Octave 3.8, jcobi/1

Percentage Accurate: 75.8% → 99.8%

Time: 11.1s

Alternatives: 13

Speedup: 0.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.995:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-1, \beta, \frac{\beta + 2}{\alpha} \cdot \left(\beta + \left(\beta + 2\right)\right) + \left(-2 - \beta\right)\right)}{\alpha}}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t\_0} + \left(1 - \frac{\alpha}{t\_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.995)
     (/
      (/
       (fma
        -1.0
        beta
        (+ (* (/ (+ beta 2.0) alpha) (+ beta (+ beta 2.0))) (- -2.0 beta)))
       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)) <= -0.995) {
		tmp = (fma(-1.0, beta, ((((beta + 2.0) / alpha) * (beta + (beta + 2.0))) + (-2.0 - beta))) / 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)) <= -0.995)
		tmp = Float64(Float64(fma(-1.0, beta, Float64(Float64(Float64(Float64(beta + 2.0) / alpha) * Float64(beta + Float64(beta + 2.0))) + Float64(-2.0 - beta))) / alpha) / Float64(-2.0));
	else
		tmp = Float64(Float64(Float64(beta / t_0) + Float64(1.0 - Float64(alpha / t_0))) / 2.0);
	end
	return 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], -0.995], N[(N[(N[(-1.0 * beta + N[(N[(N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 - beta), $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))) < -0.994999999999999996

   1. Initial program 6.6%

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

      1. +-commutative 6.6%

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

   3. Simplified 6.6%

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

   5. Taylor expanded in alpha around -inf 93.7%

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

      1. associate--l+ 93.7%

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

      2. *-commutative 93.7%

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

      3. fma-define 93.7%

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

      4. associate-/l* 93.7%

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

      5. fma-define 93.7%

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

   7. Applied egg-rr 93.7%

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

      1. fma-undefine 93.7%

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

      2. *-commutative 93.7%

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

      3. fma-undefine 93.7%

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

      4. sub-neg 93.7%

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

      5. fma-undefine 93.7%

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

      6. unpow2 93.7%

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

      7. associate-*r/ 100.0%

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

      8. distribute-rgt-out 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. unsub-neg 100.0%

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

   9. Simplified 100.0%

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

   if -0.994999999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. div-sub 99.9%

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

      2. associate-+l- 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+l+ 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.995:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-1, \beta, \frac{\beta + 2}{\alpha} \cdot \left(\beta + \left(\beta + 2\right)\right) + \left(-2 - \beta\right)\right)}{\alpha}}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.995:\\ \;\;\;\;\frac{\frac{2 - \left(4 \cdot \frac{1}{\alpha} + \beta \cdot \left(\left(2 \cdot \frac{\beta}{\alpha} + \frac{6}{\alpha}\right) - 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t\_0} + \left(1 - \frac{\alpha}{t\_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.995)
     (/
      (/
       (-
        2.0
        (+
         (* 4.0 (/ 1.0 alpha))
         (* beta (- (+ (* 2.0 (/ beta alpha)) (/ 6.0 alpha)) 2.0))))
       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)) <= -0.995) {
		tmp = ((2.0 - ((4.0 * (1.0 / alpha)) + (beta * (((2.0 * (beta / alpha)) + (6.0 / alpha)) - 2.0)))) / 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)) <= (-0.995d0)) then
        tmp = ((2.0d0 - ((4.0d0 * (1.0d0 / alpha)) + (beta * (((2.0d0 * (beta / alpha)) + (6.0d0 / alpha)) - 2.0d0)))) / 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)) <= -0.995) {
		tmp = ((2.0 - ((4.0 * (1.0 / alpha)) + (beta * (((2.0 * (beta / alpha)) + (6.0 / alpha)) - 2.0)))) / 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)) <= -0.995:
		tmp = ((2.0 - ((4.0 * (1.0 / alpha)) + (beta * (((2.0 * (beta / alpha)) + (6.0 / alpha)) - 2.0)))) / 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)) <= -0.995)
		tmp = Float64(Float64(Float64(2.0 - Float64(Float64(4.0 * Float64(1.0 / alpha)) + Float64(beta * Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(6.0 / alpha)) - 2.0)))) / 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)) <= -0.995)
		tmp = ((2.0 - ((4.0 * (1.0 / alpha)) + (beta * (((2.0 * (beta / alpha)) + (6.0 / alpha)) - 2.0)))) / 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], -0.995], N[(N[(N[(2.0 - N[(N[(4.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision] + N[(beta * N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(6.0 / alpha), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $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))) < -0.994999999999999996

