Octave 3.8, jcobi/1

Percentage Accurate: 74.7% → 99.9%

Time: 10.3s

Alternatives: 13

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: 74.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}\\ \mathbf{if}\;t\_0 \leq -0.99999:\\ \;\;\;\;\frac{-0.5 \cdot \left(\left(2 + \beta\right) \cdot \left(-1 - \frac{-2 + \beta \cdot -2}{\alpha}\right) - \beta\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ 2.0 (+ alpha beta)))))
   (if (<= t_0 -0.99999)
     (/
      (*
       -0.5
       (- (* (+ 2.0 beta) (- -1.0 (/ (+ -2.0 (* beta -2.0)) alpha))) beta))
      alpha)
     (/ (+ 1.0 t_0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (2.0 + (alpha + beta));
	double tmp;
	if (t_0 <= -0.99999) {
		tmp = (-0.5 * (((2.0 + beta) * (-1.0 - ((-2.0 + (beta * -2.0)) / alpha))) - beta)) / alpha;
	} else {
		tmp = (1.0 + t_0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / (2.0d0 + (alpha + beta))
    if (t_0 <= (-0.99999d0)) then
        tmp = ((-0.5d0) * (((2.0d0 + beta) * ((-1.0d0) - (((-2.0d0) + (beta * (-2.0d0))) / alpha))) - beta)) / alpha
    else
        tmp = (1.0d0 + t_0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (2.0 + (alpha + beta));
	double tmp;
	if (t_0 <= -0.99999) {
		tmp = (-0.5 * (((2.0 + beta) * (-1.0 - ((-2.0 + (beta * -2.0)) / alpha))) - beta)) / alpha;
	} else {
		tmp = (1.0 + t_0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / (2.0 + (alpha + beta))
	tmp = 0
	if t_0 <= -0.99999:
		tmp = (-0.5 * (((2.0 + beta) * (-1.0 - ((-2.0 + (beta * -2.0)) / alpha))) - beta)) / alpha
	else:
		tmp = (1.0 + t_0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(2.0 + Float64(alpha + beta)))
	tmp = 0.0
	if (t_0 <= -0.99999)
		tmp = Float64(Float64(-0.5 * Float64(Float64(Float64(2.0 + beta) * Float64(-1.0 - Float64(Float64(-2.0 + Float64(beta * -2.0)) / alpha))) - beta)) / alpha);
	else
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / (2.0 + (alpha + beta));
	tmp = 0.0;
	if (t_0 <= -0.99999)
		tmp = (-0.5 * (((2.0 + beta) * (-1.0 - ((-2.0 + (beta * -2.0)) / alpha))) - beta)) / alpha;
	else
		tmp = (1.0 + t_0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.99999], N[(N[(-0.5 * N[(N[(N[(2.0 + beta), $MachinePrecision] * N[(-1.0 - N[(N[(-2.0 + N[(beta * -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(1.0 + t$95$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))) < -0.999990000000000046

   1. Initial program 7.5%

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

   3. Taylor expanded in alpha around -inf

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

   5. Simplified 99.9%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)} \leq -0.99999:\\ \;\;\;\;\frac{-0.5 \cdot \left(\left(2 + \beta\right) \cdot \left(-1 - \frac{-2 + \beta \cdot -2}{\alpha}\right) - \beta\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{\beta}{2 + \beta}\\ \frac{1}{\frac{2 \cdot \left(\alpha \cdot \left(\frac{1}{2 + \beta} + \frac{\beta}{\left(2 + \beta\right) \cdot \left(2 + \beta\right)}\right)\right)}{t\_0 \cdot t\_0} + \frac{2}{t\_0}} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ beta (+ 2.0 beta)))))
   (/
    1.0
    (+
     (/
      (*
       2.0
       (*
        alpha
        (+ (/ 1.0 (+ 2.0 beta)) (/ beta (* (+ 2.0 beta) (+ 2.0 beta))))))
      (* t_0 t_0))
     (/ 2.0 t_0)))))

