Octave 3.8, jcobi/1

Percentage Accurate: 74.1% → 99.6%

Time: 22.8s

Alternatives: 11

Speedup: 1.3×

Specification

\[\alpha > -1 \land \beta > -1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ t_1 := \beta + \left(\alpha + 2\right)\\ t_2 := \frac{\alpha}{t\_1}\\ t_3 := t\_2 \cdot t\_2\\ t_4 := t\_2 \cdot t\_3\\ \mathbf{if}\;\frac{\beta - \alpha}{t\_0} \leq -0.99995:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + \left(\left(-2 - \beta\right) - \beta\right) \cdot \frac{\beta}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t\_1} + \frac{\frac{t\_4 \cdot t\_4 + -1}{t\_3 \cdot t\_3 + \left(t\_3 + 1\right)}}{-1 - \frac{\alpha}{t\_0}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0))
        (t_1 (+ beta (+ alpha 2.0)))
        (t_2 (/ alpha t_1))
        (t_3 (* t_2 t_2))
        (t_4 (* t_2 t_3)))
   (if (<= (/ (- beta alpha) t_0) -0.99995)
     (/
      (/
       (+ (+ 2.0 (* beta 2.0)) (* (- (- -2.0 beta) beta) (/ beta alpha)))
       alpha)
      2.0)
     (/
      (+
       (/ beta t_1)
       (/
        (/ (+ (* t_4 t_4) -1.0) (+ (* t_3 t_3) (+ t_3 1.0)))
        (- -1.0 (/ alpha t_0))))
      2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = beta + (alpha + 2.0);
	double t_2 = alpha / t_1;
	double t_3 = t_2 * t_2;
	double t_4 = t_2 * t_3;
	double tmp;
	if (((beta - alpha) / t_0) <= -0.99995) {
		tmp = (((2.0 + (beta * 2.0)) + (((-2.0 - beta) - beta) * (beta / alpha))) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_1) + ((((t_4 * t_4) + -1.0) / ((t_3 * t_3) + (t_3 + 1.0))) / (-1.0 - (alpha / t_0)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = (beta + alpha) + 2.0d0
    t_1 = beta + (alpha + 2.0d0)
    t_2 = alpha / t_1
    t_3 = t_2 * t_2
    t_4 = t_2 * t_3
    if (((beta - alpha) / t_0) <= (-0.99995d0)) then
        tmp = (((2.0d0 + (beta * 2.0d0)) + ((((-2.0d0) - beta) - beta) * (beta / alpha))) / alpha) / 2.0d0
    else
        tmp = ((beta / t_1) + ((((t_4 * t_4) + (-1.0d0)) / ((t_3 * t_3) + (t_3 + 1.0d0))) / ((-1.0d0) - (alpha / t_0)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = beta + (alpha + 2.0);
	double t_2 = alpha / t_1;
	double t_3 = t_2 * t_2;
	double t_4 = t_2 * t_3;
	double tmp;
	if (((beta - alpha) / t_0) <= -0.99995) {
		tmp = (((2.0 + (beta * 2.0)) + (((-2.0 - beta) - beta) * (beta / alpha))) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_1) + ((((t_4 * t_4) + -1.0) / ((t_3 * t_3) + (t_3 + 1.0))) / (-1.0 - (alpha / t_0)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	t_1 = beta + (alpha + 2.0)
	t_2 = alpha / t_1
	t_3 = t_2 * t_2
	t_4 = t_2 * t_3
	tmp = 0
	if ((beta - alpha) / t_0) <= -0.99995:
		tmp = (((2.0 + (beta * 2.0)) + (((-2.0 - beta) - beta) * (beta / alpha))) / alpha) / 2.0
	else:
		tmp = ((beta / t_1) + ((((t_4 * t_4) + -1.0) / ((t_3 * t_3) + (t_3 + 1.0))) / (-1.0 - (alpha / t_0)))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	t_1 = Float64(beta + Float64(alpha + 2.0))
	t_2 = Float64(alpha / t_1)
	t_3 = Float64(t_2 * t_2)
	t_4 = Float64(t_2 * t_3)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / t_0) <= -0.99995)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) + Float64(Float64(Float64(-2.0 - beta) - beta) * Float64(beta / alpha))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / t_1) + Float64(Float64(Float64(Float64(t_4 * t_4) + -1.0) / Float64(Float64(t_3 * t_3) + Float64(t_3 + 1.0))) / Float64(-1.0 - Float64(alpha / t_0)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	t_1 = beta + (alpha + 2.0);
	t_2 = alpha / t_1;
	t_3 = t_2 * t_2;
	t_4 = t_2 * t_3;
	tmp = 0.0;
	if (((beta - alpha) / t_0) <= -0.99995)
		tmp = (((2.0 + (beta * 2.0)) + (((-2.0 - beta) - beta) * (beta / alpha))) / alpha) / 2.0;
	else
		tmp = ((beta / t_1) + ((((t_4 * t_4) + -1.0) / ((t_3 * t_3) + (t_3 + 1.0))) / (-1.0 - (alpha / t_0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(alpha / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * t$95$3), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision], -0.99995], N[(N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / t$95$1), $MachinePrecision] + N[(N[(N[(N[(t$95$4 * t$95$4), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(t$95$3 * t$95$3), $MachinePrecision] + N[(t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(alpha / t$95$0), $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))) < -0.999950000000000006

