Octave 3.8, jcobi/1

Percentage Accurate: 75.0% → 99.7%

Time: 11.6s

Alternatives: 8

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: 75.0% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;-0.5 \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\left(\left(\beta + \alpha\right) + 3\right) + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9999999)
   (* -0.5 (/ (- (- -2.0 beta) beta) alpha))
   (+ 0.5 (* (- alpha beta) (/ -0.5 (+ (+ (+ beta alpha) 3.0) -1.0))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999) {
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	} else {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (((beta + alpha) + 3.0) + -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 - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.9999999d0)) then
        tmp = (-0.5d0) * ((((-2.0d0) - beta) - beta) / alpha)
    else
        tmp = 0.5d0 + ((alpha - beta) * ((-0.5d0) / (((beta + alpha) + 3.0d0) + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999) {
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	} else {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (((beta + alpha) + 3.0) + -1.0)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999:
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha)
	else:
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (((beta + alpha) + 3.0) + -1.0)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9999999)
		tmp = Float64(-0.5 * Float64(Float64(Float64(-2.0 - beta) - beta) / alpha));
	else
		tmp = Float64(0.5 + Float64(Float64(alpha - beta) * Float64(-0.5 / Float64(Float64(Float64(beta + alpha) + 3.0) + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999)
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	else
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (((beta + alpha) + 3.0) + -1.0)));
	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], -0.9999999], N[(-0.5 * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(N[(alpha - beta), $MachinePrecision] * N[(-0.5 / N[(N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999999900000000053

   1. Initial program 6.5%

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

      1. +-commutative 6.5%

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

      2. sub-neg 6.5%

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

      3. +-commutative 6.5%

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

      4. neg-sub0 6.5%

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

      5. associate-+l- 6.5%

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

      6. sub0-neg 6.5%

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

      7. distribute-frac-neg 6.5%

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

      8. +-commutative 6.5%

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

      9. sub-neg 6.5%

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

      10. div-sub 6.5%

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

      11. sub-neg 6.5%

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

      12. metadata-eval 6.5%

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

      13. neg-mul-1 6.5%

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

      14. *-commutative 6.5%

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

      15. +-commutative 6.5%

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

      16. associate-/l/ 6.5%

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

      17. associate-*l/ 6.5%

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

   3. Simplified 6.7%

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

   5. Taylor expanded in alpha around inf 99.4%

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

      1. associate--r+ 99.4%

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

      2. sub-neg 99.4%

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

      3. neg-mul-1 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. +-commutative 99.4%

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

      6. distribute-neg-in 99.4%

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

      7. metadata-eval 99.4%

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

      8. unsub-neg 99.4%

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

   7. Simplified 99.4%

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

   if -0.999999900000000053 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. neg-sub0 99.8%

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

      5. associate-+l- 99.8%

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

      6. sub0-neg 99.8%

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

      7. distribute-frac-neg 99.8%

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

      8. +-commutative 99.8%

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

      9. sub-neg 99.8%

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

      10. div-sub 99.8%

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

      11. sub-neg 99.8%

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

      12. metadata-eval 99.8%

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

      13. neg-mul-1 99.8%

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

      14. *-commutative 99.8%

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

      15. +-commutative 99.8%

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

      16. associate-/l/ 99.8%

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

      17. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

      1. associate-+r+ 99.8%

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

      2. expm1-log1p-u 96.7%

         \[\leadsto 0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\beta + \alpha\right) + 2\right)\right)}} \]

      3. expm1-undefine 96.6%

         \[\leadsto 0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\color{blue}{e^{\mathsf{log1p}\left(\left(\beta + \alpha\right) + 2\right)} - 1}} \]

      4. associate-+r+ 96.6%

         \[\leadsto 0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{e^{\mathsf{log1p}\left(\color{blue}{\beta + \left(\alpha + 2\right)}\right)} - 1} \]

      5. +-commutative 96.6%

         \[\leadsto 0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{e^{\mathsf{log1p}\left(\color{blue}{\left(\alpha + 2\right) + \beta}\right)} - 1} \]