C

double code(double alpha, double beta) {
	double t_0 = 1.0 + (beta / (2.0 + beta));
	return 1.0 / (((2.0 * (alpha * ((1.0 / (2.0 + beta)) + (beta / ((2.0 + beta) * (2.0 + beta)))))) / (t_0 * t_0)) + (2.0 / t_0));
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = 1.0d0 + (beta / (2.0d0 + beta))
    code = 1.0d0 / (((2.0d0 * (alpha * ((1.0d0 / (2.0d0 + beta)) + (beta / ((2.0d0 + beta) * (2.0d0 + beta)))))) / (t_0 * t_0)) + (2.0d0 / t_0))
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 1.0 + (beta / (2.0 + beta));
	return 1.0 / (((2.0 * (alpha * ((1.0 / (2.0 + beta)) + (beta / ((2.0 + beta) * (2.0 + beta)))))) / (t_0 * t_0)) + (2.0 / t_0));
}

Python

def code(alpha, beta):
	t_0 = 1.0 + (beta / (2.0 + beta))
	return 1.0 / (((2.0 * (alpha * ((1.0 / (2.0 + beta)) + (beta / ((2.0 + beta) * (2.0 + beta)))))) / (t_0 * t_0)) + (2.0 / t_0))

Julia

function code(alpha, beta)
	t_0 = Float64(1.0 + Float64(beta / Float64(2.0 + beta)))
	return Float64(1.0 / Float64(Float64(Float64(2.0 * Float64(alpha * Float64(Float64(1.0 / Float64(2.0 + beta)) + Float64(beta / Float64(Float64(2.0 + beta) * Float64(2.0 + beta)))))) / Float64(t_0 * t_0)) + Float64(2.0 / t_0)))
end

MATLAB

function tmp = code(alpha, beta)
	t_0 = 1.0 + (beta / (2.0 + beta));
	tmp = 1.0 / (((2.0 * (alpha * ((1.0 / (2.0 + beta)) + (beta / ((2.0 + beta) * (2.0 + beta)))))) / (t_0 * t_0)) + (2.0 / t_0));
end

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right), \color{blue}{1}\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2\right)\right), 1\right)\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

   10. +-lowering-+.f64 73.5%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 1\right)\right)\right) \]

4. Applied egg-rr 73.5%

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

5. Taylor expanded in alpha around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 \cdot \frac{\alpha \cdot \left(\frac{1}{2 + \beta} + \frac{\beta}{{\left(2 + \beta\right)}^{2}}\right)}{{\left(1 + \frac{\beta}{2 + \beta}\right)}^{2}} + 2 \cdot \frac{1}{1 + \frac{\beta}{2 + \beta}}\right)}\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(2 \cdot \frac{\alpha \cdot \left(\frac{1}{2 + \beta} + \frac{\beta}{{\left(2 + \beta\right)}^{2}}\right)}{{\left(1 + \frac{\beta}{2 + \beta}\right)}^{2}}\right), \color{blue}{\left(2 \cdot \frac{1}{1 + \frac{\beta}{2 + \beta}}\right)}\right)\right) \]

7. Simplified 99.2%

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

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)} \leq -0.9999998:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \beta \cdot \left(1 + \frac{\alpha}{\beta}\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ 2.0 (+ alpha beta))) -0.9999998)
   (/ (+ 1.0 beta) alpha)
   (/ (+ 1.0 (/ (- beta alpha) (+ 2.0 (* beta (+ 1.0 (/ alpha beta)))))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / (2.0 + (alpha + beta))) <= -0.9999998) {
		tmp = (1.0 + beta) / alpha;
	} else {
		tmp = (1.0 + ((beta - alpha) / (2.0 + (beta * (1.0 + (alpha / beta)))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (((beta - alpha) / (2.0d0 + (alpha + beta))) <= (-0.9999998d0)) then
        tmp = (1.0d0 + beta) / alpha
    else
        tmp = (1.0d0 + ((beta - alpha) / (2.0d0 + (beta * (1.0d0 + (alpha / beta)))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / (2.0 + (alpha + beta))) <= -0.9999998) {
		tmp = (1.0 + beta) / alpha;
	} else {
		tmp = (1.0 + ((beta - alpha) / (2.0 + (beta * (1.0 + (alpha / beta)))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / (2.0 + (alpha + beta))) <= -0.9999998:
		tmp = (1.0 + beta) / alpha
	else:
		tmp = (1.0 + ((beta - alpha) / (2.0 + (beta * (1.0 + (alpha / beta)))))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(2.0 + Float64(alpha + beta))) <= -0.9999998)
		tmp = Float64(Float64(1.0 + beta) / alpha);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta * Float64(1.0 + Float64(alpha / beta)))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / (2.0 + (alpha + beta))) <= -0.9999998)
		tmp = (1.0 + beta) / alpha;
	else
		tmp = (1.0 + ((beta - alpha) / (2.0 + (beta * (1.0 + (alpha / beta)))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.9999998], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta * N[(1.0 + N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999999799999999994

