Octave 3.8, jcobi/1

Percentage Accurate: 74.1% → 99.8%

Time: 12.1s

Alternatives: 11

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: 74.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\beta + 2\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{-0.5 \cdot \left(\left(\beta + 2\right) \cdot \frac{t\_0}{\alpha} - t\_0\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t\_1} - \log \left(e^{\frac{\alpha}{t\_1} + -1}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ beta 2.0))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99995)
     (/ (* -0.5 (- (* (+ beta 2.0) (/ t_0 alpha)) t_0)) alpha)
     (/ (- (/ beta t_1) (log (exp (+ (/ alpha t_1) -1.0)))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (beta + 2.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995) {
		tmp = (-0.5 * (((beta + 2.0) * (t_0 / alpha)) - t_0)) / alpha;
	} else {
		tmp = ((beta / t_1) - log(exp(((alpha / t_1) + -1.0)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = beta + (beta + 2.0d0)
    t_1 = alpha + (beta + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.99995d0)) then
        tmp = ((-0.5d0) * (((beta + 2.0d0) * (t_0 / alpha)) - t_0)) / alpha
    else
        tmp = ((beta / t_1) - log(exp(((alpha / t_1) + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = beta + (beta + 2.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995) {
		tmp = (-0.5 * (((beta + 2.0) * (t_0 / alpha)) - t_0)) / alpha;
	} else {
		tmp = ((beta / t_1) - Math.log(Math.exp(((alpha / t_1) + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (beta + 2.0)
	t_1 = alpha + (beta + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995:
		tmp = (-0.5 * (((beta + 2.0) * (t_0 / alpha)) - t_0)) / alpha
	else:
		tmp = ((beta / t_1) - math.log(math.exp(((alpha / t_1) + -1.0)))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(beta + 2.0))
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99995)
		tmp = Float64(Float64(-0.5 * Float64(Float64(Float64(beta + 2.0) * Float64(t_0 / alpha)) - t_0)) / alpha);
	else
		tmp = Float64(Float64(Float64(beta / t_1) - log(exp(Float64(Float64(alpha / t_1) + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (beta + 2.0);
	t_1 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995)
		tmp = (-0.5 * (((beta + 2.0) * (t_0 / alpha)) - t_0)) / alpha;
	else
		tmp = ((beta / t_1) - log(exp(((alpha / t_1) + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99995], N[(N[(-0.5 * N[(N[(N[(beta + 2.0), $MachinePrecision] * N[(t$95$0 / alpha), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta / t$95$1), $MachinePrecision] - N[Log[N[Exp[N[(N[(alpha / t$95$1), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.1%