      6. associate-+l+ 96.6%

         \[\leadsto 0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{e^{\mathsf{log1p}\left(\color{blue}{\alpha + \left(2 + \beta\right)}\right)} - 1} \]

   6. Applied egg-rr 96.6%

      \[\leadsto 0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\color{blue}{e^{\mathsf{log1p}\left(\alpha + \left(2 + \beta\right)\right)} - 1}} \]
   7. Step-by-step derivation

      1. sub-neg 96.6%

         \[\leadsto 0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\color{blue}{e^{\mathsf{log1p}\left(\alpha + \left(2 + \beta\right)\right)} + \left(-1\right)}} \]

      2. log1p-undefine 96.6%

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

      3. rem-exp-log 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. associate-+r+ 99.8%

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

      8. metadata-eval 99.8%

         \[\leadsto 0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\left(\color{blue}{3} + \left(\alpha + \beta\right)\right) + \left(-1\right)} \]

      9. +-commutative 99.8%

         \[\leadsto 0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\left(3 + \color{blue}{\left(\beta + \alpha\right)}\right) + \left(-1\right)} \]

      10. metadata-eval 99.8%

          \[\leadsto 0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\left(3 + \left(\beta + \alpha\right)\right) + \color{blue}{-1}} \]

   8. Simplified 99.8%

      \[\leadsto 0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\color{blue}{\left(3 + \left(\beta + \alpha\right)\right) + -1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;-0.5 \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\left(\left(\beta + \alpha\right) + 3\right) + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;-0.5 \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9999999)
   (* -0.5 (/ (- (- -2.0 beta) beta) alpha))
   (+ 0.5 (* (- alpha beta) (/ -0.5 (+ beta (+ alpha 2.0)))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999) {
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	} else {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 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)) <= (-0.9999999d0)) then
        tmp = (-0.5d0) * ((((-2.0d0) - beta) - beta) / alpha)
    else
        tmp = 0.5d0 + ((alpha - beta) * ((-0.5d0) / (beta + (alpha + 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)) <= -0.9999999) {
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	} else {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999:
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha)
	else:
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9999999)
		tmp = Float64(-0.5 * Float64(Float64(Float64(-2.0 - beta) - beta) / alpha));
	else
		tmp = Float64(0.5 + Float64(Float64(alpha - beta) * Float64(-0.5 / Float64(beta + Float64(alpha + 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999)
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	else
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 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], -0.9999999], N[(-0.5 * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(N[(alpha - beta), $MachinePrecision] * N[(-0.5 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999999900000000053

   1. Initial program 6.5%

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

      1. +-commutative 6.5%

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

      2. sub-neg 6.5%

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

      3. +-commutative 6.5%

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

      4. neg-sub0 6.5%

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

      5. associate-+l- 6.5%

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

      6. sub0-neg 6.5%

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

      7. distribute-frac-neg 6.5%

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

      8. +-commutative 6.5%

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

      9. sub-neg 6.5%

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

      10. div-sub 6.5%

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

      11. sub-neg 6.5%

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

      12. metadata-eval 6.5%

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

      13. neg-mul-1 6.5%

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

      14. *-commutative 6.5%

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

      15. +-commutative 6.5%

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

      16. associate-/l/ 6.5%

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

      17. associate-*l/ 6.5%

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

   3. Simplified 6.7%

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

   5. Taylor expanded in alpha around inf 99.4%

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

      1. associate--r+ 99.4%

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

      2. sub-neg 99.4%

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

      3. neg-mul-1 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. +-commutative 99.4%

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

      6. distribute-neg-in 99.4%

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

      7. metadata-eval 99.4%

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

      8. unsub-neg 99.4%

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

   7. Simplified 99.4%

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

   if -0.999999900000000053 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. neg-sub0 99.8%