   1. Initial program 6.7%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right)\right), \color{blue}{\alpha}\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \alpha\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \alpha\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(\frac{1}{2} \cdot 2\right) \cdot \beta\right), \alpha\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + 1 \cdot \beta\right), \alpha\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \alpha\right) \]

      8. +-lowering-+.f64 99.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \alpha\right) \]

   5. Simplified 99.5%

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

   if -0.999999799999999994 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.7%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\color{blue}{\left(\beta \cdot \left(1 + \frac{\alpha}{\beta}\right)\right)}, 2\right)\right), 1\right), 2\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \left(1 + \frac{\alpha}{\beta}\right)\right), 2\right)\right), 1\right), 2\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(1, \left(\frac{\alpha}{\beta}\right)\right)\right), 2\right)\right), 1\right), 2\right) \]

      3. /-lowering-/.f64 99.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\alpha, \beta\right)\right)\right), 2\right)\right), 1\right), 2\right) \]

   5. Simplified 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)} \leq -0.9999998:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \beta \cdot \left(1 + \frac{\alpha}{\beta}\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}\\ \mathbf{if}\;t\_0 \leq -0.9999998:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ 2.0 (+ alpha beta)))))
   (if (<= t_0 -0.9999998) (/ (+ 1.0 beta) alpha) (/ (+ 1.0 t_0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (2.0 + (alpha + beta));
	double tmp;
	if (t_0 <= -0.9999998) {
		tmp = (1.0 + beta) / alpha;
	} else {
		tmp = (1.0 + t_0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / (2.0d0 + (alpha + beta))
    if (t_0 <= (-0.9999998d0)) then
        tmp = (1.0d0 + beta) / alpha
    else
        tmp = (1.0d0 + t_0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (2.0 + (alpha + beta));
	double tmp;
	if (t_0 <= -0.9999998) {
		tmp = (1.0 + beta) / alpha;
	} else {
		tmp = (1.0 + t_0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / (2.0 + (alpha + beta))
	tmp = 0
	if t_0 <= -0.9999998:
		tmp = (1.0 + beta) / alpha
	else:
		tmp = (1.0 + t_0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(2.0 + Float64(alpha + beta)))
	tmp = 0.0
	if (t_0 <= -0.9999998)
		tmp = Float64(Float64(1.0 + beta) / alpha);
	else
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / (2.0 + (alpha + beta));
	tmp = 0.0;
	if (t_0 <= -0.9999998)
		tmp = (1.0 + beta) / alpha;
	else
		tmp = (1.0 + t_0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.9999998], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(1.0 + t$95$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))) < -0.999999799999999994

   1. Initial program 6.7%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right)\right), \color{blue}{\alpha}\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \alpha\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \alpha\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(\frac{1}{2} \cdot 2\right) \cdot \beta\right), \alpha\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + 1 \cdot \beta\right), \alpha\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \alpha\right) \]

      8. +-lowering-+.f64 99.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \alpha\right) \]

   5. Simplified 99.5%

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

   if -0.999999799999999994 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)} \leq -0.9999998:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 1

   1. Initial program 66.5%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right), \color{blue}{1}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2\right)\right), 1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      10. +-lowering-+.f64 66.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 66.5%