   1. Initial program 6.1%

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

   3. Taylor expanded in alpha around inf 96.9%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in beta around inf 100.0%

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

   if -0.999950000000000006 < (/.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. Step-by-step derivation

      1. div-sub 99.8%

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

      2. associate-+l- 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. flip-- 99.8%

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

      2. associate-+r+ 99.8%

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

      3. associate-+r+ 99.8%

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

      4. metadata-eval 99.8%

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

      5. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. flip3-- 99.8%

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + \left(\left(-2 - \beta\right) - \beta\right) \cdot \frac{\beta}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\frac{\left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} \cdot \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} \cdot \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)\right) \cdot \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} \cdot \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} \cdot \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)\right) + -1}{\left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} \cdot \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} \cdot \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right) + \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} \cdot \frac{\alpha}{\beta + \left(\alpha + 2\right)} + 1\right)}}{-1 - \frac{\alpha}{\left(\beta + \alpha\right) + 2}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + \left(\left(-2 - \beta\right) - \beta\right) \cdot \frac{\beta}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\beta + \left(\alpha + 2\right)}, \beta - \alpha, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99995)
   (/
    (/
     (+ (+ 2.0 (* beta 2.0)) (* (- (- -2.0 beta) beta) (/ beta alpha)))
     alpha)
    2.0)
   (/ (fma (/ 1.0 (+ beta (+ alpha 2.0))) (- beta alpha) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995) {
		tmp = (((2.0 + (beta * 2.0)) + (((-2.0 - beta) - beta) * (beta / alpha))) / alpha) / 2.0;
	} else {
		tmp = fma((1.0 / (beta + (alpha + 2.0))), (beta - alpha), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99995)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) + Float64(Float64(Float64(-2.0 - beta) - beta) * Float64(beta / alpha))) / alpha) / 2.0);
	else
		tmp = Float64(fma(Float64(1.0 / Float64(beta + Float64(alpha + 2.0))), Float64(beta - alpha), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99995], N[(N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $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))) < -0.999950000000000006

   1. Initial program 6.1%

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

   3. Taylor expanded in alpha around inf 96.9%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in beta around inf 100.0%

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

   if -0.999950000000000006 < (/.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. Step-by-step derivation