   1. Initial program 6.6%

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

      1. +-commutative 6.6%

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

   3. Simplified 6.6%

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

   5. Taylor expanded in alpha around -inf 93.7%

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

   6. Taylor expanded in beta around 0 99.9%

      \[\leadsto \frac{-1 \cdot \frac{\color{blue}{\left(4 \cdot \frac{1}{\alpha} + \beta \cdot \left(\left(2 \cdot \frac{\beta}{\alpha} + 6 \cdot \frac{1}{\alpha}\right) - 2\right)\right) - 2}}{\alpha}}{2} \]

   7. Taylor expanded in alpha around 0 99.9%

      \[\leadsto \frac{-1 \cdot \frac{\left(4 \cdot \frac{1}{\alpha} + \beta \cdot \left(\left(2 \cdot \frac{\beta}{\alpha} + \color{blue}{\frac{6}{\alpha}}\right) - 2\right)\right) - 2}{\alpha}}{2} \]

   if -0.994999999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. div-sub 99.9%

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

      2. associate-+l- 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+l+ 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.995:\\ \;\;\;\;\frac{\frac{2 - \left(4 \cdot \frac{1}{\alpha} + \beta \cdot \left(\left(2 \cdot \frac{\beta}{\alpha} + \frac{6}{\alpha}\right) - 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.995:\\ \;\;\;\;\frac{-0.5 \cdot \left(\left(\beta + 2\right) \cdot \frac{2 + \left(\beta + \beta\right)}{\alpha} + \left(\left(-2 - \beta\right) - \beta\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t\_0} + \left(1 - \frac{\alpha}{t\_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.995)
     (/
      (*
       -0.5
       (+
        (* (+ beta 2.0) (/ (+ 2.0 (+ beta beta)) alpha))
        (- (- -2.0 beta) beta)))
      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)) <= -0.995) {
		tmp = (-0.5 * (((beta + 2.0) * ((2.0 + (beta + beta)) / alpha)) + ((-2.0 - beta) - beta))) / 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)) <= (-0.995d0)) then
        tmp = ((-0.5d0) * (((beta + 2.0d0) * ((2.0d0 + (beta + beta)) / alpha)) + (((-2.0d0) - beta) - beta))) / 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)) <= -0.995) {
		tmp = (-0.5 * (((beta + 2.0) * ((2.0 + (beta + beta)) / alpha)) + ((-2.0 - beta) - beta))) / 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)) <= -0.995:
		tmp = (-0.5 * (((beta + 2.0) * ((2.0 + (beta + beta)) / alpha)) + ((-2.0 - beta) - beta))) / 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)) <= -0.995)
		tmp = Float64(Float64(-0.5 * Float64(Float64(Float64(beta + 2.0) * Float64(Float64(2.0 + Float64(beta + beta)) / alpha)) + Float64(Float64(-2.0 - beta) - beta))) / 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)) <= -0.995)
		tmp = (-0.5 * (((beta + 2.0) * ((2.0 + (beta + beta)) / alpha)) + ((-2.0 - beta) - beta))) / 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], -0.995], N[(N[(-0.5 * N[(N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(2.0 + N[(beta + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision]), $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))) < -0.994999999999999996

   1. Initial program 6.6%

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

      1. +-commutative 6.6%

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

   3. Simplified 6.6%

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

   5. Taylor expanded in alpha around -inf 93.7%

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

   6. Taylor expanded in alpha around inf 93.7%

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

      1. +-commutative 93.7%

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

      2. distribute-lft-out 93.7%

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

      3. +-commutative 93.7%

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

      4. unpow2 93.7%

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

      5. distribute-rgt-out 93.7%

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

      6. associate--r+ 93.7%

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

      7. mul-1-neg 93.7%

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

      8. neg-sub0 93.7%

         \[\leadsto \frac{-0.5 \cdot \left(\frac{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) + \beta\right)}{\alpha} + \left(\left(\color{blue}{\left(0 - \beta\right)} - 2\right) - \beta\right)\right)}{\alpha} \]

      9. associate--r+ 93.7%

         \[\leadsto \frac{-0.5 \cdot \left(\frac{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) + \beta\right)}{\alpha} + \left(\color{blue}{\left(0 - \left(\beta + 2\right)\right)} - \beta\right)\right)}{\alpha} \]

      10. +-commutative 93.7%

          \[\leadsto \frac{-0.5 \cdot \left(\frac{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) + \beta\right)}{\alpha} + \left(\left(0 - \color{blue}{\left(2 + \beta\right)}\right) - \beta\right)\right)}{\alpha} \]