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

   5. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 \cdot \frac{\alpha \cdot \left(\frac{1}{2 + \beta} + \frac{\beta}{{\left(2 + \beta\right)}^{2}}\right)}{{\left(1 + \frac{\beta}{2 + \beta}\right)}^{2}} + 2 \cdot \frac{1}{1 + \frac{\beta}{2 + \beta}}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(2 \cdot \frac{\alpha \cdot \left(\frac{1}{2 + \beta} + \frac{\beta}{{\left(2 + \beta\right)}^{2}}\right)}{{\left(1 + \frac{\beta}{2 + \beta}\right)}^{2}}\right), \color{blue}{\left(2 \cdot \frac{1}{1 + \frac{\beta}{2 + \beta}}\right)}\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + \alpha\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 98.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\alpha}\right)\right) \]

   10. Simplified 98.5%

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

   if 1 < beta

   1. Initial program 90.7%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right), \color{blue}{1}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2\right)\right), 1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      10. +-lowering-+.f64 90.7%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 90.7%

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

   5. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 \cdot \frac{\alpha \cdot \left(\frac{1}{2 + \beta} + \frac{\beta}{{\left(2 + \beta\right)}^{2}}\right)}{{\left(1 + \frac{\beta}{2 + \beta}\right)}^{2}} + 2 \cdot \frac{1}{1 + \frac{\beta}{2 + \beta}}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(2 \cdot \frac{\alpha \cdot \left(\frac{1}{2 + \beta} + \frac{\beta}{{\left(2 + \beta\right)}^{2}}\right)}{{\left(1 + \frac{\beta}{2 + \beta}\right)}^{2}}\right), \color{blue}{\left(2 \cdot \frac{1}{1 + \frac{\beta}{2 + \beta}}\right)}\right)\right) \]

   7. Simplified 97.2%

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

   8. Taylor expanded in beta around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \alpha - 1}{\beta}\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{-1 \cdot \left(-1 \cdot \alpha - 1\right)}{\color{blue}{\beta}}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{-1 \cdot \left(-1 \cdot \alpha + \left(\mathsf{neg}\left(1\right)\right)\right)}{\beta}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{-1 \cdot \left(-1 \cdot \alpha + -1\right)}{\beta}\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{-1 \cdot \left(-1 \cdot \alpha\right) + -1 \cdot -1}{\beta}\right)\right) \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{\left(\mathsf{neg}\left(-1 \cdot \alpha\right)\right) + -1 \cdot -1}{\beta}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\alpha\right)\right)\right)\right) + -1 \cdot -1}{\beta}\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{\alpha + -1 \cdot -1}{\beta}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{\alpha + 1}{\beta}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{1 + \alpha}{\beta}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\beta}\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\alpha + 1\right), \beta\right)\right)\right) \]

      13. +-lowering-+.f64 98.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \beta\right)\right)\right) \]

   10. Simplified 98.1%

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

4. Final simplification 98.4%

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

Alternative 6: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.35:\\ \;\;\;\;\frac{1}{2 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{\alpha}{\beta}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.35) (/ 1.0 (+ 2.0 alpha)) (/ 1.0 (+ 1.0 (/ alpha beta)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.35) {
		tmp = 1.0 / (2.0 + alpha);
	} else {
		tmp = 1.0 / (1.0 + (alpha / beta));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.35d0) then
        tmp = 1.0d0 / (2.0d0 + alpha)
    else
        tmp = 1.0d0 / (1.0d0 + (alpha / beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.35) {
		tmp = 1.0 / (2.0 + alpha);
	} else {
		tmp = 1.0 / (1.0 + (alpha / beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.35:
		tmp = 1.0 / (2.0 + alpha)
	else:
		tmp = 1.0 / (1.0 + (alpha / beta))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.35)
		tmp = Float64(1.0 / Float64(2.0 + alpha));
	else
		tmp = Float64(1.0 / Float64(1.0 + Float64(alpha / beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.35)
		tmp = 1.0 / (2.0 + alpha);
	else
		tmp = 1.0 / (1.0 + (alpha / beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.35], N[(1.0 / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.3500000000000001

   1. Initial program 66.5%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right), \color{blue}{1}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2\right)\right), 1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      10. +-lowering-+.f64 66.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 66.5%