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

      1. +-commutative 6.1%

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

      2. sub-neg 6.1%

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

      3. +-commutative 6.1%

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

      4. neg-sub0 6.1%

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

      5. associate-+l- 6.1%

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

      6. sub0-neg 6.1%

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

      7. distribute-frac-neg 6.1%

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

      8. +-commutative 6.1%

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

      9. sub-neg 6.1%

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

      10. div-sub 6.1%

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

      11. metadata-eval 6.1%

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

      12. +-commutative 6.1%

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

      13. associate-/l/ 6.1%

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

      14. sub-neg 6.1%

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

      15. remove-double-neg 6.1%

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

      16. distribute-neg-out 6.1%

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

      17. neg-mul-1 6.1%

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

      18. *-commutative 6.1%

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

   3. Simplified 5.7%

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

   5. Taylor expanded in alpha around inf 96.9%

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

      1. distribute-lft-out-- 96.9%

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

      2. associate-/l* 100.0%

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

      3. mul-1-neg 100.0%

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

      4. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

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

   1. Initial program 99.8%

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

      1. div-sub 99.8%

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

      2. associate-+l- 99.8%

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

      3. associate-+l+ 99.8%

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

      4. associate-+l+ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. add-log-exp 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{-0.5 \cdot \left(\left(\beta + 2\right) \cdot \frac{\beta + \left(\beta + 2\right)}{\alpha} - \left(\beta + \left(\beta + 2\right)\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\alpha + \left(\beta + 2\right)} - \log \left(e^{\frac{\alpha}{\alpha + \left(\beta + 2\right)} + -1}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\beta + 2\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{-0.5 \cdot \left(\left(\beta + 2\right) \cdot \frac{t\_0}{\alpha} - t\_0\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t\_1} - \mathsf{fma}\left(\alpha, \frac{1}{t\_1}, -1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ beta 2.0))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99995)
     (/ (* -0.5 (- (* (+ beta 2.0) (/ t_0 alpha)) t_0)) alpha)
     (/ (- (/ beta t_1) (fma alpha (/ 1.0 t_1) -1.0)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (beta + 2.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995) {
		tmp = (-0.5 * (((beta + 2.0) * (t_0 / alpha)) - t_0)) / alpha;
	} else {
		tmp = ((beta / t_1) - fma(alpha, (1.0 / t_1), -1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(beta + 2.0))
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99995)
		tmp = Float64(Float64(-0.5 * Float64(Float64(Float64(beta + 2.0) * Float64(t_0 / alpha)) - t_0)) / alpha);
	else
		tmp = Float64(Float64(Float64(beta / t_1) - fma(alpha, Float64(1.0 / t_1), -1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99995], N[(N[(-0.5 * N[(N[(N[(beta + 2.0), $MachinePrecision] * N[(t$95$0 / alpha), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta / t$95$1), $MachinePrecision] - N[(alpha * N[(1.0 / t$95$1), $MachinePrecision] + -1.0), $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))) < -0.999950000000000006

   1. Initial program 6.1%

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

      1. +-commutative 6.1%

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

      2. sub-neg 6.1%

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

      3. +-commutative 6.1%

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

      4. neg-sub0 6.1%

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

      5. associate-+l- 6.1%

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

      6. sub0-neg 6.1%

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

      7. distribute-frac-neg 6.1%

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

      8. +-commutative 6.1%

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

      9. sub-neg 6.1%

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

      10. div-sub 6.1%

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

      11. metadata-eval 6.1%

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

      12. +-commutative 6.1%

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

      13. associate-/l/ 6.1%

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

      14. sub-neg 6.1%

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

      15. remove-double-neg 6.1%

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

      16. distribute-neg-out 6.1%

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

      17. neg-mul-1 6.1%

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

      18. *-commutative 6.1%

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

   3. Simplified 5.7%

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

   5. Taylor expanded in alpha around inf 96.9%

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

      1. distribute-lft-out-- 96.9%

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

      2. associate-/l* 100.0%

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

      3. mul-1-neg 100.0%

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

      4. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

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

   1. Initial program 99.8%

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

      1. div-sub 99.8%

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

      2. associate-+l- 99.8%

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

      3. associate-+l+ 99.8%

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

      4. associate-+l+ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. div-inv 99.8%

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

      2. fma-neg 99.8%

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

      3. metadata-eval 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{-0.5 \cdot \left(\left(\beta + 2\right) \cdot \frac{\beta + \left(\beta + 2\right)}{\alpha} - \left(\beta + \left(\beta + 2\right)\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\alpha + \left(\beta + 2\right)} - \mathsf{fma}\left(\alpha, \frac{1}{\alpha + \left(\beta + 2\right)}, -1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\beta + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{-0.5 \cdot \left(\left(\beta + 2\right) \cdot \frac{t\_0}{\alpha} - t\_0\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\alpha + \left(\beta + 2\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ beta 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99995)
     (/ (* -0.5 (- (* (+ beta 2.0) (/ t_0 alpha)) t_0)) alpha)
     (/ (fma (- beta alpha) (/ 1.0 (+ alpha (+ beta 2.0))) 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995) {
		tmp = (-0.5 * (((beta + 2.0) * (t_0 / alpha)) - t_0)) / alpha;
	} else {
		tmp = fma((beta - alpha), (1.0 / (alpha + (beta + 2.0))), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(beta + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99995)
		tmp = Float64(Float64(-0.5 * Float64(Float64(Float64(beta + 2.0) * Float64(t_0 / alpha)) - t_0)) / alpha);
	else
		tmp = Float64(fma(Float64(beta - alpha), Float64(1.0 / Float64(alpha + Float64(beta + 2.0))), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99995], N[(N[(-0.5 * N[(N[(N[(beta + 2.0), $MachinePrecision] * N[(t$95$0 / alpha), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.1%