      1. clear-num 99.8%

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

      2. associate-/r/ 99.8%

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

      3. fma-define 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+l+ 99.8%

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + \left(\left(-2 - \beta\right) - \beta\right) \cdot \frac{\beta}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\beta + \left(\alpha + 2\right)}, \beta - \alpha, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ t_1 := \frac{\alpha}{t\_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{t\_0} \leq -0.99995:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + \left(\left(-2 - \beta\right) - \beta\right) \cdot \frac{\beta}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{t\_1 \cdot t\_1 + -1}{-1 - t\_1}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)) (t_1 (/ alpha t_0)))
   (if (<= (/ (- beta alpha) t_0) -0.99995)
     (/
      (/
       (+ (+ 2.0 (* beta 2.0)) (* (- (- -2.0 beta) beta) (/ beta alpha)))
       alpha)
      2.0)
     (/
      (+ (/ beta (+ beta (+ alpha 2.0))) (/ (+ (* t_1 t_1) -1.0) (- -1.0 t_1)))
      2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = alpha / t_0;
	double tmp;
	if (((beta - alpha) / t_0) <= -0.99995) {
		tmp = (((2.0 + (beta * 2.0)) + (((-2.0 - beta) - beta) * (beta / alpha))) / alpha) / 2.0;
	} else {
		tmp = ((beta / (beta + (alpha + 2.0))) + (((t_1 * t_1) + -1.0) / (-1.0 - t_1))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (beta + alpha) + 2.0d0
    t_1 = alpha / t_0
    if (((beta - alpha) / t_0) <= (-0.99995d0)) then
        tmp = (((2.0d0 + (beta * 2.0d0)) + ((((-2.0d0) - beta) - beta) * (beta / alpha))) / alpha) / 2.0d0
    else
        tmp = ((beta / (beta + (alpha + 2.0d0))) + (((t_1 * t_1) + (-1.0d0)) / ((-1.0d0) - t_1))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = alpha / t_0;
	double tmp;
	if (((beta - alpha) / t_0) <= -0.99995) {
		tmp = (((2.0 + (beta * 2.0)) + (((-2.0 - beta) - beta) * (beta / alpha))) / alpha) / 2.0;
	} else {
		tmp = ((beta / (beta + (alpha + 2.0))) + (((t_1 * t_1) + -1.0) / (-1.0 - t_1))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	t_1 = alpha / t_0
	tmp = 0
	if ((beta - alpha) / t_0) <= -0.99995:
		tmp = (((2.0 + (beta * 2.0)) + (((-2.0 - beta) - beta) * (beta / alpha))) / alpha) / 2.0
	else:
		tmp = ((beta / (beta + (alpha + 2.0))) + (((t_1 * t_1) + -1.0) / (-1.0 - t_1))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	t_1 = Float64(alpha / t_0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / t_0) <= -0.99995)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) + Float64(Float64(Float64(-2.0 - beta) - beta) * Float64(beta / alpha))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / Float64(beta + Float64(alpha + 2.0))) + Float64(Float64(Float64(t_1 * t_1) + -1.0) / Float64(-1.0 - t_1))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	t_1 = alpha / t_0;
	tmp = 0.0;
	if (((beta - alpha) / t_0) <= -0.99995)
		tmp = (((2.0 + (beta * 2.0)) + (((-2.0 - beta) - beta) * (beta / alpha))) / alpha) / 2.0;
	else
		tmp = ((beta / (beta + (alpha + 2.0))) + (((t_1 * t_1) + -1.0) / (-1.0 - t_1))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(alpha / t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision], -0.99995], N[(N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] + -1.0), $MachinePrecision] / N[(-1.0 - t$95$1), $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))) < -0.999950000000000006

   1. Initial program 6.1%

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

   3. Taylor expanded in alpha around inf 96.9%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in beta around inf 100.0%

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

   if -0.999950000000000006 < (/.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. Step-by-step derivation

      1. div-sub 99.8%

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

      2. associate-+l- 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. flip-- 99.8%