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

   5. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 \cdot \frac{\alpha \cdot \left(\frac{1}{2 + \beta} + \frac{\beta}{{\left(2 + \beta\right)}^{2}}\right)}{{\left(1 + \frac{\beta}{2 + \beta}\right)}^{2}} + 2 \cdot \frac{1}{1 + \frac{\beta}{2 + \beta}}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(2 \cdot \frac{\alpha \cdot \left(\frac{1}{2 + \beta} + \frac{\beta}{{\left(2 + \beta\right)}^{2}}\right)}{{\left(1 + \frac{\beta}{2 + \beta}\right)}^{2}}\right), \color{blue}{\left(2 \cdot \frac{1}{1 + \frac{\beta}{2 + \beta}}\right)}\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + \alpha\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 98.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\alpha}\right)\right) \]

   10. Simplified 98.5%

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

   if 1.3500000000000001 < beta

   1. Initial program 90.7%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right), \color{blue}{1}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2\right)\right), 1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      10. +-lowering-+.f64 90.7%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 90.7%

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

   5. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 \cdot \frac{\alpha \cdot \left(\frac{1}{2 + \beta} + \frac{\beta}{{\left(2 + \beta\right)}^{2}}\right)}{{\left(1 + \frac{\beta}{2 + \beta}\right)}^{2}} + 2 \cdot \frac{1}{1 + \frac{\beta}{2 + \beta}}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(2 \cdot \frac{\alpha \cdot \left(\frac{1}{2 + \beta} + \frac{\beta}{{\left(2 + \beta\right)}^{2}}\right)}{{\left(1 + \frac{\beta}{2 + \beta}\right)}^{2}}\right), \color{blue}{\left(2 \cdot \frac{1}{1 + \frac{\beta}{2 + \beta}}\right)}\right)\right) \]

   7. Simplified 97.2%

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

   8. Taylor expanded in beta around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \alpha - 1}{\beta}\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{-1 \cdot \left(-1 \cdot \alpha - 1\right)}{\color{blue}{\beta}}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{-1 \cdot \left(-1 \cdot \alpha + \left(\mathsf{neg}\left(1\right)\right)\right)}{\beta}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{-1 \cdot \left(-1 \cdot \alpha + -1\right)}{\beta}\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{-1 \cdot \left(-1 \cdot \alpha\right) + -1 \cdot -1}{\beta}\right)\right) \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{\left(\mathsf{neg}\left(-1 \cdot \alpha\right)\right) + -1 \cdot -1}{\beta}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\alpha\right)\right)\right)\right) + -1 \cdot -1}{\beta}\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{\alpha + -1 \cdot -1}{\beta}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{\alpha + 1}{\beta}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{1 + \alpha}{\beta}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\beta}\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\alpha + 1\right), \beta\right)\right)\right) \]

      13. +-lowering-+.f64 98.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \beta\right)\right)\right) \]

   10. Simplified 98.1%

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

   11. Taylor expanded in alpha around inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\alpha}{\beta}\right)}\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 97.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\alpha, \color{blue}{\beta}\right)\right)\right) \]

   13. Simplified 97.0%

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

Alternative 7: 93.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7:\\ \;\;\;\;\frac{1}{2 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{\beta}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.0) (/ 1.0 (+ 2.0 alpha)) (/ 1.0 (+ 1.0 (/ 1.0 beta)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.0) {
		tmp = 1.0 / (2.0 + alpha);
	} else {
		tmp = 1.0 / (1.0 + (1.0 / beta));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 7.0d0) then
        tmp = 1.0d0 / (2.0d0 + alpha)
    else
        tmp = 1.0d0 / (1.0d0 + (1.0d0 / beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.0) {
		tmp = 1.0 / (2.0 + alpha);
	} else {
		tmp = 1.0 / (1.0 + (1.0 / beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 7.0:
		tmp = 1.0 / (2.0 + alpha)
	else:
		tmp = 1.0 / (1.0 + (1.0 / beta))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.0)
		tmp = Float64(1.0 / Float64(2.0 + alpha));
	else
		tmp = Float64(1.0 / Float64(1.0 + Float64(1.0 / beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.0)
		tmp = 1.0 / (2.0 + alpha);
	else
		tmp = 1.0 / (1.0 + (1.0 / beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 7.0], N[(1.0 / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7