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

      1. +-commutative 6.1%

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

      2. sub-neg 6.1%

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

      3. +-commutative 6.1%

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

      4. neg-sub0 6.1%

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

      5. associate-+l- 6.1%

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

      6. sub0-neg 6.1%

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

      7. distribute-frac-neg 6.1%

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

      8. +-commutative 6.1%

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

      9. sub-neg 6.1%

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

      10. div-sub 6.1%

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

      11. metadata-eval 6.1%

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

      12. +-commutative 6.1%

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

      13. associate-/l/ 6.1%

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

      14. sub-neg 6.1%

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

      15. remove-double-neg 6.1%

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

      16. distribute-neg-out 6.1%

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

      17. neg-mul-1 6.1%

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

      18. *-commutative 6.1%

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

   3. Simplified 5.7%

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

   5. Taylor expanded in alpha around inf 96.9%

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

      1. distribute-lft-out-- 96.9%

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

      2. associate-/l* 100.0%

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

      3. mul-1-neg 100.0%

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

      4. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

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

   1. Initial program 99.8%

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

      1. div-inv 99.8%

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

      2. fma-define 99.8%

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

      3. associate-+l+ 99.8%

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{-0.5 \cdot \left(\left(\beta + 2\right) \cdot \frac{\beta + \left(\beta + 2\right)}{\alpha} - \left(\beta + \left(\beta + 2\right)\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\alpha + \left(\beta + 2\right)}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\beta + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{-0.5 \cdot \left(\left(\beta + 2\right) \cdot \frac{t\_0}{\alpha} - t\_0\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{-0.5}{\beta + \left(\alpha + 2\right)} \cdot \left(\alpha - \beta\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ beta 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99995)
     (/ (* -0.5 (- (* (+ beta 2.0) (/ t_0 alpha)) t_0)) alpha)
     (+ 0.5 (* (/ -0.5 (+ beta (+ alpha 2.0))) (- alpha beta))))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995) {
		tmp = (-0.5 * (((beta + 2.0) * (t_0 / alpha)) - t_0)) / alpha;
	} else {
		tmp = 0.5 + ((-0.5 / (beta + (alpha + 2.0))) * (alpha - beta));
	}
	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 + (beta + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.99995d0)) then
        tmp = ((-0.5d0) * (((beta + 2.0d0) * (t_0 / alpha)) - t_0)) / alpha
    else
        tmp = 0.5d0 + (((-0.5d0) / (beta + (alpha + 2.0d0))) * (alpha - beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = beta + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995) {
		tmp = (-0.5 * (((beta + 2.0) * (t_0 / alpha)) - t_0)) / alpha;
	} else {
		tmp = 0.5 + ((-0.5 / (beta + (alpha + 2.0))) * (alpha - beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (beta + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995:
		tmp = (-0.5 * (((beta + 2.0) * (t_0 / alpha)) - t_0)) / alpha
	else:
		tmp = 0.5 + ((-0.5 / (beta + (alpha + 2.0))) * (alpha - beta))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(beta + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99995)
		tmp = Float64(Float64(-0.5 * Float64(Float64(Float64(beta + 2.0) * Float64(t_0 / alpha)) - t_0)) / alpha);
	else
		tmp = Float64(0.5 + Float64(Float64(-0.5 / Float64(beta + Float64(alpha + 2.0))) * Float64(alpha - beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (beta + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995)
		tmp = (-0.5 * (((beta + 2.0) * (t_0 / alpha)) - t_0)) / alpha;
	else
		tmp = 0.5 + ((-0.5 / (beta + (alpha + 2.0))) * (alpha - beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99995], N[(N[(-0.5 * N[(N[(N[(beta + 2.0), $MachinePrecision] * N[(t$95$0 / alpha), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(0.5 + N[(N[(-0.5 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(alpha - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.1%

