Octave 3.8, jcobi/1

Percentage Accurate: 74.4% → 99.7%

Time: 11.6s

Alternatives: 11

Speedup: 1.1×

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: 74.4% 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, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.95:\\ \;\;\;\;\frac{\frac{\beta + 2}{{\alpha}^{2}} \cdot \left(\left(-2 - \beta\right) - \beta\right) + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt{e}\right)}^{\left(2 \cdot \mathsf{log1p}\left(\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.95)
   (/
    (+
     (* (/ (+ beta 2.0) (pow alpha 2.0)) (- (- -2.0 beta) beta))
     (/ (+ beta (- beta -2.0)) alpha))
    2.0)
   (/
    (pow (sqrt E) (* 2.0 (log1p (/ (- beta alpha) (+ alpha (+ beta 2.0))))))
    2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.95) {
		tmp = ((((beta + 2.0) / pow(alpha, 2.0)) * ((-2.0 - beta) - beta)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = pow(sqrt(((double) M_E)), (2.0 * log1p(((beta - alpha) / (alpha + (beta + 2.0)))))) / 2.0;
	}
	return tmp;
}

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.95) {
		tmp = ((((beta + 2.0) / Math.pow(alpha, 2.0)) * ((-2.0 - beta) - beta)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = Math.pow(Math.sqrt(Math.E), (2.0 * Math.log1p(((beta - alpha) / (alpha + (beta + 2.0)))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.95:
		tmp = ((((beta + 2.0) / math.pow(alpha, 2.0)) * ((-2.0 - beta) - beta)) + ((beta + (beta - -2.0)) / alpha)) / 2.0
	else:
		tmp = math.pow(math.sqrt(math.e), (2.0 * math.log1p(((beta - alpha) / (alpha + (beta + 2.0)))))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.95)
		tmp = Float64(Float64(Float64(Float64(Float64(beta + 2.0) / (alpha ^ 2.0)) * Float64(Float64(-2.0 - beta) - beta)) + Float64(Float64(beta + Float64(beta - -2.0)) / alpha)) / 2.0);
	else
		tmp = Float64((sqrt(exp(1)) ^ Float64(2.0 * log1p(Float64(Float64(beta - alpha) / Float64(alpha + Float64(beta + 2.0)))))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.95], N[(N[(N[(N[(N[(beta + 2.0), $MachinePrecision] / N[Power[alpha, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision] + N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Power[N[Sqrt[E], $MachinePrecision], N[(2.0 * N[Log[1 + N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.94999999999999996