   1. Initial program 66.5%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right), \color{blue}{1}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2\right)\right), 1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      10. +-lowering-+.f64 66.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 66.5%

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

   5. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 \cdot \frac{\alpha \cdot \left(\frac{1}{2 + \beta} + \frac{\beta}{{\left(2 + \beta\right)}^{2}}\right)}{{\left(1 + \frac{\beta}{2 + \beta}\right)}^{2}} + 2 \cdot \frac{1}{1 + \frac{\beta}{2 + \beta}}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(2 \cdot \frac{\alpha \cdot \left(\frac{1}{2 + \beta} + \frac{\beta}{{\left(2 + \beta\right)}^{2}}\right)}{{\left(1 + \frac{\beta}{2 + \beta}\right)}^{2}}\right), \color{blue}{\left(2 \cdot \frac{1}{1 + \frac{\beta}{2 + \beta}}\right)}\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + \alpha\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 98.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\alpha}\right)\right) \]

   10. Simplified 98.5%

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

   if 7 < beta

   1. Initial program 90.7%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right), \color{blue}{1}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2\right)\right), 1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      10. +-lowering-+.f64 90.7%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 90.7%

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

   5. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 \cdot \frac{\alpha \cdot \left(\frac{1}{2 + \beta} + \frac{\beta}{{\left(2 + \beta\right)}^{2}}\right)}{{\left(1 + \frac{\beta}{2 + \beta}\right)}^{2}} + 2 \cdot \frac{1}{1 + \frac{\beta}{2 + \beta}}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(2 \cdot \frac{\alpha \cdot \left(\frac{1}{2 + \beta} + \frac{\beta}{{\left(2 + \beta\right)}^{2}}\right)}{{\left(1 + \frac{\beta}{2 + \beta}\right)}^{2}}\right), \color{blue}{\left(2 \cdot \frac{1}{1 + \frac{\beta}{2 + \beta}}\right)}\right)\right) \]

   7. Simplified 97.2%

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

   8. Taylor expanded in beta around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \alpha - 1}{\beta}\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{-1 \cdot \left(-1 \cdot \alpha - 1\right)}{\color{blue}{\beta}}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{-1 \cdot \left(-1 \cdot \alpha + \left(\mathsf{neg}\left(1\right)\right)\right)}{\beta}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{-1 \cdot \left(-1 \cdot \alpha + -1\right)}{\beta}\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{-1 \cdot \left(-1 \cdot \alpha\right) + -1 \cdot -1}{\beta}\right)\right) \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{\left(\mathsf{neg}\left(-1 \cdot \alpha\right)\right) + -1 \cdot -1}{\beta}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\alpha\right)\right)\right)\right) + -1 \cdot -1}{\beta}\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{\alpha + -1 \cdot -1}{\beta}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{\alpha + 1}{\beta}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \frac{1 + \alpha}{\beta}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\beta}\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\alpha + 1\right), \beta\right)\right)\right) \]

      13. +-lowering-+.f64 98.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \beta\right)\right)\right) \]

   10. Simplified 98.1%

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

   11. Taylor expanded in alpha around 0

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + \frac{1}{\beta}\right)}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\beta}\right)}\right)\right) \]

       3. /-lowering-/.f64 88.2%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\beta}\right)\right)\right) \]

   13. Simplified 88.2%

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

Alternative 8: 92.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.5:\\ \;\;\;\;\frac{1}{2 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 - \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.5) (/ 1.0 (+ 2.0 alpha)) (+ 1.0 (/ (- -1.0 alpha) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.5) {
		tmp = 1.0 / (2.0 + alpha);
	} else {
		tmp = 1.0 + ((-1.0 - alpha) / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.5d0) then
        tmp = 1.0d0 / (2.0d0 + alpha)
    else
        tmp = 1.0d0 + (((-1.0d0) - alpha) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.5) {
		tmp = 1.0 / (2.0 + alpha);
	} else {
		tmp = 1.0 + ((-1.0 - alpha) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 8.5:
		tmp = 1.0 / (2.0 + alpha)
	else:
		tmp = 1.0 + ((-1.0 - alpha) / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.5)
		tmp = Float64(1.0 / Float64(2.0 + alpha));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.5)
		tmp = 1.0 / (2.0 + alpha);
	else
		tmp = 1.0 + ((-1.0 - alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 8.5], N[(1.0 / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.5