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

      1. +-commutative 6.1%

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

      2. sub-neg 6.1%

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

      3. +-commutative 6.1%

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

      4. neg-sub0 6.1%

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

      5. associate-+l- 6.1%

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

      6. sub0-neg 6.1%

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

      7. distribute-frac-neg 6.1%

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

      8. +-commutative 6.1%

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

      9. sub-neg 6.1%

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

      10. div-sub 6.1%

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

      11. metadata-eval 6.1%

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

      12. +-commutative 6.1%

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

      13. associate-/l/ 6.1%

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

      14. sub-neg 6.1%

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

      15. remove-double-neg 6.1%

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

      16. distribute-neg-out 6.1%

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

      17. neg-mul-1 6.1%

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

      18. *-commutative 6.1%

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

   3. Simplified 5.7%

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

   5. Taylor expanded in alpha around inf 96.9%

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

      1. distribute-lft-out-- 96.9%

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

      2. associate-/l* 100.0%

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

      3. mul-1-neg 100.0%

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

      4. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

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

   1. Initial program 99.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. 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. metadata-eval 99.8%

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

      12. +-commutative 99.8%

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

      13. associate-/l/ 99.8%

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

      14. sub-neg 99.8%

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

      15. remove-double-neg 99.8%

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

      16. distribute-neg-out 99.8%

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

      17. neg-mul-1 99.8%

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

      18. *-commutative 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{-0.5 \cdot \left(\left(\beta + 2\right) \cdot \frac{\beta + \left(\beta + 2\right)}{\alpha} - \left(\beta + \left(\beta + 2\right)\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{-0.5}{\beta + \left(\alpha + 2\right)} \cdot \left(\alpha - \beta\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\left(\beta + \left(\beta + 2\right)\right) \cdot 0.5}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{-0.5}{\beta + \left(\alpha + 2\right)} \cdot \left(\alpha - \beta\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99995)
   (/ (* (+ beta (+ beta 2.0)) 0.5) alpha)
   (+ 0.5 (* (/ -0.5 (+ beta (+ alpha 2.0))) (- alpha beta)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995) {
		tmp = ((beta + (beta + 2.0)) * 0.5) / alpha;
	} else {
		tmp = 0.5 + ((-0.5 / (beta + (alpha + 2.0))) * (alpha - beta));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.99995d0)) then
        tmp = ((beta + (beta + 2.0d0)) * 0.5d0) / alpha
    else
        tmp = 0.5d0 + (((-0.5d0) / (beta + (alpha + 2.0d0))) * (alpha - beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995) {
		tmp = ((beta + (beta + 2.0)) * 0.5) / alpha;
	} else {
		tmp = 0.5 + ((-0.5 / (beta + (alpha + 2.0))) * (alpha - beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995:
		tmp = ((beta + (beta + 2.0)) * 0.5) / alpha
	else:
		tmp = 0.5 + ((-0.5 / (beta + (alpha + 2.0))) * (alpha - beta))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99995)
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) * 0.5) / alpha);
	else
		tmp = Float64(0.5 + Float64(Float64(-0.5 / Float64(beta + Float64(alpha + 2.0))) * Float64(alpha - beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995)
		tmp = ((beta + (beta + 2.0)) * 0.5) / alpha;
	else
		tmp = 0.5 + ((-0.5 / (beta + (alpha + 2.0))) * (alpha - beta));
	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], -0.99995], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / alpha), $MachinePrecision], N[(0.5 + N[(N[(-0.5 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(alpha - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.1%

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

      1. +-commutative 6.1%

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

      2. sub-neg 6.1%

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

      3. +-commutative 6.1%

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

      4. neg-sub0 6.1%

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

      5. associate-+l- 6.1%

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

      6. sub0-neg 6.1%

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

      7. distribute-frac-neg 6.1%

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

      8. +-commutative 6.1%

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

      9. sub-neg 6.1%

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

      10. div-sub 6.1%

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

      11. metadata-eval 6.1%

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

      12. +-commutative 6.1%

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

      13. associate-/l/ 6.1%

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

      14. sub-neg 6.1%

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

      15. remove-double-neg 6.1%

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

      16. distribute-neg-out 6.1%

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

      17. neg-mul-1 6.1%

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

      18. *-commutative 6.1%

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

   3. Simplified 5.7%

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

   5. Taylor expanded in alpha around inf 99.9%

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

      1. associate-*r/ 99.9%

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

      2. cancel-sign-sub-inv 99.9%

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

      3. metadata-eval 99.9%

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

      4. *-lft-identity 99.9%

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

   7. Simplified 99.9%

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

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

   1. Initial program 99.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. 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. metadata-eval 99.8%