      11. associate--r+ 93.7%

          \[\leadsto \frac{-0.5 \cdot \left(\frac{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) + \beta\right)}{\alpha} + \left(\color{blue}{\left(\left(0 - 2\right) - \beta\right)} - \beta\right)\right)}{\alpha} \]

      12. metadata-eval 93.7%

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

   8. Simplified 93.7%

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

      1. associate-/l* 99.9%

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

      2. associate-+l+ 99.9%

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

   10. Applied egg-rr 99.9%

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

   if -0.994999999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. div-sub 99.9%

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

      2. associate-+l- 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+l+ 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.995:\\ \;\;\;\;\frac{-0.5 \cdot \left(\left(\beta + 2\right) \cdot \frac{2 + \left(\beta + \beta\right)}{\alpha} + \left(\left(-2 - \beta\right) - \beta\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \frac{\beta}{t\_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.995:\\ \;\;\;\;\frac{t\_1 + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 + \left(1 - \frac{\alpha}{t\_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))) (t_1 (/ beta t_0)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.995)
     (/ (+ t_1 (/ (- beta -2.0) alpha)) 2.0)
     (/ (+ t_1 (- 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 / t_0;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.995) {
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (t_1 + (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) :: tmp
    t_0 = beta + (alpha + 2.0d0)
    t_1 = beta / t_0
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.995d0)) then
        tmp = (t_1 + ((beta - (-2.0d0)) / alpha)) / 2.0d0
    else
        tmp = (t_1 + (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 / t_0;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.995) {
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (t_1 + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	t_1 = beta / t_0
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.995:
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0
	else:
		tmp = (t_1 + (1.0 - (alpha / t_0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	t_1 = Float64(beta / t_0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.995)
		tmp = Float64(Float64(t_1 + Float64(Float64(beta - -2.0) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(t_1 + 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 / t_0;
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.995)
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0;
	else
		tmp = (t_1 + (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]}, Block[{t$95$1 = N[(beta / t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.995], N[(N[(t$95$1 + N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$1 + 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))) < -0.994999999999999996

   1. Initial program 6.6%

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

      1. +-commutative 6.6%

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

   3. Simplified 6.6%

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

      1. div-sub 6.7%

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

      2. associate-+l- 8.6%

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

      3. associate-+l+ 8.6%

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

      4. associate-+l+ 8.6%

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

   6. Applied egg-rr 8.6%

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

   7. Taylor expanded in alpha around inf 99.4%

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

      1. associate-*r/ 99.4%

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

      2. neg-mul-1 99.4%

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

      3. distribute-neg-in 99.4%

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

      4. metadata-eval 99.4%

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

      5. unsub-neg 99.4%

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

   9. Simplified 99.4%

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

   if -0.994999999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. div-sub 99.9%

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

      2. associate-+l- 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+l+ 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.995:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.995:\\ \;\;\;\;\frac{\frac{\beta}{t\_0} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1}{\frac{t\_0}{\alpha - \beta}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.995)
     (/ (+ (/ beta t_0) (/ (- beta -2.0) alpha)) 2.0)
     (/ (+ 1.0 (/ -1.0 (/ t_0 (- alpha beta)))) 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)) <= -0.995) {
		tmp = ((beta / t_0) + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (1.0 + (-1.0 / (t_0 / (alpha - beta)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = beta + (alpha + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.995d0)) then
        tmp = ((beta / t_0) + ((beta - (-2.0d0)) / alpha)) / 2.0d0
    else
        tmp = (1.0d0 + ((-1.0d0) / (t_0 / (alpha - beta)))) / 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)) <= -0.995) {
		tmp = ((beta / t_0) + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (1.0 + (-1.0 / (t_0 / (alpha - beta)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.995:
		tmp = ((beta / t_0) + ((beta - -2.0) / alpha)) / 2.0
	else:
		tmp = (1.0 + (-1.0 / (t_0 / (alpha - beta)))) / 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)) <= -0.995)
		tmp = Float64(Float64(Float64(beta / t_0) + Float64(Float64(beta - -2.0) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(-1.0 / Float64(t_0 / Float64(alpha - beta)))) / 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)) <= -0.995)
		tmp = ((beta / t_0) + ((beta - -2.0) / alpha)) / 2.0;
	else
		tmp = (1.0 + (-1.0 / (t_0 / (alpha - beta)))) / 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], -0.995], N[(N[(N[(beta / t$95$0), $MachinePrecision] + N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(-1.0 / N[(t$95$0 / N[(alpha - beta), $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))) < -0.994999999999999996