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

      5. associate-+l- 99.8%

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

      6. sub0-neg 99.8%

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

      7. distribute-frac-neg 99.8%

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

      8. +-commutative 99.8%

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

      9. sub-neg 99.8%

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

      10. div-sub 99.8%

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

      11. sub-neg 99.8%

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

      12. metadata-eval 99.8%

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

      13. neg-mul-1 99.8%

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

      14. *-commutative 99.8%

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

      15. +-commutative 99.8%

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

      16. associate-/l/ 99.8%

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

      17. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;-0.5 \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \alpha \cdot -0.25\\ \mathbf{if}\;\alpha \leq -5.2 \cdot 10^{-204}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\alpha \leq -3 \cdot 10^{-258}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* alpha -0.25))))
   (if (<= alpha -5.2e-204)
     t_0
     (if (<= alpha -3e-258)
       1.0
       (if (<= alpha 2.0) t_0 (* -0.5 (/ (- (- -2.0 beta) beta) alpha)))))))

C

double code(double alpha, double beta) {
	double t_0 = 0.5 + (alpha * -0.25);
	double tmp;
	if (alpha <= -5.2e-204) {
		tmp = t_0;
	} else if (alpha <= -3e-258) {
		tmp = 1.0;
	} else if (alpha <= 2.0) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 + (alpha * (-0.25d0))
    if (alpha <= (-5.2d-204)) then
        tmp = t_0
    else if (alpha <= (-3d-258)) then
        tmp = 1.0d0
    else if (alpha <= 2.0d0) then
        tmp = t_0
    else
        tmp = (-0.5d0) * ((((-2.0d0) - beta) - beta) / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 0.5 + (alpha * -0.25);
	double tmp;
	if (alpha <= -5.2e-204) {
		tmp = t_0;
	} else if (alpha <= -3e-258) {
		tmp = 1.0;
	} else if (alpha <= 2.0) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 0.5 + (alpha * -0.25)
	tmp = 0
	if alpha <= -5.2e-204:
		tmp = t_0
	elif alpha <= -3e-258:
		tmp = 1.0
	elif alpha <= 2.0:
		tmp = t_0
	else:
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(0.5 + Float64(alpha * -0.25))
	tmp = 0.0
	if (alpha <= -5.2e-204)
		tmp = t_0;
	elseif (alpha <= -3e-258)
		tmp = 1.0;
	elseif (alpha <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(-0.5 * Float64(Float64(Float64(-2.0 - beta) - beta) / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 0.5 + (alpha * -0.25);
	tmp = 0.0;
	if (alpha <= -5.2e-204)
		tmp = t_0;
	elseif (alpha <= -3e-258)
		tmp = 1.0;
	elseif (alpha <= 2.0)
		tmp = t_0;
	else
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, -5.2e-204], t$95$0, If[LessEqual[alpha, -3e-258], 1.0, If[LessEqual[alpha, 2.0], t$95$0, N[(-0.5 * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -5.19999999999999965e-204 or -3.00000000000000021e-258 < alpha < 2

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-mul-1 100.0%

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

      14. *-commutative 100.0%

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

      15. +-commutative 100.0%

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

      16. associate-/l/ 100.0%

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

      17. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 98.9%

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

      1. +-commutative 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in beta around 0 72.3%