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

      12. +-commutative 99.8%

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

      13. associate-/l/ 99.8%

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

      14. sub-neg 99.8%

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

      15. remove-double-neg 99.8%

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

      16. distribute-neg-out 99.8%

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

      17. neg-mul-1 99.8%

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

      18. *-commutative 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\left(\beta + \left(\beta + 2\right)\right) \cdot 0.5}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{-0.5}{\beta + \left(\alpha + 2\right)} \cdot \left(\alpha - \beta\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta \cdot 0.5 + 1}{2}\\ \mathbf{if}\;\beta \leq -1.5 \cdot 10^{-59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\beta \leq -4 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (* beta 0.5) 1.0) 2.0)))
   (if (<= beta -1.5e-59)
     t_0
     (if (<= beta -4e-82) (/ 1.0 alpha) (if (<= beta 2.0) t_0 1.0)))))

C

double code(double alpha, double beta) {
	double t_0 = ((beta * 0.5) + 1.0) / 2.0;
	double tmp;
	if (beta <= -1.5e-59) {
		tmp = t_0;
	} else if (beta <= -4e-82) {
		tmp = 1.0 / alpha;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = ((beta * 0.5d0) + 1.0d0) / 2.0d0
    if (beta <= (-1.5d-59)) then
        tmp = t_0
    else if (beta <= (-4d-82)) then
        tmp = 1.0d0 / alpha
    else if (beta <= 2.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = ((beta * 0.5) + 1.0) / 2.0;
	double tmp;
	if (beta <= -1.5e-59) {
		tmp = t_0;
	} else if (beta <= -4e-82) {
		tmp = 1.0 / alpha;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = ((beta * 0.5) + 1.0) / 2.0
	tmp = 0
	if beta <= -1.5e-59:
		tmp = t_0
	elif beta <= -4e-82:
		tmp = 1.0 / alpha
	elif beta <= 2.0:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(Float64(beta * 0.5) + 1.0) / 2.0)
	tmp = 0.0
	if (beta <= -1.5e-59)
		tmp = t_0;
	elseif (beta <= -4e-82)
		tmp = Float64(1.0 / alpha);
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = ((beta * 0.5) + 1.0) / 2.0;
	tmp = 0.0;
	if (beta <= -1.5e-59)
		tmp = t_0;
	elseif (beta <= -4e-82)
		tmp = 1.0 / alpha;
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(N[(beta * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[beta, -1.5e-59], t$95$0, If[LessEqual[beta, -4e-82], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[beta, 2.0], t$95$0, 1.0]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.7%