   1. Initial program 6.6%

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

      1. +-commutative 6.6%

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

   3. Simplified 6.6%

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

      1. div-sub 6.7%

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

      2. associate-+l- 8.6%

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

      3. associate-+l+ 8.6%

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

      4. associate-+l+ 8.6%

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

   6. Applied egg-rr 8.6%

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

   7. Taylor expanded in alpha around inf 99.4%

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

      1. associate-*r/ 99.4%

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

      2. neg-mul-1 99.4%

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

      3. distribute-neg-in 99.4%

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

      4. metadata-eval 99.4%

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

      5. unsub-neg 99.4%

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

   9. Simplified 99.4%

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

   if -0.994999999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. *-rgt-identity 99.9%

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

      2. log1p-expm1-u 99.9%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot 1\right)\right)} + 1}{2} \]

      3. *-rgt-identity 99.9%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}\right)\right) + 1}{2} \]

      4. associate-+l+ 99.9%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\beta - \alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}}\right)\right) + 1}{2} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)\right)} + 1}{2} \]
   7. Step-by-step derivation

      1. log1p-expm1-u 99.9%

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

      2. clear-num 99.9%

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

      3. +-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.995:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1}{\frac{\beta + \left(\alpha + 2\right)}{\alpha - \beta}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1}{\frac{\beta + \left(\alpha + 2\right)}{\alpha - \beta}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.995)
   (/ (+ beta 1.0) alpha)
   (/ (+ 1.0 (/ -1.0 (/ (+ beta (+ alpha 2.0)) (- alpha beta)))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.995) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (1.0 + (-1.0 / ((beta + (alpha + 2.0)) / (alpha - beta)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.995d0)) then
        tmp = (beta + 1.0d0) / alpha
    else
        tmp = (1.0d0 + ((-1.0d0) / ((beta + (alpha + 2.0d0)) / (alpha - beta)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.995) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (1.0 + (-1.0 / ((beta + (alpha + 2.0)) / (alpha - beta)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.995:
		tmp = (beta + 1.0) / alpha
	else:
		tmp = (1.0 + (-1.0 / ((beta + (alpha + 2.0)) / (alpha - beta)))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.995)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = Float64(Float64(1.0 + Float64(-1.0 / Float64(Float64(beta + Float64(alpha + 2.0)) / Float64(alpha - beta)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.995)
		tmp = (beta + 1.0) / alpha;
	else
		tmp = (1.0 + (-1.0 / ((beta + (alpha + 2.0)) / (alpha - beta)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.995], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(1.0 + N[(-1.0 / N[(N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(alpha - beta), $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))) < -0.994999999999999996

   1. Initial program 6.6%

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

      1. +-commutative 6.6%

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

   3. Simplified 6.6%

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

   5. Taylor expanded in alpha around -inf 93.7%

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

   6. Taylor expanded in alpha around inf 93.7%

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

      1. +-commutative 93.7%

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

      2. distribute-lft-out 93.7%

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

      3. +-commutative 93.7%

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

      4. unpow2 93.7%

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

      5. distribute-rgt-out 93.7%

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

      6. associate--r+ 93.7%

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

      7. mul-1-neg 93.7%

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

      8. neg-sub0 93.7%

         \[\leadsto \frac{-0.5 \cdot \left(\frac{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) + \beta\right)}{\alpha} + \left(\left(\color{blue}{\left(0 - \beta\right)} - 2\right) - \beta\right)\right)}{\alpha} \]

      9. associate--r+ 93.7%

         \[\leadsto \frac{-0.5 \cdot \left(\frac{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) + \beta\right)}{\alpha} + \left(\color{blue}{\left(0 - \left(\beta + 2\right)\right)} - \beta\right)\right)}{\alpha} \]

      10. +-commutative 93.7%

          \[\leadsto \frac{-0.5 \cdot \left(\frac{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) + \beta\right)}{\alpha} + \left(\left(0 - \color{blue}{\left(2 + \beta\right)}\right) - \beta\right)\right)}{\alpha} \]

      11. associate--r+ 93.7%

          \[\leadsto \frac{-0.5 \cdot \left(\frac{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) + \beta\right)}{\alpha} + \left(\color{blue}{\left(\left(0 - 2\right) - \beta\right)} - \beta\right)\right)}{\alpha} \]

      12. metadata-eval 93.7%

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

   8. Simplified 93.7%

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

   9. Taylor expanded in alpha around inf 99.3%

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

       1. associate-*r/ 99.3%

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

       2. distribute-lft-in 99.3%

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

       3. metadata-eval 99.3%

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

       4. associate-*r* 99.3%

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

       5. metadata-eval 99.3%

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

   11. Simplified 99.3%

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

   if -0.994999999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. *-rgt-identity 99.9%