   1. Initial program 10.1%

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

   3. Taylor expanded in alpha around -inf 96.6%

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

   5. Simplified 99.6%

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

   if -0.94999999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 100.0%

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

      1. flip3-+ 100.0%

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

      2. add-exp-log 100.0%

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

      3. flip3-+ 100.0%

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

      4. +-commutative 100.0%

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

      5. log1p-udef 100.0%

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

      6. +-commutative 100.0%

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

      7. associate-+l+ 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

   6. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 99.3%

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

      3. associate-+l+ 99.3%

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

      4. +-commutative 99.3%

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

   8. Applied egg-rr 99.3%

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

      1. exp-1-e 99.3%

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

      2. +-commutative 99.3%

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

   10. Simplified 99.3%

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

       1. add-sqr-sqrt 100.0%

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

       2. unpow-prod-down 99.3%

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

       3. +-commutative 99.3%

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

       4. +-commutative 99.3%

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

   12. Applied egg-rr 99.3%

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

       1. pow-sqr 100.0%

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

       2. +-commutative 100.0%

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

   14. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{{\left(\sqrt{e}\right)}^{\left(2 \cdot \mathsf{log1p}\left(\frac{\beta - \alpha}{\alpha + \left(2 + \beta\right)}\right)\right)}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.95:\\ \;\;\;\;\frac{\frac{\beta + 2}{{\alpha}^{2}} \cdot \left(\left(-2 - \beta\right) - \beta\right) + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt{e}\right)}^{\left(2 \cdot \mathsf{log1p}\left(\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.95:\\ \;\;\;\;\frac{\frac{\beta + 2}{{\alpha}^{2}} \cdot \left(\left(-2 - \beta\right) - \beta\right) + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.95)
   (/
    (+
     (* (/ (+ beta 2.0) (pow alpha 2.0)) (- (- -2.0 beta) beta))
     (/ (+ beta (- beta -2.0)) alpha))
    2.0)
   (/ (exp (log1p (/ (- beta alpha) (+ beta (+ alpha 2.0))))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.95) {
		tmp = ((((beta + 2.0) / pow(alpha, 2.0)) * ((-2.0 - beta) - beta)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = exp(log1p(((beta - alpha) / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.95) {
		tmp = ((((beta + 2.0) / Math.pow(alpha, 2.0)) * ((-2.0 - beta) - beta)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = Math.exp(Math.log1p(((beta - alpha) / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.95:
		tmp = ((((beta + 2.0) / math.pow(alpha, 2.0)) * ((-2.0 - beta) - beta)) + ((beta + (beta - -2.0)) / alpha)) / 2.0
	else:
		tmp = math.exp(math.log1p(((beta - alpha) / (beta + (alpha + 2.0))))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.95)
		tmp = Float64(Float64(Float64(Float64(Float64(beta + 2.0) / (alpha ^ 2.0)) * Float64(Float64(-2.0 - beta) - beta)) + Float64(Float64(beta + Float64(beta - -2.0)) / alpha)) / 2.0);
	else
		tmp = Float64(exp(log1p(Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0))))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.95], N[(N[(N[(N[(N[(beta + 2.0), $MachinePrecision] / N[Power[alpha, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision] + N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Exp[N[Log[1 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.94999999999999996

   1. Initial program 10.1%

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

   3. Taylor expanded in alpha around -inf 96.6%

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

   5. Simplified 99.6%

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

   if -0.94999999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 100.0%

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

      1. flip3-+ 100.0%

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

      2. add-exp-log 100.0%

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

      3. flip3-+ 100.0%

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

      4. +-commutative 100.0%

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

      5. log1p-udef 100.0%

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

      6. +-commutative 100.0%

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

      7. associate-+l+ 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

   6. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.95:\\ \;\;\;\;\frac{\frac{\beta + 2}{{\alpha}^{2}} \cdot \left(\left(-2 - \beta\right) - \beta\right) + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.95:\\ \;\;\;\;\frac{\frac{\beta + 2}{{\alpha}^{2}} \cdot \left(\left(-2 - \beta\right) - \beta\right) + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.95)
     (/
      (+
       (* (/ (+ beta 2.0) (pow alpha 2.0)) (- (- -2.0 beta) beta))
       (/ (+ beta (- beta -2.0)) alpha))
      2.0)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.95) {
		tmp = ((((beta + 2.0) / pow(alpha, 2.0)) * ((-2.0 - beta) - beta)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.95d0)) then
        tmp = ((((beta + 2.0d0) / (alpha ** 2.0d0)) * (((-2.0d0) - beta) - beta)) + ((beta + (beta - (-2.0d0))) / alpha)) / 2.0d0
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.95) {
		tmp = ((((beta + 2.0) / Math.pow(alpha, 2.0)) * ((-2.0 - beta) - beta)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.95:
		tmp = ((((beta + 2.0) / math.pow(alpha, 2.0)) * ((-2.0 - beta) - beta)) + ((beta + (beta - -2.0)) / alpha)) / 2.0
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.95)
		tmp = Float64(Float64(Float64(Float64(Float64(beta + 2.0) / (alpha ^ 2.0)) * Float64(Float64(-2.0 - beta) - beta)) + Float64(Float64(beta + Float64(beta - -2.0)) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.95)
		tmp = ((((beta + 2.0) / (alpha ^ 2.0)) * ((-2.0 - beta) - beta)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.95], N[(N[(N[(N[(N[(beta + 2.0), $MachinePrecision] / N[Power[alpha, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision] + N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.94999999999999996

