Octave 3.8, jcobi/1

Percentage Accurate: 74.3% → 99.9%

Time: 11.3s

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: 74.3% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta + 2}{\alpha}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.998:\\ \;\;\;\;\frac{\beta + \left(0.5 \cdot \left(t\_0 \cdot \left(\left(-2 - \beta \cdot 2\right) + t\_0 \cdot \left(2 + \beta \cdot 2\right)\right)\right) + 1\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\beta + \left(\alpha + 2\right)}, \beta - \alpha, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ beta 2.0) alpha)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.998)
     (/
      (+
       beta
       (+
        (* 0.5 (* t_0 (+ (- -2.0 (* beta 2.0)) (* t_0 (+ 2.0 (* beta 2.0))))))
        1.0))
      alpha)
     (/ (fma (/ 1.0 (+ beta (+ alpha 2.0))) (- beta alpha) 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta + 2.0) / alpha;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.998) {
		tmp = (beta + ((0.5 * (t_0 * ((-2.0 - (beta * 2.0)) + (t_0 * (2.0 + (beta * 2.0)))))) + 1.0)) / alpha;
	} else {
		tmp = fma((1.0 / (beta + (alpha + 2.0))), (beta - alpha), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta + 2.0) / alpha)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.998)
		tmp = Float64(Float64(beta + Float64(Float64(0.5 * Float64(t_0 * Float64(Float64(-2.0 - Float64(beta * 2.0)) + Float64(t_0 * Float64(2.0 + Float64(beta * 2.0)))))) + 1.0)) / alpha);
	else
		tmp = Float64(fma(Float64(1.0 / Float64(beta + Float64(alpha + 2.0))), Float64(beta - alpha), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.998], N[(N[(beta + N[(N[(0.5 * N[(t$95$0 * N[(N[(-2.0 - N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta - alpha), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.998

   1. Initial program 9.0%

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

   3. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right) + \left(\frac{1}{2} \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha} + \frac{1}{2} \cdot \frac{{\left(2 + \beta\right)}^{3} - -1 \cdot \left(\beta \cdot {\left(2 + \beta\right)}^{2}\right)}{{\alpha}^{2}}\right)}{\alpha}} \]

   5. Simplified 82.5%

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

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

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

   7. Applied egg-rr 99.8%

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

   if -0.998 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. fma-define N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

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

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

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

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

      10. --lowering--.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.998:\\ \;\;\;\;\frac{\beta + \left(0.5 \cdot \left(\frac{\beta + 2}{\alpha} \cdot \left(\left(-2 - \beta \cdot 2\right) + \frac{\beta + 2}{\alpha} \cdot \left(2 + \beta \cdot 2\right)\right)\right) + 1\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\beta + \left(\alpha + 2\right)}, \beta - \alpha, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta + 2}{\alpha}\\ t_1 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_1 \leq -0.998:\\ \;\;\;\;\frac{\beta + \left(0.5 \cdot \left(t\_0 \cdot \left(\left(-2 - \beta \cdot 2\right) + t\_0 \cdot \left(2 + \beta \cdot 2\right)\right)\right) + 1\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ beta 2.0) alpha))
        (t_1 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_1 -0.998)
     (/
      (+
       beta
       (+
        (* 0.5 (* t_0 (+ (- -2.0 (* beta 2.0)) (* t_0 (+ 2.0 (* beta 2.0))))))
        1.0))
      alpha)
     (/ (+ t_1 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta + 2.0) / alpha;
	double t_1 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_1 <= -0.998) {
		tmp = (beta + ((0.5 * (t_0 * ((-2.0 - (beta * 2.0)) + (t_0 * (2.0 + (beta * 2.0)))))) + 1.0)) / alpha;
	} else {
		tmp = (t_1 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (beta + 2.0d0) / alpha
    t_1 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_1 <= (-0.998d0)) then
        tmp = (beta + ((0.5d0 * (t_0 * (((-2.0d0) - (beta * 2.0d0)) + (t_0 * (2.0d0 + (beta * 2.0d0)))))) + 1.0d0)) / alpha
    else
        tmp = (t_1 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta + 2.0) / alpha;
	double t_1 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_1 <= -0.998) {
		tmp = (beta + ((0.5 * (t_0 * ((-2.0 - (beta * 2.0)) + (t_0 * (2.0 + (beta * 2.0)))))) + 1.0)) / alpha;
	} else {
		tmp = (t_1 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta + 2.0) / alpha
	t_1 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_1 <= -0.998:
		tmp = (beta + ((0.5 * (t_0 * ((-2.0 - (beta * 2.0)) + (t_0 * (2.0 + (beta * 2.0)))))) + 1.0)) / alpha
	else:
		tmp = (t_1 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta + 2.0) / alpha)
	t_1 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_1 <= -0.998)
		tmp = Float64(Float64(beta + Float64(Float64(0.5 * Float64(t_0 * Float64(Float64(-2.0 - Float64(beta * 2.0)) + Float64(t_0 * Float64(2.0 + Float64(beta * 2.0)))))) + 1.0)) / alpha);
	else
		tmp = Float64(Float64(t_1 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta + 2.0) / alpha;
	t_1 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_1 <= -0.998)
		tmp = (beta + ((0.5 * (t_0 * ((-2.0 - (beta * 2.0)) + (t_0 * (2.0 + (beta * 2.0)))))) + 1.0)) / alpha;
	else
		tmp = (t_1 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.998], N[(N[(beta + N[(N[(0.5 * N[(t$95$0 * N[(N[(-2.0 - N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(t$95$1 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.998