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

      2. log1p-expm1-u 99.9%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot 1\right)\right)} + 1}{2} \]

      3. *-rgt-identity 99.9%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}\right)\right) + 1}{2} \]

      4. associate-+l+ 99.9%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\beta - \alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}}\right)\right) + 1}{2} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)\right)} + 1}{2} \]
   7. Step-by-step derivation

      1. log1p-expm1-u 99.9%

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

      2. clear-num 99.9%

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

      3. +-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1}{\frac{\beta + \left(\alpha + 2\right)}{\alpha - \beta}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t\_0} \leq -0.995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{t\_0}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= (/ (- beta alpha) t_0) -0.995)
     (/ (+ beta 1.0) alpha)
     (/ (- 1.0 (/ (- alpha beta) t_0)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (((beta - alpha) / t_0) <= -0.995) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (1.0 - ((alpha - beta) / 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) / t_0) <= (-0.995d0)) then
        tmp = (beta + 1.0d0) / alpha
    else
        tmp = (1.0d0 - ((alpha - beta) / 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) / t_0) <= -0.995) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (1.0 - ((alpha - beta) / t_0)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	tmp = 0
	if ((beta - alpha) / t_0) <= -0.995:
		tmp = (beta + 1.0) / alpha
	else:
		tmp = (1.0 - ((alpha - beta) / t_0)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / t_0) <= -0.995)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(alpha - beta) / 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) / t_0) <= -0.995)
		tmp = (beta + 1.0) / alpha;
	else
		tmp = (1.0 - ((alpha - beta) / 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]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision], -0.995], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(1.0 - N[(N[(alpha - beta), $MachinePrecision] / t$95$0), $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))) < -0.994999999999999996

   1. Initial program 6.6%

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

      1. +-commutative 6.6%

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

   3. Simplified 6.6%

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

   5. Taylor expanded in alpha around -inf 93.7%

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

   6. Taylor expanded in alpha around inf 93.7%

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

      1. +-commutative 93.7%

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

      2. distribute-lft-out 93.7%

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

      3. +-commutative 93.7%

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

      4. unpow2 93.7%

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

      5. distribute-rgt-out 93.7%

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

      6. associate--r+ 93.7%

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

      7. mul-1-neg 93.7%

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

      8. neg-sub0 93.7%

         \[\leadsto \frac{-0.5 \cdot \left(\frac{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) + \beta\right)}{\alpha} + \left(\left(\color{blue}{\left(0 - \beta\right)} - 2\right) - \beta\right)\right)}{\alpha} \]

      9. associate--r+ 93.7%

         \[\leadsto \frac{-0.5 \cdot \left(\frac{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) + \beta\right)}{\alpha} + \left(\color{blue}{\left(0 - \left(\beta + 2\right)\right)} - \beta\right)\right)}{\alpha} \]

      10. +-commutative 93.7%

          \[\leadsto \frac{-0.5 \cdot \left(\frac{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) + \beta\right)}{\alpha} + \left(\left(0 - \color{blue}{\left(2 + \beta\right)}\right) - \beta\right)\right)}{\alpha} \]

      11. associate--r+ 93.7%

          \[\leadsto \frac{-0.5 \cdot \left(\frac{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) + \beta\right)}{\alpha} + \left(\color{blue}{\left(\left(0 - 2\right) - \beta\right)} - \beta\right)\right)}{\alpha} \]