   1. Initial program 10.1%

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

   3. Taylor expanded in alpha around -inf 96.6%

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

   5. Simplified 99.6%

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

   if -0.94999999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.95:\\ \;\;\;\;\frac{\frac{\beta + 2}{{\alpha}^{2}} \cdot \left(\left(-2 - \beta\right) - \beta\right) + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.95:\\ \;\;\;\;\frac{\left(\left(-2 - \beta\right) - \beta\right) \cdot \frac{-1}{\alpha} + \frac{\frac{-1}{\alpha} \cdot \left(\beta + \left(\beta - -2\right)\right)}{\frac{\alpha}{\beta + 2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.95)
     (/
      (+
       (* (- (- -2.0 beta) beta) (/ -1.0 alpha))
       (/ (* (/ -1.0 alpha) (+ beta (- beta -2.0))) (/ alpha (+ beta 2.0))))
      2.0)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.95) {
		tmp = ((((-2.0 - beta) - beta) * (-1.0 / alpha)) + (((-1.0 / alpha) * (beta + (beta - -2.0))) / (alpha / (beta + 2.0)))) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.95d0)) then
        tmp = (((((-2.0d0) - beta) - beta) * ((-1.0d0) / alpha)) + ((((-1.0d0) / alpha) * (beta + (beta - (-2.0d0)))) / (alpha / (beta + 2.0d0)))) / 2.0d0
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.95) {
		tmp = ((((-2.0 - beta) - beta) * (-1.0 / alpha)) + (((-1.0 / alpha) * (beta + (beta - -2.0))) / (alpha / (beta + 2.0)))) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.95:
		tmp = ((((-2.0 - beta) - beta) * (-1.0 / alpha)) + (((-1.0 / alpha) * (beta + (beta - -2.0))) / (alpha / (beta + 2.0)))) / 2.0
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.95)
		tmp = Float64(Float64(Float64(Float64(Float64(-2.0 - beta) - beta) * Float64(-1.0 / alpha)) + Float64(Float64(Float64(-1.0 / alpha) * Float64(beta + Float64(beta - -2.0))) / Float64(alpha / Float64(beta + 2.0)))) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.95)
		tmp = ((((-2.0 - beta) - beta) * (-1.0 / alpha)) + (((-1.0 / alpha) * (beta + (beta - -2.0))) / (alpha / (beta + 2.0)))) / 2.0;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.95], N[(N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] * N[(-1.0 / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1.0 / alpha), $MachinePrecision] * N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.94999999999999996

   1. Initial program 10.1%

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

      1. flip3-+ 10.0%

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

      2. add-exp-log 10.0%

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

      3. flip3-+ 10.1%

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

      4. +-commutative 10.1%

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

      5. log1p-udef 10.1%

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

      6. +-commutative 10.1%

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

      7. associate-+l+ 10.1%

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

   4. Applied egg-rr 10.1%

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

      1. +-commutative 10.1%

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

      2. +-commutative 10.1%

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

      3. +-commutative 10.1%

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

   6. Simplified 10.1%

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

   7. Taylor expanded in alpha around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. unsub-neg 0.0%

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

      3. exp-sum 0.0%

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

      4. rem-exp-log 0.0%

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

      5. neg-mul-1 0.0%

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

      6. associate--r+ 0.0%

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

      7. sub-neg 0.0%

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

      8. metadata-eval 0.0%

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

      9. +-commutative 0.0%

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

      10. unsub-neg 0.0%

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

      11. rem-exp-log 0.0%

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

      12. associate-/l* 0.0%

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

   9. Simplified 99.6%

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

   if -0.94999999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.95:\\ \;\;\;\;\frac{\left(\left(-2 - \beta\right) - \beta\right) \cdot \frac{-1}{\alpha} + \frac{\frac{-1}{\alpha} \cdot \left(\beta + \left(\beta - -2\right)\right)}{\frac{\alpha}{\beta + 2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.999999998:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.999999998)
     (+ (/ 1.0 alpha) (/ beta alpha))
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.999999998) {
		tmp = (1.0 / alpha) + (beta / alpha);
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.999999998d0)) then
        tmp = (1.0d0 / alpha) + (beta / alpha)
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.999999998) {
		tmp = (1.0 / alpha) + (beta / alpha);
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.999999998:
		tmp = (1.0 / alpha) + (beta / alpha)
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.999999998)
		tmp = Float64(Float64(1.0 / alpha) + Float64(beta / alpha));
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.999999998)
		tmp = (1.0 / alpha) + (beta / alpha);
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.999999998], N[(N[(1.0 / alpha), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999999997999999946