   1. Initial program 9.0%

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

   3. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right) + \left(\frac{1}{2} \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha} + \frac{1}{2} \cdot \frac{{\left(2 + \beta\right)}^{3} - -1 \cdot \left(\beta \cdot {\left(2 + \beta\right)}^{2}\right)}{{\alpha}^{2}}\right)}{\alpha}} \]

   5. Simplified 82.5%

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

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

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

   7. Applied egg-rr 99.8%

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

   if -0.998 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.998:\\ \;\;\;\;\frac{\beta + \left(0.5 \cdot \left(\frac{\beta + 2}{\alpha} \cdot \left(\left(-2 - \beta \cdot 2\right) + \frac{\beta + 2}{\alpha} \cdot \left(2 + \beta \cdot 2\right)\right)\right) + 1\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.9%

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

   3. Taylor expanded in alpha around inf

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

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

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

   5. Simplified 99.9%

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

   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 99.8%

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

Alternative 4: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.998:\\ \;\;\;\;\frac{\beta + \left(1 + \frac{-2 + \frac{4}{\alpha}}{\alpha}\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.998)
     (/ (+ beta (+ 1.0 (/ (+ -2.0 (/ 4.0 alpha)) alpha))) alpha)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.998) {
		tmp = (beta + (1.0 + ((-2.0 + (4.0 / alpha)) / alpha))) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.998d0)) then
        tmp = (beta + (1.0d0 + (((-2.0d0) + (4.0d0 / alpha)) / alpha))) / alpha
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.998) {
		tmp = (beta + (1.0 + ((-2.0 + (4.0 / alpha)) / alpha))) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.998:
		tmp = (beta + (1.0 + ((-2.0 + (4.0 / alpha)) / alpha))) / alpha
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.998)
		tmp = Float64(Float64(beta + Float64(1.0 + Float64(Float64(-2.0 + Float64(4.0 / alpha)) / alpha))) / alpha);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.998)
		tmp = (beta + (1.0 + ((-2.0 + (4.0 / alpha)) / alpha))) / alpha;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.998], N[(N[(beta + N[(1.0 + N[(N[(-2.0 + N[(4.0 / alpha), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.998

   1. Initial program 9.0%

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

   3. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right) + \left(\frac{1}{2} \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha} + \frac{1}{2} \cdot \frac{{\left(2 + \beta\right)}^{3} - -1 \cdot \left(\beta \cdot {\left(2 + \beta\right)}^{2}\right)}{{\alpha}^{2}}\right)}{\alpha}} \]

   5. Simplified 82.5%

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

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

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in beta around 0

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

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

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

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(4 \cdot \frac{1}{\alpha} + \left(\mathsf{neg}\left(2\right)\right)\right), \alpha\right), 1\right), \beta\right), \alpha\right) \]

      3. metadata-eval N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 98.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, \alpha\right), -2\right), \alpha\right), 1\right), \beta\right), \alpha\right) \]