      12. metadata-eval 93.7%

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

   8. Simplified 93.7%

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

   9. Taylor expanded in alpha around inf 99.3%

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

       1. associate-*r/ 99.3%

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

       2. distribute-lft-in 99.3%

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

       3. metadata-eval 99.3%

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

       4. associate-*r* 99.3%

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

       5. metadata-eval 99.3%

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

   11. Simplified 99.3%

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

   if -0.994999999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.35 \cdot 10^{-147}:\\ \;\;\;\;\frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{elif}\;\beta \leq 2 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.35e-147)
   (/ (+ 1.0 (* alpha -0.5)) 2.0)
   (if (<= beta 2e-98) (/ (/ 2.0 alpha) 2.0) (if (<= beta 2.0) 0.5 1.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.35e-147) {
		tmp = (1.0 + (alpha * -0.5)) / 2.0;
	} else if (beta <= 2e-98) {
		tmp = (2.0 / alpha) / 2.0;
	} else 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 <= 2.35d-147) then
        tmp = (1.0d0 + (alpha * (-0.5d0))) / 2.0d0
    else if (beta <= 2d-98) then
        tmp = (2.0d0 / alpha) / 2.0d0
    else 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 <= 2.35e-147) {
		tmp = (1.0 + (alpha * -0.5)) / 2.0;
	} else if (beta <= 2e-98) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.35e-147:
		tmp = (1.0 + (alpha * -0.5)) / 2.0
	elif beta <= 2e-98:
		tmp = (2.0 / alpha) / 2.0
	elif beta <= 2.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.35e-147)
		tmp = Float64(Float64(1.0 + Float64(alpha * -0.5)) / 2.0);
	elseif (beta <= 2e-98)
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	elseif (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 <= 2.35e-147)
		tmp = (1.0 + (alpha * -0.5)) / 2.0;
	elseif (beta <= 2e-98)
		tmp = (2.0 / alpha) / 2.0;
	elseif (beta <= 2.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.35e-147], N[(N[(1.0 + N[(alpha * -0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 2e-98], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 2.0], 0.5, 1.0]]]

Derivation

1. Split input into 4 regimes

2. if beta < 2.34999999999999994e-147

   1. Initial program 74.0%

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

      1. +-commutative 74.0%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in beta around 0 73.6%

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

      1. +-commutative 73.6%

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

   7. Simplified 73.6%

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

   8. Taylor expanded in alpha around 0 71.5%

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

   if 2.34999999999999994e-147 < beta < 1.99999999999999988e-98

   1. Initial program 26.9%

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

      1. +-commutative 26.9%

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

   3. Simplified 26.9%

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

   5. Taylor expanded in beta around 0 26.9%

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

      1. +-commutative 26.9%

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

   7. Simplified 26.9%

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

   8. Taylor expanded in alpha around inf 79.4%

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

   if 1.99999999999999988e-98 < beta < 2

   1. Initial program 73.6%

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

      1. +-commutative 73.6%

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

   3. Simplified 73.6%

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

   5. Taylor expanded in beta around 0 71.7%

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

      1. +-commutative 71.7%

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

   7. Simplified 71.7%

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

   8. Taylor expanded in alpha around 0 71.2%

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

   if 2 < beta

   1. Initial program 88.7%

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

      1. +-commutative 88.7%

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

   3. Simplified 88.7%

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

      1. *-rgt-identity 88.7%

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

      2. log1p-expm1-u 88.7%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot 1\right)\right)} + 1}{2} \]

      3. *-rgt-identity 88.7%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}\right)\right) + 1}{2} \]

      4. associate-+l+ 88.7%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\beta - \alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}}\right)\right) + 1}{2} \]

   6. Applied egg-rr 88.7%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)\right)} + 1}{2} \]
   7. Step-by-step derivation

      1. log1p-expm1-u 88.7%

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

      2. clear-num 88.7%

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

      3. +-commutative 88.7%

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

   8. Applied egg-rr 88.7%

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

   9. Taylor expanded in beta around inf 86.6%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.35 \cdot 10^{-147}:\\ \;\;\;\;\frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{elif}\;\beta \leq 2 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{-147}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 1.45 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.1e-147)
   0.5
   (if (<= beta 1.45e-96) (/ (/ 2.0 alpha) 2.0) (if (<= beta 2.0) 0.5 1.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.1e-147) {
		tmp = 0.5;
	} else if (beta <= 1.45e-96) {
		tmp = (2.0 / alpha) / 2.0;
	} else 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 <= 4.1d-147) then
        tmp = 0.5d0
    else if (beta <= 1.45d-96) then
        tmp = (2.0d0 / alpha) / 2.0d0
    else 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 <= 4.1e-147) {
		tmp = 0.5;
	} else if (beta <= 1.45e-96) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.1e-147:
		tmp = 0.5
	elif beta <= 1.45e-96:
		tmp = (2.0 / alpha) / 2.0
	elif beta <= 2.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.1e-147)
		tmp = 0.5;
	elseif (beta <= 1.45e-96)
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	elseif (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 <= 4.1e-147)
		tmp = 0.5;
	elseif (beta <= 1.45e-96)
		tmp = (2.0 / alpha) / 2.0;
	elseif (beta <= 2.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.1e-147], 0.5, If[LessEqual[beta, 1.45e-96], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 2.0], 0.5, 1.0]]]

Derivation

1. Split input into 3 regimes

2. if beta < 4.1e-147 or 1.44999999999999997e-96 < beta < 2

   1. Initial program 74.0%

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

      1. +-commutative 74.0%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in beta around 0 73.3%