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

   3. Taylor expanded in alpha around 0 72.4%

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

   4. Taylor expanded in beta around 0 72.0%

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

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

   1. Initial program 34.9%

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

   3. Taylor expanded in beta around 0 34.9%

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

      1. +-commutative 34.9%

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

   5. Simplified 34.9%

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

   6. Taylor expanded in alpha around inf 70.0%

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

   if 2 < beta

   1. Initial program 87.1%

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

      1. +-commutative 87.1%

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

      2. sub-neg 87.1%

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

      3. +-commutative 87.1%

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

      4. neg-sub0 87.1%

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

      5. associate-+l- 87.1%

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

      6. sub0-neg 87.1%

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

      7. distribute-frac-neg 87.1%

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

      8. +-commutative 87.1%

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

      9. sub-neg 87.1%

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

      10. div-sub 87.1%

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

      11. metadata-eval 87.1%

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

      12. +-commutative 87.1%

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

      13. associate-/l/ 87.1%

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

      14. sub-neg 87.1%

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

      15. remove-double-neg 87.1%

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

      16. distribute-neg-out 87.1%

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

      17. neg-mul-1 87.1%

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

      18. *-commutative 87.1%

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

   3. Simplified 87.2%

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

   5. Taylor expanded in beta around inf 83.9%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -1.5 \cdot 10^{-59}:\\ \;\;\;\;\frac{\beta \cdot 0.5 + 1}{2}\\ \mathbf{elif}\;\beta \leq -4 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;\frac{\beta \cdot 0.5 + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq -1.45 \cdot 10^{-59}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -4 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 4200:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta -1.45e-59)
   0.5
   (if (<= beta -4e-82) (/ 1.0 alpha) (if (<= beta 4200.0) 0.5 1.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= -1.45e-59) {
		tmp = 0.5;
	} else if (beta <= -4e-82) {
		tmp = 1.0 / alpha;
	} else if (beta <= 4200.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= (-1.45d-59)) then
        tmp = 0.5d0
    else if (beta <= (-4d-82)) then
        tmp = 1.0d0 / alpha
    else if (beta <= 4200.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= -1.45e-59) {
		tmp = 0.5;
	} else if (beta <= -4e-82) {
		tmp = 1.0 / alpha;
	} else if (beta <= 4200.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= -1.45e-59:
		tmp = 0.5
	elif beta <= -4e-82:
		tmp = 1.0 / alpha
	elif beta <= 4200.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= -1.45e-59)
		tmp = 0.5;
	elseif (beta <= -4e-82)
		tmp = Float64(1.0 / alpha);
	elseif (beta <= 4200.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= -1.45e-59)
		tmp = 0.5;
	elseif (beta <= -4e-82)
		tmp = 1.0 / alpha;
	elseif (beta <= 4200.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, -1.45e-59], 0.5, If[LessEqual[beta, -4e-82], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[beta, 4200.0], 0.5, 1.0]]]

Derivation

1. Split input into 3 regimes

2. if beta < -1.45000000000000008e-59 or -4e-82 < beta < 4200

   1. Initial program 73.3%

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

   3. Taylor expanded in beta around 0 71.9%

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

      1. +-commutative 71.9%

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

   5. Simplified 71.9%

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

   6. Taylor expanded in alpha around 0 70.7%

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

   if -1.45000000000000008e-59 < beta < -4e-82

   1. Initial program 34.9%

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

   3. Taylor expanded in beta around 0 34.9%

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

      1. +-commutative 34.9%

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

   5. Simplified 34.9%

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

   6. Taylor expanded in alpha around inf 70.0%

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

   if 4200 < beta

   1. Initial program 88.1%

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

      1. +-commutative 88.1%

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

      2. sub-neg 88.1%

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

      3. +-commutative 88.1%

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

      4. neg-sub0 88.1%

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

      5. associate-+l- 88.1%

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

      6. sub0-neg 88.1%

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

      7. distribute-frac-neg 88.1%

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

      8. +-commutative 88.1%

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

      9. sub-neg 88.1%

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

      10. div-sub 88.1%

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

      11. metadata-eval 88.1%

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

      12. +-commutative 88.1%

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

      13. associate-/l/ 88.1%

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

      14. sub-neg 88.1%

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

      15. remove-double-neg 88.1%

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

      16. distribute-neg-out 88.1%

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

      17. neg-mul-1 88.1%

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

      18. *-commutative 88.1%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in beta around inf 84.9%

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

Alternative 8: 92.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.75 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta + \left(\beta + 2\right)\right) \cdot 0.5}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.75e+28)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (/ (* (+ beta (+ beta 2.0)) 0.5) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.75e+28) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = ((beta + (beta + 2.0)) * 0.5) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.75d+28) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else
        tmp = ((beta + (beta + 2.0d0)) * 0.5d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.75e+28) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = ((beta + (beta + 2.0)) * 0.5) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.75e+28:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = ((beta + (beta + 2.0)) * 0.5) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.75e+28)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) * 0.5) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.75e+28)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = ((beta + (beta + 2.0)) * 0.5) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.75e+28], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.75e28