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

      1. *-commutative 72.3%

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

   10. Simplified 72.3%

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

   if -5.19999999999999965e-204 < alpha < -3.00000000000000021e-258

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-mul-1 100.0%

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

      14. *-commutative 100.0%

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

      15. +-commutative 100.0%

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

      16. associate-/l/ 100.0%

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

      17. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 74.6%

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

   if 2 < alpha

   1. Initial program 19.6%

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

      1. +-commutative 19.6%

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

      2. sub-neg 19.6%

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

      3. +-commutative 19.6%

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

      4. neg-sub0 19.6%

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

      5. associate-+l- 19.6%

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

      6. sub0-neg 19.6%

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

      7. distribute-frac-neg 19.6%

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

      8. +-commutative 19.6%

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

      9. sub-neg 19.6%

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

      10. div-sub 19.6%

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

      11. sub-neg 19.6%

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

      12. metadata-eval 19.6%

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

      13. neg-mul-1 19.6%

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

      14. *-commutative 19.6%

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

      15. +-commutative 19.6%

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

      16. associate-/l/ 19.6%

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

      17. associate-*l/ 19.6%

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

   3. Simplified 19.8%

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

   5. Taylor expanded in alpha around inf 86.6%

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

      1. associate--r+ 86.6%

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

      2. sub-neg 86.6%

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

      3. neg-mul-1 86.6%

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

      4. distribute-neg-in 86.6%

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

      5. +-commutative 86.6%

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

      6. distribute-neg-in 86.6%

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

      7. metadata-eval 86.6%

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

      8. unsub-neg 86.6%

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

   7. Simplified 86.6%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -5.2 \cdot 10^{-204}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq -3 \cdot 10^{-258}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \alpha \cdot -0.25\\ \mathbf{if}\;\alpha \leq -5 \cdot 10^{-203}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\alpha \leq -5 \cdot 10^{-258}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 0.88:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* alpha -0.25))))
   (if (<= alpha -5e-203)
     t_0
     (if (<= alpha -5e-258)
       1.0
       (if (<= alpha 0.88) t_0 (/ (/ 2.0 alpha) 2.0))))))

C

double code(double alpha, double beta) {
	double t_0 = 0.5 + (alpha * -0.25);
	double tmp;
	if (alpha <= -5e-203) {
		tmp = t_0;
	} else if (alpha <= -5e-258) {
		tmp = 1.0;
	} else if (alpha <= 0.88) {
		tmp = t_0;
	} else {
		tmp = (2.0 / alpha) / 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 = 0.5d0 + (alpha * (-0.25d0))
    if (alpha <= (-5d-203)) then
        tmp = t_0
    else if (alpha <= (-5d-258)) then
        tmp = 1.0d0
    else if (alpha <= 0.88d0) then
        tmp = t_0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 0.5 + (alpha * -0.25);
	double tmp;
	if (alpha <= -5e-203) {
		tmp = t_0;
	} else if (alpha <= -5e-258) {
		tmp = 1.0;
	} else if (alpha <= 0.88) {
		tmp = t_0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 0.5 + (alpha * -0.25)
	tmp = 0
	if alpha <= -5e-203:
		tmp = t_0
	elif alpha <= -5e-258:
		tmp = 1.0
	elif alpha <= 0.88:
		tmp = t_0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(0.5 + Float64(alpha * -0.25))
	tmp = 0.0
	if (alpha <= -5e-203)
		tmp = t_0;
	elseif (alpha <= -5e-258)
		tmp = 1.0;
	elseif (alpha <= 0.88)
		tmp = t_0;
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 0.5 + (alpha * -0.25);
	tmp = 0.0;
	if (alpha <= -5e-203)
		tmp = t_0;
	elseif (alpha <= -5e-258)
		tmp = 1.0;
	elseif (alpha <= 0.88)
		tmp = t_0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, -5e-203], t$95$0, If[LessEqual[alpha, -5e-258], 1.0, If[LessEqual[alpha, 0.88], t$95$0, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -5.0000000000000002e-203 or -4.9999999999999999e-258 < alpha < 0.880000000000000004

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-mul-1 100.0%

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

      14. *-commutative 100.0%

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

      15. +-commutative 100.0%

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

      16. associate-/l/ 100.0%

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

      17. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 98.9%

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

      1. +-commutative 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in beta around 0 72.3%