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

      1. +-commutative 73.3%

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

   7. Simplified 73.3%

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

   8. Taylor expanded in alpha around 0 71.5%

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

   if 4.1e-147 < beta < 1.44999999999999997e-96

   1. Initial program 26.9%

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

      1. +-commutative 26.9%

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

   3. Simplified 26.9%

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

   5. Taylor expanded in beta around 0 26.9%

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

      1. +-commutative 26.9%

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

   7. Simplified 26.9%

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

   8. Taylor expanded in alpha around inf 79.4%

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

   if 2 < beta

   1. Initial program 88.7%

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

      1. +-commutative 88.7%

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

   3. Simplified 88.7%

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

      1. *-rgt-identity 88.7%

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

      2. log1p-expm1-u 88.7%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot 1\right)\right)} + 1}{2} \]

      3. *-rgt-identity 88.7%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}\right)\right) + 1}{2} \]

      4. associate-+l+ 88.7%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\beta - \alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}}\right)\right) + 1}{2} \]

   6. Applied egg-rr 88.7%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)\right)} + 1}{2} \]
   7. Step-by-step derivation

      1. log1p-expm1-u 88.7%

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

      2. clear-num 88.7%

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

      3. +-commutative 88.7%

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

   8. Applied egg-rr 88.7%

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

   9. Taylor expanded in beta around inf 86.6%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.1 \cdot 10^{-147}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 1.45 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.15 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.15e+15)
   (/ (+ 1.0 (/ (- beta alpha) (+ beta 2.0))) 2.0)
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.15e+15) {
		tmp = (1.0 + ((beta - alpha) / (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 <= 1.15d+15) then
        tmp = (1.0d0 + ((beta - alpha) / (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 <= 1.15e+15) {
		tmp = (1.0 + ((beta - alpha) / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.15e+15:
		tmp = (1.0 + ((beta - alpha) / (beta + 2.0))) / 2.0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.15e+15)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / 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 <= 1.15e+15)
		tmp = (1.0 + ((beta - alpha) / (beta + 2.0))) / 2.0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.15e+15], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / 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 < 1.15e15

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in beta around inf 98.2%

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

   if 1.15e15 < alpha

   1. Initial program 22.2%

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

      1. +-commutative 22.2%

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

   3. Simplified 22.2%

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

   5. Taylor expanded in alpha around -inf 77.8%

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

   6. Taylor expanded in alpha around inf 77.7%

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

      1. +-commutative 77.7%

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

      2. distribute-lft-out 77.7%

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

      3. +-commutative 77.7%

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

      4. unpow2 77.7%

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

      5. distribute-rgt-out 77.7%

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

      6. associate--r+ 77.7%

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

      7. mul-1-neg 77.7%

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

      8. neg-sub0 77.7%

         \[\leadsto \frac{-0.5 \cdot \left(\frac{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) + \beta\right)}{\alpha} + \left(\left(\color{blue}{\left(0 - \beta\right)} - 2\right) - \beta\right)\right)}{\alpha} \]

      9. associate--r+ 77.7%

         \[\leadsto \frac{-0.5 \cdot \left(\frac{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) + \beta\right)}{\alpha} + \left(\color{blue}{\left(0 - \left(\beta + 2\right)\right)} - \beta\right)\right)}{\alpha} \]

      10. +-commutative 77.7%

          \[\leadsto \frac{-0.5 \cdot \left(\frac{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) + \beta\right)}{\alpha} + \left(\left(0 - \color{blue}{\left(2 + \beta\right)}\right) - \beta\right)\right)}{\alpha} \]

      11. associate--r+ 77.7%

          \[\leadsto \frac{-0.5 \cdot \left(\frac{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) + \beta\right)}{\alpha} + \left(\color{blue}{\left(\left(0 - 2\right) - \beta\right)} - \beta\right)\right)}{\alpha} \]

      12. metadata-eval 77.7%

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

   8. Simplified 77.7%

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

   9. Taylor expanded in alpha around inf 83.7%

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

       1. associate-*r/ 83.7%

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

       2. distribute-lft-in 83.7%

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

       3. metadata-eval 83.7%

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

       4. associate-*r* 83.7%

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

       5. metadata-eval 83.7%

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

   11. Simplified 83.7%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.15 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.8e+15)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.8e+15) {
		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 <= 1.8d+15) 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 <= 1.8e+15) {
		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 <= 1.8e+15:
		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 <= 1.8e+15)
		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 <= 1.8e+15)
		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, 1.8e+15], 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 < 1.8e15