   1. Initial program 66.5%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right), \color{blue}{1}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2\right)\right), 1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      10. +-lowering-+.f64 66.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 66.5%

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

   5. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 \cdot \frac{\alpha \cdot \left(\frac{1}{2 + \beta} + \frac{\beta}{{\left(2 + \beta\right)}^{2}}\right)}{{\left(1 + \frac{\beta}{2 + \beta}\right)}^{2}} + 2 \cdot \frac{1}{1 + \frac{\beta}{2 + \beta}}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(2 \cdot \frac{\alpha \cdot \left(\frac{1}{2 + \beta} + \frac{\beta}{{\left(2 + \beta\right)}^{2}}\right)}{{\left(1 + \frac{\beta}{2 + \beta}\right)}^{2}}\right), \color{blue}{\left(2 \cdot \frac{1}{1 + \frac{\beta}{2 + \beta}}\right)}\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + \alpha\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 98.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\alpha}\right)\right) \]

   10. Simplified 98.5%

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

   if 8.5 < beta

   1. Initial program 90.7%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right), \color{blue}{1}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2\right)\right), 1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      10. +-lowering-+.f64 90.7%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 90.7%

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

   5. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 \cdot \frac{\alpha \cdot \left(\frac{1}{2 + \beta} + \frac{\beta}{{\left(2 + \beta\right)}^{2}}\right)}{{\left(1 + \frac{\beta}{2 + \beta}\right)}^{2}} + 2 \cdot \frac{1}{1 + \frac{\beta}{2 + \beta}}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(2 \cdot \frac{\alpha \cdot \left(\frac{1}{2 + \beta} + \frac{\beta}{{\left(2 + \beta\right)}^{2}}\right)}{{\left(1 + \frac{\beta}{2 + \beta}\right)}^{2}}\right), \color{blue}{\left(2 \cdot \frac{1}{1 + \frac{\beta}{2 + \beta}}\right)}\right)\right) \]

   7. Simplified 97.2%

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

   8. Taylor expanded in beta around inf

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

      1. mul-1-neg N/A

         \[\leadsto 1 + \left(\mathsf{neg}\left(\frac{1 + \alpha}{\beta}\right)\right) \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\beta}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\alpha + 1\right), \beta\right)\right) \]

      6. +-lowering-+.f64 88.0%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \beta\right)\right) \]

   10. Simplified 88.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.5:\\ \;\;\;\;\frac{1}{2 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 - \alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.8% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 10.0) (/ 1.0 (+ 2.0 alpha)) 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 10.0) {
		tmp = 1.0 / (2.0 + alpha);
	} 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 <= 10.0d0) then
        tmp = 1.0d0 / (2.0d0 + alpha)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 10.0) {
		tmp = 1.0 / (2.0 + alpha);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 10.0:
		tmp = 1.0 / (2.0 + alpha)
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 10.0)
		tmp = Float64(1.0 / Float64(2.0 + alpha));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 10.0)
		tmp = 1.0 / (2.0 + alpha);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 10.0], N[(1.0 / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 10

   1. Initial program 66.5%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right), \color{blue}{1}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2\right)\right), 1\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), 1\right)\right)\right) \]

      10. +-lowering-+.f64 66.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 66.5%

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

   5. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 \cdot \frac{\alpha \cdot \left(\frac{1}{2 + \beta} + \frac{\beta}{{\left(2 + \beta\right)}^{2}}\right)}{{\left(1 + \frac{\beta}{2 + \beta}\right)}^{2}} + 2 \cdot \frac{1}{1 + \frac{\beta}{2 + \beta}}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(2 \cdot \frac{\alpha \cdot \left(\frac{1}{2 + \beta} + \frac{\beta}{{\left(2 + \beta\right)}^{2}}\right)}{{\left(1 + \frac{\beta}{2 + \beta}\right)}^{2}}\right), \color{blue}{\left(2 \cdot \frac{1}{1 + \frac{\beta}{2 + \beta}}\right)}\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in beta around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + \alpha\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 98.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\alpha}\right)\right) \]