   1. Initial program 99.4%

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

   3. Taylor expanded in alpha around 0 98.6%

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

   if 1.75e28 < alpha

   1. Initial program 22.6%

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

      1. +-commutative 22.6%

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

      2. sub-neg 22.6%

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

      3. +-commutative 22.6%

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

      4. neg-sub0 22.6%

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

      5. associate-+l- 22.6%

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

      6. sub0-neg 22.6%

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

      7. distribute-frac-neg 22.6%

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

      8. +-commutative 22.6%

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

      9. sub-neg 22.6%

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

      10. div-sub 22.6%

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

      11. metadata-eval 22.6%

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

      12. +-commutative 22.6%

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

      13. associate-/l/ 22.6%

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

      14. sub-neg 22.6%

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

      15. remove-double-neg 22.6%

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

      16. distribute-neg-out 22.6%

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

      17. neg-mul-1 22.6%

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

      18. *-commutative 22.6%

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

   3. Simplified 22.3%

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

   5. Taylor expanded in alpha around inf 84.0%

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

      1. associate-*r/ 84.0%

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

      2. cancel-sign-sub-inv 84.0%

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

      3. metadata-eval 84.0%

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

      4. *-lft-identity 84.0%

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

   7. Simplified 84.0%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.75 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta + \left(\beta + 2\right)\right) \cdot 0.5}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if alpha < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0 73.1%

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

      1. +-commutative 73.1%

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

   5. Simplified 73.1%

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

   6. Taylor expanded in alpha around 0 72.7%

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

      1. *-commutative 72.7%

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

   8. Simplified 72.7%

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

   if 2 < alpha

   1. Initial program 25.1%

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

      1. +-commutative 25.1%

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

      2. sub-neg 25.1%

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

      3. +-commutative 25.1%

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

      4. neg-sub0 25.1%

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

      5. associate-+l- 25.1%

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

      6. sub0-neg 25.1%

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

      7. distribute-frac-neg 25.1%

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

      8. +-commutative 25.1%

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

      9. sub-neg 25.1%

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

      10. div-sub 25.1%

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

      11. metadata-eval 25.1%

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

      12. +-commutative 25.1%

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

      13. associate-/l/ 25.1%

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

      14. sub-neg 25.1%

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

      15. remove-double-neg 25.1%

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

      16. distribute-neg-out 25.1%

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

      17. neg-mul-1 25.1%

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

      18. *-commutative 25.1%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in alpha around inf 81.5%

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

      1. associate-*r/ 81.6%

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

      2. cancel-sign-sub-inv 81.6%

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

      3. metadata-eval 81.6%

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

      4. *-lft-identity 81.6%

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

   7. Simplified 81.6%

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

4. Final simplification 75.5%

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

Alternative 10: 70.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4200.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 4200

   1. Initial program 70.4%

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

   3. Taylor expanded in beta around 0 69.2%

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

      1. +-commutative 69.2%

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

   5. Simplified 69.2%

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

   6. Taylor expanded in alpha around 0 67.9%

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

   if 4200 < beta

   1. Initial program 88.1%

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

      1. +-commutative 88.1%

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

      2. sub-neg 88.1%

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

      3. +-commutative 88.1%

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

      4. neg-sub0 88.1%

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

      5. associate-+l- 88.1%

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

      6. sub0-neg 88.1%

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

      7. distribute-frac-neg 88.1%

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

      8. +-commutative 88.1%

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

      9. sub-neg 88.1%

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

      10. div-sub 88.1%

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

      11. metadata-eval 88.1%

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

      12. +-commutative 88.1%

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

      13. associate-/l/ 88.1%

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

      14. sub-neg 88.1%

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

      15. remove-double-neg 88.1%

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

      16. distribute-neg-out 88.1%

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

      17. neg-mul-1 88.1%

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

      18. *-commutative 88.1%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in beta around inf 84.9%

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

Alternative 11: 49.0% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.5

Julia

function code(alpha, beta)
	return 0.5
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.5

Derivation

1. Initial program 76.0%

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

3. Taylor expanded in beta around 0 51.7%

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

   1. +-commutative 51.7%

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

5. Simplified 51.7%

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

6. Taylor expanded in alpha around 0 51.8%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))