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in alpha around 0 97.3%

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

   if 1.8e15 < alpha

   1. Initial program 22.2%

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

      1. +-commutative 22.2%

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

   3. Simplified 22.2%

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

   5. Taylor expanded in alpha around -inf 77.8%

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

   6. Taylor expanded in alpha around inf 77.7%

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

      1. +-commutative 77.7%

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

      2. distribute-lft-out 77.7%

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

      3. +-commutative 77.7%

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

      4. unpow2 77.7%

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

      5. distribute-rgt-out 77.7%

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

      6. associate--r+ 77.7%

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

      7. mul-1-neg 77.7%

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

      8. neg-sub0 77.7%

         \[\leadsto \frac{-0.5 \cdot \left(\frac{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) + \beta\right)}{\alpha} + \left(\left(\color{blue}{\left(0 - \beta\right)} - 2\right) - \beta\right)\right)}{\alpha} \]

      9. associate--r+ 77.7%

         \[\leadsto \frac{-0.5 \cdot \left(\frac{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) + \beta\right)}{\alpha} + \left(\color{blue}{\left(0 - \left(\beta + 2\right)\right)} - \beta\right)\right)}{\alpha} \]

      10. +-commutative 77.7%

          \[\leadsto \frac{-0.5 \cdot \left(\frac{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) + \beta\right)}{\alpha} + \left(\left(0 - \color{blue}{\left(2 + \beta\right)}\right) - \beta\right)\right)}{\alpha} \]

      11. associate--r+ 77.7%

          \[\leadsto \frac{-0.5 \cdot \left(\frac{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) + \beta\right)}{\alpha} + \left(\color{blue}{\left(\left(0 - 2\right) - \beta\right)} - \beta\right)\right)}{\alpha} \]

      12. metadata-eval 77.7%

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

   8. Simplified 77.7%

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

   9. Taylor expanded in alpha around inf 83.7%

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

       1. associate-*r/ 83.7%

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

       2. distribute-lft-in 83.7%

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

       3. metadata-eval 83.7%

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

       4. associate-*r* 83.7%

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

       5. metadata-eval 83.7%

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

   11. Simplified 83.7%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.4% 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 70.4%

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

      1. +-commutative 70.4%

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

   3. Simplified 70.4%

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

   5. Taylor expanded in beta around 0 69.7%

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

      1. +-commutative 69.7%

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

   7. Simplified 69.7%

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

   8. Taylor expanded in alpha around 0 67.3%

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

   if 2 < beta

   1. Initial program 88.7%

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

      1. +-commutative 88.7%

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

   3. Simplified 88.7%

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

      1. *-rgt-identity 88.7%

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

      2. log1p-expm1-u 88.7%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot 1\right)\right)} + 1}{2} \]

      3. *-rgt-identity 88.7%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}\right)\right) + 1}{2} \]

      4. associate-+l+ 88.7%

         \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\beta - \alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}}\right)\right) + 1}{2} \]

   6. Applied egg-rr 88.7%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)\right)} + 1}{2} \]
   7. Step-by-step derivation

      1. log1p-expm1-u 88.7%

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

      2. clear-num 88.7%

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

      3. +-commutative 88.7%

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

   8. Applied egg-rr 88.7%

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

   9. Taylor expanded in beta around inf 86.6%

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

4. Final simplification 73.9%

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

Alternative 13: 36.9% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 1.0)

C

double code(double alpha, double beta) {
	return 1.0;
}

Fortran

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

Java

public static double code(double alpha, double beta) {
	return 1.0;
}

Python

def code(alpha, beta):
	return 1.0

Julia

function code(alpha, beta)
	return 1.0
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 1.0;
end

Wolfram

code[alpha_, beta_] := 1.0

Derivation

1. Initial program 76.6%

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

   1. +-commutative 76.6%

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

3. Simplified 76.6%

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

   1. *-rgt-identity 76.6%

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

   2. log1p-expm1-u 76.6%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot 1\right)\right)} + 1}{2} \]

   3. *-rgt-identity 76.6%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}\right)\right) + 1}{2} \]

   4. associate-+l+ 76.6%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\beta - \alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}}\right)\right) + 1}{2} \]

6. Applied egg-rr 76.6%

   \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)\right)} + 1}{2} \]
7. Step-by-step derivation

   1. log1p-expm1-u 76.6%

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

   2. clear-num 76.6%

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

   3. +-commutative 76.6%

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

8. Applied egg-rr 76.6%

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

9. Taylor expanded in beta around inf 39.0%

   \[\leadsto \color{blue}{1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(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))