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

      2. associate-+r+ 99.8%

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

      3. associate-+r+ 99.8%

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

      4. metadata-eval 99.8%

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

      5. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + \left(\left(-2 - \beta\right) - \beta\right) \cdot \frac{\beta}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\frac{\alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\alpha}{\left(\beta + \alpha\right) + 2} + -1}{-1 - \frac{\alpha}{\left(\beta + \alpha\right) + 2}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + \left(\left(-2 - \beta\right) - \beta\right) \cdot \frac{\beta}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t\_0} + \left(1 - \frac{\alpha}{t\_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99995)
     (/
      (/
       (+ (+ 2.0 (* beta 2.0)) (* (- (- -2.0 beta) beta) (/ beta alpha)))
       alpha)
      2.0)
     (/ (+ (/ beta t_0) (- 1.0 (/ alpha t_0))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995) {
		tmp = (((2.0 + (beta * 2.0)) + (((-2.0 - beta) - beta) * (beta / alpha))) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = beta + (alpha + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.99995d0)) then
        tmp = (((2.0d0 + (beta * 2.0d0)) + ((((-2.0d0) - beta) - beta) * (beta / alpha))) / alpha) / 2.0d0
    else
        tmp = ((beta / t_0) + (1.0d0 - (alpha / t_0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995) {
		tmp = (((2.0 + (beta * 2.0)) + (((-2.0 - beta) - beta) * (beta / alpha))) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995:
		tmp = (((2.0 + (beta * 2.0)) + (((-2.0 - beta) - beta) * (beta / alpha))) / alpha) / 2.0
	else:
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99995)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) + Float64(Float64(Float64(-2.0 - beta) - beta) * Float64(beta / alpha))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / t_0) + Float64(1.0 - Float64(alpha / t_0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995)
		tmp = (((2.0 + (beta * 2.0)) + (((-2.0 - beta) - beta) * (beta / alpha))) / alpha) / 2.0;
	else
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99995], N[(N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] + N[(1.0 - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999950000000000006

   1. Initial program 6.1%

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

   3. Taylor expanded in alpha around inf 96.9%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in beta around inf 100.0%

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

   if -0.999950000000000006 < (/.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. Step-by-step derivation

      1. div-sub 99.8%

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

      2. associate-+l- 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + \left(\left(-2 - \beta\right) - \beta\right) \cdot \frac{\beta}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t\_0} + \left(1 - \frac{\alpha}{t\_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99995)
     (/ (+ beta 1.0) alpha)
     (/ (+ (/ beta t_0) (- 1.0 (/ alpha t_0))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = beta + (alpha + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.99995d0)) then
        tmp = (beta + 1.0d0) / alpha
    else
        tmp = ((beta / t_0) + (1.0d0 - (alpha / t_0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995:
		tmp = (beta + 1.0) / alpha
	else:
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99995)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = Float64(Float64(Float64(beta / t_0) + Float64(1.0 - Float64(alpha / t_0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995)
		tmp = (beta + 1.0) / alpha;
	else
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99995], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] + N[(1.0 - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999950000000000006

   1. Initial program 6.1%

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

   3. Taylor expanded in alpha around inf 99.9%

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

      1. associate-*r/ 99.9%

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

      2. distribute-lft-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. associate-*r* 99.9%

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

      5. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   if -0.999950000000000006 < (/.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. Step-by-step derivation

      1. div-sub 99.8%

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

      2. associate-+l- 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.99995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.99995) (/ (+ beta 1.0) alpha) (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.99995) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.99995d0)) then
        tmp = (beta + 1.0d0) / alpha
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.99995) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.99995:
		tmp = (beta + 1.0) / alpha
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.99995)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.99995)
		tmp = (beta + 1.0) / alpha;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.99995], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999950000000000006

   1. Initial program 6.1%

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

   3. Taylor expanded in alpha around inf 99.9%

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

      1. associate-*r/ 99.9%

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

      2. distribute-lft-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. associate-*r* 99.9%

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

      5. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   if -0.999950000000000006 < (/.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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq -1.5 \cdot 10^{-59}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -4 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 4200:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta -1.5e-59)
   0.5
   (if (<= beta -4e-82) (/ 1.0 alpha) (if (<= beta 4200.0) 0.5 1.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= -1.5e-59) {
		tmp = 0.5;
	} else if (beta <= -4e-82) {
		tmp = 1.0 / alpha;
	} else if (beta <= 4200.0) {
		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 <= (-1.5d-59)) then
        tmp = 0.5d0
    else if (beta <= (-4d-82)) then
        tmp = 1.0d0 / alpha
    else if (beta <= 4200.0d0) 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 <= -1.5e-59) {
		tmp = 0.5;
	} else if (beta <= -4e-82) {
		tmp = 1.0 / alpha;
	} else if (beta <= 4200.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= -1.5e-59:
		tmp = 0.5
	elif beta <= -4e-82:
		tmp = 1.0 / alpha
	elif beta <= 4200.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= -1.5e-59)
		tmp = 0.5;
	elseif (beta <= -4e-82)
		tmp = Float64(1.0 / alpha);
	elseif (beta <= 4200.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= -1.5e-59)
		tmp = 0.5;
	elseif (beta <= -4e-82)
		tmp = 1.0 / alpha;
	elseif (beta <= 4200.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, -1.5e-59], 0.5, If[LessEqual[beta, -4e-82], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[beta, 4200.0], 0.5, 1.0]]]