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

      1. *-commutative 72.3%

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

   10. Simplified 72.3%

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

   if -5.0000000000000002e-203 < alpha < -4.9999999999999999e-258

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-mul-1 100.0%

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

      14. *-commutative 100.0%

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

      15. +-commutative 100.0%

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

      16. associate-/l/ 100.0%

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

      17. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 74.6%

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

   if 0.880000000000000004 < alpha

   1. Initial program 19.6%

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

      1. +-commutative 19.6%

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

   3. Simplified 19.6%

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

      1. expm1-log1p-u 19.4%

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

      2. expm1-undefine 19.4%

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

      3. associate-+r+ 19.4%

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

      4. +-commutative 19.4%

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

      5. associate-+l+ 19.4%

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

   6. Applied egg-rr 19.4%

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

      1. sub-neg 19.4%

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

      2. log1p-undefine 19.5%

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

      3. rem-exp-log 19.7%

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

      4. +-commutative 19.7%

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

      5. associate-+r+ 19.7%

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

      6. metadata-eval 19.7%

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

      7. +-commutative 19.7%

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

      8. +-commutative 19.7%

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

      9. associate-+r+ 19.7%

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

      10. metadata-eval 19.7%

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

   8. Simplified 19.7%

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

   9. Taylor expanded in beta around 0 6.2%

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

       1. +-commutative 6.2%

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

   11. Simplified 6.2%

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

   12. Taylor expanded in alpha around inf 65.0%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -5 \cdot 10^{-203}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq -5 \cdot 10^{-258}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 0.88:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1200000000:\\ \;\;\;\;0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1200000000.0)
   (+ 0.5 (* (- alpha beta) (/ -0.5 (+ beta 2.0))))
   (* -0.5 (/ (- (- -2.0 beta) beta) alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1200000000.0) {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + 2.0)));
	} else {
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1200000000.0d0) then
        tmp = 0.5d0 + ((alpha - beta) * ((-0.5d0) / (beta + 2.0d0)))
    else
        tmp = (-0.5d0) * ((((-2.0d0) - beta) - beta) / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1200000000.0) {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + 2.0)));
	} else {
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1200000000.0:
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + 2.0)))
	else:
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1200000000.0)
		tmp = Float64(0.5 + Float64(Float64(alpha - beta) * Float64(-0.5 / Float64(beta + 2.0))));
	else
		tmp = Float64(-0.5 * Float64(Float64(Float64(-2.0 - beta) - beta) / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1200000000.0)
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + 2.0)));
	else
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1200000000.0], N[(0.5 + N[(N[(alpha - beta), $MachinePrecision] * N[(-0.5 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.2e9

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-mul-1 100.0%

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

      14. *-commutative 100.0%

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

      15. +-commutative 100.0%

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

      16. associate-/l/ 100.0%

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

      17. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 99.0%

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

      1. +-commutative 99.0%

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

   7. Simplified 99.0%

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

   if 1.2e9 < alpha

   1. Initial program 18.9%

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

      1. +-commutative 18.9%

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

      2. sub-neg 18.9%

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

      3. +-commutative 18.9%

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

      4. neg-sub0 18.9%

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

      5. associate-+l- 18.9%

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

      6. sub0-neg 18.9%

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

      7. distribute-frac-neg 18.9%

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

      8. +-commutative 18.9%

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

      9. sub-neg 18.9%

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

      10. div-sub 18.9%

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

      11. sub-neg 18.9%

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

      12. metadata-eval 18.9%

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

      13. neg-mul-1 18.9%

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

      14. *-commutative 18.9%

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

      15. +-commutative 18.9%

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

      16. associate-/l/ 18.9%

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

      17. associate-*l/ 18.9%

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

   3. Simplified 19.0%

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

   5. Taylor expanded in alpha around inf 87.4%

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

      1. associate--r+ 87.4%

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

      2. sub-neg 87.4%

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

      3. neg-mul-1 87.4%

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

      4. distribute-neg-in 87.4%

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

      5. +-commutative 87.4%

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

      6. distribute-neg-in 87.4%

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

      7. metadata-eval 87.4%

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

      8. unsub-neg 87.4%

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

   7. Simplified 87.4%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1200000000:\\ \;\;\;\;0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 900000000:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 900000000.0)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (* -0.5 (/ (- (- -2.0 beta) beta) alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 900000000.0) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 900000000.0d0) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else
        tmp = (-0.5d0) * ((((-2.0d0) - beta) - beta) / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 900000000.0) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 900000000.0:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 900000000.0)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	else
		tmp = Float64(-0.5 * Float64(Float64(Float64(-2.0 - beta) - beta) / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 900000000.0)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 900000000.0], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(-0.5 * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 9e8