   10. Simplified 98.8%

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

   if -0.998 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.998:\\ \;\;\;\;\frac{\beta + \left(1 + \frac{-2 + \frac{4}{\alpha}}{\alpha}\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 330:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + \left(1 + \frac{-2 + \frac{4}{\alpha}}{\alpha}\right)}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 330.0)
   (/ (+ 1.0 (/ (- beta alpha) (+ beta 2.0))) 2.0)
   (/ (+ beta (+ 1.0 (/ (+ -2.0 (/ 4.0 alpha)) alpha))) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 330.0) {
		tmp = (1.0 + ((beta - alpha) / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + (1.0 + ((-2.0 + (4.0 / alpha)) / alpha))) / 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 <= 330.0d0) then
        tmp = (1.0d0 + ((beta - alpha) / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (beta + (1.0d0 + (((-2.0d0) + (4.0d0 / alpha)) / alpha))) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 330.0) {
		tmp = (1.0 + ((beta - alpha) / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + (1.0 + ((-2.0 + (4.0 / alpha)) / alpha))) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 330.0:
		tmp = (1.0 + ((beta - alpha) / (beta + 2.0))) / 2.0
	else:
		tmp = (beta + (1.0 + ((-2.0 + (4.0 / alpha)) / alpha))) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 330.0)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(beta + Float64(1.0 + Float64(Float64(-2.0 + Float64(4.0 / alpha)) / alpha))) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 330.0)
		tmp = (1.0 + ((beta - alpha) / (beta + 2.0))) / 2.0;
	else
		tmp = (beta + (1.0 + ((-2.0 + (4.0 / alpha)) / alpha))) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 330.0], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta + N[(1.0 + N[(N[(-2.0 + N[(4.0 / alpha), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 330

   1. Initial program 100.0%

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

   3. Taylor expanded in alpha around 0

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

   5. Simplified 98.7%

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

   if 330 < alpha

   1. Initial program 24.2%

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

   3. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right) + \left(\frac{1}{2} \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha} + \frac{1}{2} \cdot \frac{{\left(2 + \beta\right)}^{3} - -1 \cdot \left(\beta \cdot {\left(2 + \beta\right)}^{2}\right)}{{\alpha}^{2}}\right)}{\alpha}} \]

   5. Simplified 69.5%

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

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

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

   7. Applied egg-rr 84.3%

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

   8. Taylor expanded in beta around 0

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

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

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

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(4 \cdot \frac{1}{\alpha} + \left(\mathsf{neg}\left(2\right)\right)\right), \alpha\right), 1\right), \beta\right), \alpha\right) \]

      3. metadata-eval N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 83.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, \alpha\right), -2\right), \alpha\right), 1\right), \beta\right), \alpha\right) \]

   10. Simplified 83.8%

       \[\leadsto \frac{\left(\color{blue}{\frac{\frac{4}{\alpha} + -2}{\alpha}} + 1\right) + \beta}{\alpha} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 330:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + \left(1 + \frac{-2 + \frac{4}{\alpha}}{\alpha}\right)}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 11000000000:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 11000000000.0)
   (/ (+ 1.0 (/ (- beta alpha) (+ beta 2.0))) 2.0)
   (* -0.5 (/ (- (- -2.0 beta) beta) alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 11000000000.0) {
		tmp = (1.0 + ((beta - alpha) / (beta + 2.0))) / 2.0;
	} else {
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 11000000000.0d0) then
        tmp = (1.0d0 + ((beta - alpha) / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (-0.5d0) * ((((-2.0d0) - beta) - beta) / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 11000000000.0) {
		tmp = (1.0 + ((beta - alpha) / (beta + 2.0))) / 2.0;
	} else {
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 11000000000.0:
		tmp = (1.0 + ((beta - alpha) / (beta + 2.0))) / 2.0
	else:
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 11000000000.0)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(-0.5 * Float64(Float64(Float64(-2.0 - beta) - beta) / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 11000000000.0)
		tmp = (1.0 + ((beta - alpha) / (beta + 2.0))) / 2.0;
	else
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 11000000000.0], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(-0.5 * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.1e10

   1. Initial program 99.8%

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

   3. Taylor expanded in alpha around 0

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

   5. Simplified 97.6%

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

   if 1.1e10 < alpha

   1. Initial program 21.7%

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

   3. Taylor expanded in alpha around -inf

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

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

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

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

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

      3. associate--r+ N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. --lowering--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. metadata-eval 84.5%

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

   5. Simplified 84.5%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 11000000000:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 7900000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 7900000000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (* -0.5 (/ (- (- -2.0 beta) beta) alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7900000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 7900000000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (-0.5d0) * ((((-2.0d0) - beta) - beta) / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7900000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 7900000000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 7900000000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(-0.5 * Float64(Float64(Float64(-2.0 - beta) - beta) / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 7900000000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 7900000000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(-0.5 * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 7.9e9