Derivation

1. Split input into 3 regimes

2. if beta < -1.5e-59 or -4e-82 < beta < 4200

   1. Initial program 73.3%

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

   3. Taylor expanded in beta around 0 71.9%

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

      1. +-commutative 71.9%

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

   5. Simplified 71.9%

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

   6. Taylor expanded in alpha around 0 70.7%

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

   if -1.5e-59 < beta < -4e-82

   1. Initial program 34.9%

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

      1. flip-+ 34.9%

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

      2. metadata-eval 34.9%

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

      3. div-sub 34.9%

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

   4. Applied egg-rr 31.7%

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

   5. Taylor expanded in beta around 0 31.7%

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

      1. associate-*r/ 31.7%

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

      2. unpow2 31.7%

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

      3. *-commutative 31.7%

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

      4. unpow2 31.7%

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

      5. +-commutative 31.7%

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

      6. +-commutative 31.7%

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

      7. +-commutative 31.7%

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

      8. +-commutative 31.7%

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

   7. Simplified 31.7%

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

   8. Taylor expanded in alpha around inf 70.0%

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

   9. Taylor expanded in alpha around 0 70.0%

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

   if 4200 < beta

   1. Initial program 88.1%

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

   3. Taylor expanded in beta around inf 84.9%

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

Alternative 8: 92.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.75 \cdot 10^{+28}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.75e+28)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.75e+28) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.75d+28) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.75e+28) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.75e+28:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.75e+28)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.75e+28)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.75e+28], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.75e28

   1. Initial program 99.4%

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

   3. Taylor expanded in alpha around 0 98.6%

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

   if 1.75e28 < alpha

   1. Initial program 22.6%

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

   3. Taylor expanded in alpha around inf 84.0%

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

      1. associate-*r/ 84.0%

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

      2. distribute-lft-in 84.0%

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

      3. metadata-eval 84.0%

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

      4. associate-*r* 84.0%

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

      5. metadata-eval 84.0%

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

   5. Simplified 84.0%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.75 \cdot 10^{+28}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.8% 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 beta around 0 73.1%

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

      1. +-commutative 73.1%

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

   5. Simplified 73.1%

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

   6. Taylor expanded in alpha around 0 72.7%

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

      1. *-commutative 72.7%

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

   8. Simplified 72.7%

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

   if 2 < alpha

   1. Initial program 25.1%

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

   3. Taylor expanded in alpha around inf 81.5%

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

      1. associate-*r/ 81.6%

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

      2. distribute-lft-in 81.6%

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

      3. metadata-eval 81.6%

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

      4. associate-*r* 81.6%

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

      5. metadata-eval 81.6%

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

   5. Simplified 81.6%

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

4. Final simplification 75.5%

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

Alternative 10: 70.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4200:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 4200.0) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4200.0) {
		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 <= 4200.0d0) 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 <= 4200.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4200.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4200.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4200.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4200.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 4200

   1. Initial program 70.4%

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

   3. Taylor expanded in beta around 0 69.2%

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

      1. +-commutative 69.2%

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

   5. Simplified 69.2%

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

   6. Taylor expanded in alpha around 0 67.9%

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

   if 4200 < beta

   1. Initial program 88.1%

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

   3. Taylor expanded in beta around inf 84.9%

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

Alternative 11: 49.0% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.5

Julia

function code(alpha, beta)
	return 0.5
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.5

Derivation

1. Initial program 76.0%

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

3. Taylor expanded in beta around 0 51.7%

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

   1. +-commutative 51.7%

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

5. Simplified 51.7%

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

6. Taylor expanded in alpha around 0 51.8%

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

Reproduce

herbie shell --seed 2024107 
(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))