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 98.4%

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

      1. +-commutative 98.4%

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

      2. +-commutative 98.4%

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

   7. Simplified 98.4%

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

   if 9e8 < alpha

   1. Initial program 18.9%

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

      1. +-commutative 18.9%

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

      2. sub-neg 18.9%

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

      3. +-commutative 18.9%

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

      4. neg-sub0 18.9%

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

      5. associate-+l- 18.9%

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

      6. sub0-neg 18.9%

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

      7. distribute-frac-neg 18.9%

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

      8. +-commutative 18.9%

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

      9. sub-neg 18.9%

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

      10. div-sub 18.9%

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

      11. sub-neg 18.9%

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

      12. metadata-eval 18.9%

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

      13. neg-mul-1 18.9%

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

      14. *-commutative 18.9%

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

      15. +-commutative 18.9%

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

      16. associate-/l/ 18.9%

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

      17. associate-*l/ 18.9%

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

   3. Simplified 19.0%

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

   5. Taylor expanded in alpha around inf 87.4%

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

      1. associate--r+ 87.4%

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

      2. sub-neg 87.4%

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

      3. neg-mul-1 87.4%

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

      4. distribute-neg-in 87.4%

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

      5. +-commutative 87.4%

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

      6. distribute-neg-in 87.4%

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

      7. metadata-eval 87.4%

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

      8. unsub-neg 87.4%

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

   7. Simplified 87.4%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 900000000:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 225:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 225.0) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 225.0) {
		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 <= 225.0d0) 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 <= 225.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 225.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 225.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 225.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 225.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 225

   1. Initial program 62.2%

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

      1. +-commutative 62.2%

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

   3. Simplified 62.2%

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

      1. expm1-log1p-u 62.2%

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

      2. expm1-undefine 62.2%

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

      3. associate-+r+ 62.2%

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

      4. +-commutative 62.2%

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

      5. associate-+l+ 62.2%

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

   6. Applied egg-rr 62.2%

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

      1. sub-neg 62.2%

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

      2. log1p-undefine 62.2%

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

      3. rem-exp-log 62.2%

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

      4. +-commutative 62.2%

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

      5. associate-+r+ 62.2%

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

      6. metadata-eval 62.2%

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

      7. +-commutative 62.2%

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

      8. +-commutative 62.2%

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

      9. associate-+r+ 62.2%

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

      10. metadata-eval 62.2%

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

   8. Simplified 62.2%

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

   9. Taylor expanded in beta around 0 61.0%

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

       1. +-commutative 61.0%

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

   11. Simplified 61.0%

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

   12. Taylor expanded in alpha around 0 58.7%

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

   if 225 < beta

   1. Initial program 77.0%

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

      1. +-commutative 77.0%

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

      2. sub-neg 77.0%

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

      3. +-commutative 77.0%

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

      4. neg-sub0 77.0%

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

      5. associate-+l- 77.0%

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

      6. sub0-neg 77.0%

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

      7. distribute-frac-neg 77.0%

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

      8. +-commutative 77.0%

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

      9. sub-neg 77.0%

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

      10. div-sub 77.0%

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

      11. sub-neg 77.0%

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

      12. metadata-eval 77.0%

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

      13. neg-mul-1 77.0%

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

      14. *-commutative 77.0%

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

      15. +-commutative 77.0%

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

      16. associate-/l/ 77.0%

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

      17. associate-*l/ 77.0%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in beta around inf 74.5%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 225:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 37.7% 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 67.3%

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

   1. +-commutative 67.3%

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

   2. sub-neg 67.3%

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

   3. +-commutative 67.3%

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

   4. neg-sub0 67.3%

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

   5. associate-+l- 67.3%

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

   6. sub0-neg 67.3%

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

   7. distribute-frac-neg 67.3%

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

   8. +-commutative 67.3%

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

   9. sub-neg 67.3%

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

   10. div-sub 67.3%

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

   11. sub-neg 67.3%

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

   12. metadata-eval 67.3%

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

   13. neg-mul-1 67.3%

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

   14. *-commutative 67.3%

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

   15. +-commutative 67.3%

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

   16. associate-/l/ 67.3%

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

   17. associate-*l/ 67.3%

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

3. Simplified 67.4%

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

5. Taylor expanded in beta around inf 34.5%

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

6. Final simplification 34.5%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024079 
(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))