7%

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

5. Taylor expanded in beta around 0 46.9%

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

   1. metadata-eval 46.9%

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

   2. distribute-neg-frac 46.9%

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

   3. distribute-neg-frac2 46.9%

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

   4. distribute-neg-in 46.9%

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

   5. metadata-eval 46.9%

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

   6. unsub-neg 46.9%

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

7. Simplified 46.9%

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

8. Taylor expanded in alpha around 0 44.8%

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

   1. *-commutative 44.8%

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

10. Simplified 44.8%

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

11. Taylor expanded in beta around 0 48.3%

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

Alternative 12: 3.7% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.0

Julia

function code(alpha, beta)
	return 0.0
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.0

Derivation

1. Initial program 71.8%

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

   1. +-commutative 71.8%

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

   2. sub-neg 71.8%

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

   3. +-commutative 71.8%

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

   4. neg-sub0 71.8%

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

   5. associate-+l- 71.8%

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

   6. sub0-neg 71.8%

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

   7. distribute-frac-neg 71.8%

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

   8. +-commutative 71.8%

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

   9. sub-neg 71.8%

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

   10. div-sub 71.8%

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

   11. sub-neg 71.8%

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

   12. metadata-eval 71.8%

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

   13. neg-mul-1 71.8%

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

   14. *-commutative 71.8%

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

   15. +-commutative 71.8%

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

   16. associate-/l/ 71.8%

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

   17. associate-*l/ 71.8%

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

3. Simplified 71.7%

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

5. Taylor expanded in alpha around inf 3.8%

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

   1. metadata-eval 3.8%

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

7. Applied egg-rr 3.8%

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

Reproduce

herbie shell --seed 2024163 
(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))