   10. Simplified 98.5%

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

   if 10 < beta

   1. Initial program 90.7%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 87.2%

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

Alternative 10: 69.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 0.98:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 0.98) (+ 0.5 (* alpha -0.25)) (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.98) {
		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 <= 0.98d0) 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 <= 0.98) {
		tmp = 0.5 + (alpha * -0.25);
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 0.98:
		tmp = 0.5 + (alpha * -0.25)
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 0.98)
		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 <= 0.98)
		tmp = 0.5 + (alpha * -0.25);
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 0.98], N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision], N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 0.97999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 - \frac{\alpha}{2 + \alpha}\right)}, 2\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\alpha}{2 + \alpha}\right)\right), 2\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \left(2 + \alpha\right)\right)\right), 2\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \left(\alpha + 2\right)\right)\right), 2\right) \]

      4. +-lowering-+.f64 72.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 2\right) \]

   5. Simplified 72.7%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{4} \cdot \alpha} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{4} \cdot \alpha\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\alpha \cdot \color{blue}{\frac{-1}{4}}\right)\right) \]

      3. *-lowering-*.f64 72.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\alpha, \color{blue}{\frac{-1}{4}}\right)\right) \]

   8. Simplified 72.0%

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

   if 0.97999999999999998 < alpha

   1. Initial program 21.2%

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

   3. Taylor expanded in beta around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 - \frac{\alpha}{2 + \alpha}\right)}, 2\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\alpha}{2 + \alpha}\right)\right), 2\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \left(2 + \alpha\right)\right)\right), 2\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \left(\alpha + 2\right)\right)\right), 2\right) \]

      4. +-lowering-+.f64 7.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 2\right) \]

   5. Simplified 7.9%

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

   6. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 76.1%

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\alpha}\right) \]

   8. Simplified 76.1%

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

Alternative 11: 68.9% 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. Add Preprocessing

   3. Taylor expanded in beta around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 - \frac{\alpha}{2 + \alpha}\right)}, 2\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\alpha}{2 + \alpha}\right)\right), 2\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \left(2 + \alpha\right)\right)\right), 2\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \left(\alpha + 2\right)\right)\right), 2\right) \]

      4. +-lowering-+.f64 72.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 2\right) \]

   5. Simplified 72.7%

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

   6. Taylor expanded in alpha around 0

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

   8. Simplified 71.4%

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

   if 2 < alpha

   1. Initial program 21.2%

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

   3. Taylor expanded in beta around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 - \frac{\alpha}{2 + \alpha}\right)}, 2\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\alpha}{2 + \alpha}\right)\right), 2\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \left(2 + \alpha\right)\right)\right), 2\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \left(\alpha + 2\right)\right)\right), 2\right) \]

      4. +-lowering-+.f64 7.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 2\right) \]

   5. Simplified 7.9%

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

   6. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 76.1%

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\alpha}\right) \]

   8. Simplified 76.1%

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

Alternative 12: 71.5% 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 66.5%

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

   3. Taylor expanded in beta around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 - \frac{\alpha}{2 + \alpha}\right)}, 2\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\alpha}{2 + \alpha}\right)\right), 2\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \left(2 + \alpha\right)\right)\right), 2\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \left(\alpha + 2\right)\right)\right), 2\right) \]

      4. +-lowering-+.f64 65.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 2\right) \]

   5. Simplified 65.6%

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

   6. Taylor expanded in alpha around 0

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

   8. Simplified 63.1%

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

   if 2 < beta

   1. Initial program 90.7%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 87.2%

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

Alternative 13: 49.6% 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 73.5%

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

3. Taylor expanded in beta around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 - \frac{\alpha}{2 + \alpha}\right)}, 2\right) \]
4. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\alpha}{2 + \alpha}\right)\right), 2\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \left(2 + \alpha\right)\right)\right), 2\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \left(\alpha + 2\right)\right)\right), 2\right) \]

   4. +-lowering-+.f64 50.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 2\right) \]

5. Simplified 50.9%

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

6. Taylor expanded in alpha around 0

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

8. Simplified 49.9%

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

Reproduce

herbie shell --seed 2024191 
(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))