   1. Initial program 99.8%

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

   3. Taylor expanded in alpha around 0

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

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

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

      2. +-lowering-+.f64 97.0%

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

   5. Simplified 97.0%

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

   if 7.9e9 < alpha

   1. Initial program 21.7%

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

   3. Taylor expanded in alpha around -inf

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

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

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

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

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

      3. associate--r+ N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. --lowering--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. metadata-eval 84.5%

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

   5. Simplified 84.5%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 7900000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.96:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.96)
   (+ 0.5 (* alpha -0.25))
   (* -0.5 (/ (- (- -2.0 beta) beta) alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.96) {
		tmp = 0.5 + (alpha * -0.25);
	} else {
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.96d0) then
        tmp = 0.5d0 + (alpha * (-0.25d0))
    else
        tmp = (-0.5d0) * ((((-2.0d0) - beta) - beta) / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.96) {
		tmp = 0.5 + (alpha * -0.25);
	} else {
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.96:
		tmp = 0.5 + (alpha * -0.25)
	else:
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.96)
		tmp = Float64(0.5 + Float64(alpha * -0.25));
	else
		tmp = Float64(-0.5 * Float64(Float64(Float64(-2.0 - beta) - beta) / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.96)
		tmp = 0.5 + (alpha * -0.25);
	else
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.96], N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.96

   1. Initial program 100.0%

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

   3. Taylor expanded in alpha around 0

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

   5. Simplified 99.2%

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

   6. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 73.0%

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

   8. Simplified 73.0%

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

   9. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{4} \cdot \alpha} \]
   10. 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 73.0%

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

   11. Simplified 73.0%

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

   if 1.96 < alpha

   1. Initial program 26.8%

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

   3. Taylor expanded in alpha around -inf

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

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

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

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

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

      3. associate--r+ N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. --lowering--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. metadata-eval 79.9%

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

   5. Simplified 79.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 74.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.0], N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / 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 alpha around 0

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

   5. Simplified 99.2%

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

   6. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 73.0%

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

   8. Simplified 73.0%

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

   9. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{4} \cdot \alpha} \]
   10. 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 73.0%

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

   11. Simplified 73.0%

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

   if 2 < alpha

   1. Initial program 26.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 79.8%

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

   5. Simplified 79.8%

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

4. Final simplification 75.3%

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

Alternative 10: 71.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5 + (beta * 0.25);
	} else {
		tmp = 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 <= 2.0d0) then
        tmp = 0.5d0 + (beta * 0.25d0)
    else
        tmp = 1.0d0 + ((-1.0d0) / beta)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 73.4%

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

   3. Taylor expanded in alpha around 0

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

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

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

      2. +-lowering-+.f64 69.9%

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

   5. Simplified 69.9%

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

   6. Taylor expanded in beta around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 69.4%

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

   8. Simplified 69.4%

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

   if 2 < beta

   1. Initial program 79.6%

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

   3. Taylor expanded in alpha around 0

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

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

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

      2. +-lowering-+.f64 78.1%

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

   5. Simplified 78.1%

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

   6. Taylor expanded in beta around inf

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

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

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

      2. /-lowering-/.f64 77.6%

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

   8. Simplified 77.6%

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

4. Final simplification 72.1%

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

Alternative 11: 71.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5 + (beta * 0.25);
	} 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 + (beta * 0.25d0)
    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 + (beta * 0.25);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

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

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(0.5 + Float64(beta * 0.25));
	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 + (beta * 0.25);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 73.4%

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

   3. Taylor expanded in alpha around 0

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

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

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

      2. +-lowering-+.f64 69.9%

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

   5. Simplified 69.9%

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

   6. Taylor expanded in beta around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 69.4%

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

   8. Simplified 69.4%

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

   if 2 < beta

   1. Initial program 79.6%

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

   3. Taylor expanded in beta around inf

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

   5. Simplified 76.9%

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

Alternative 12: 71.0% 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 73.4%

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

   3. Taylor expanded in alpha around 0

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

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

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

      2. +-lowering-+.f64 69.9%

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

   5. Simplified 69.9%

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

   6. Taylor expanded in beta around 0

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

   8. Simplified 67.6%

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

   if 2 < beta

   1. Initial program 79.6%

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

   3. Taylor expanded in beta around inf

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

   5. Simplified 76.9%

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

Alternative 13: 49.2% 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 75.4%

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

3. Taylor expanded in alpha around 0

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

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

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

   2. +-lowering-+.f64 72.6%

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

5. Simplified 72.6%

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

6. Taylor expanded in beta around 0

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

8. Simplified 50.7%

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

Reproduce

herbie shell --seed 2024144 
(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))