   1. Initial program 6.4%

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

   3. Taylor expanded in alpha around inf 99.5%

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

   4. Taylor expanded in beta around 0 99.5%

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

   if -0.999999997999999946 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999998:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq 4 \cdot 10^{-210}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 2.5 \cdot 10^{-183}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (* beta 0.5)) 2.0)))
   (if (<= beta 4e-210)
     t_0
     (if (<= beta 2.5e-183) (/ 1.0 alpha) (if (<= beta 2.0) t_0 1.0)))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double tmp;
	if (beta <= 4e-210) {
		tmp = t_0;
	} else if (beta <= 2.5e-183) {
		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 = (1.0d0 + (beta * 0.5d0)) / 2.0d0
    if (beta <= 4d-210) then
        tmp = t_0
    else if (beta <= 2.5d-183) 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 = (1.0 + (beta * 0.5)) / 2.0;
	double tmp;
	if (beta <= 4e-210) {
		tmp = t_0;
	} else if (beta <= 2.5e-183) {
		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 = (1.0 + (beta * 0.5)) / 2.0
	tmp = 0
	if beta <= 4e-210:
		tmp = t_0
	elif beta <= 2.5e-183:
		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(1.0 + Float64(beta * 0.5)) / 2.0)
	tmp = 0.0
	if (beta <= 4e-210)
		tmp = t_0;
	elseif (beta <= 2.5e-183)
		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 = (1.0 + (beta * 0.5)) / 2.0;
	tmp = 0.0;
	if (beta <= 4e-210)
		tmp = t_0;
	elseif (beta <= 2.5e-183)
		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[(1.0 + N[(beta * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[beta, 4e-210], t$95$0, If[LessEqual[beta, 2.5e-183], 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 < 4.0000000000000002e-210 or 2.5000000000000001e-183 < beta < 2

   1. Initial program 70.7%

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

   3. Taylor expanded in alpha around 0 67.3%

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

   4. Taylor expanded in beta around 0 67.3%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \beta} + 1}{2} \]
   5. Step-by-step derivation

      1. *-commutative 67.3%

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

   6. Simplified 67.3%

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

   if 4.0000000000000002e-210 < beta < 2.5000000000000001e-183

   1. Initial program 33.8%

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

   3. Taylor expanded in alpha around inf 72.4%

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

   4. Taylor expanded in beta around 0 72.4%

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

   if 2 < beta

   1. Initial program 93.2%

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

   3. Taylor expanded in beta around inf 90.7%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{-210}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 2.5 \cdot 10^{-183}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{2}{\alpha + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.4e-10)
   (/ (/ 2.0 (+ alpha 2.0)) 2.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.4e-10) {
		tmp = (2.0 / (alpha + 2.0)) / 2.0;
	} else {
		tmp = (1.0 + (beta / (beta + 2.0))) / 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 <= 1.4d-10) then
        tmp = (2.0d0 / (alpha + 2.0d0)) / 2.0d0
    else
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.4e-10) {
		tmp = (2.0 / (alpha + 2.0)) / 2.0;
	} else {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.4e-10:
		tmp = (2.0 / (alpha + 2.0)) / 2.0
	else:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.4e-10)
		tmp = Float64(Float64(2.0 / Float64(alpha + 2.0)) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.4e-10)
		tmp = (2.0 / (alpha + 2.0)) / 2.0;
	else
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.4e-10], N[(N[(2.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.40000000000000008e-10

   1. Initial program 67.9%

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

      1. flip3-+ 67.9%

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

      2. add-exp-log 67.9%

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

      3. flip3-+ 67.9%

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

      4. +-commutative 67.9%

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

      5. log1p-udef 67.9%

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

      6. +-commutative 67.9%

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

      7. associate-+l+ 67.9%

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

   4. Applied egg-rr 67.9%

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

      1. +-commutative 67.9%

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

      2. +-commutative 67.9%

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

      3. +-commutative 67.9%

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

   6. Simplified 67.9%

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

   7. Taylor expanded in alpha around inf 33.8%

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

      1. sub-neg 33.8%

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

      2. mul-1-neg 33.8%

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

      3. remove-double-neg 33.8%

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

      4. +-commutative 33.8%

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

      5. log-rec 33.8%

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

      6. unsub-neg 33.8%

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

      7. associate-*r/ 33.8%

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

      8. distribute-lft-in 33.8%

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

      9. metadata-eval 33.8%

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

      10. neg-mul-1 33.8%

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

      11. unsub-neg 33.8%

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

   9. Simplified 33.8%

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

   10. Taylor expanded in beta around 0 33.7%

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

       1. exp-diff 34.0%

          \[\leadsto \frac{\color{blue}{\frac{e^{\log 2}}{e^{\log \alpha + 2 \cdot \frac{1}{\alpha}}}}}{2} \]

       2. rem-exp-log 34.0%

          \[\leadsto \frac{\frac{\color{blue}{2}}{e^{\log \alpha + 2 \cdot \frac{1}{\alpha}}}}{2} \]

       3. exp-sum 34.0%

          \[\leadsto \frac{\frac{2}{\color{blue}{e^{\log \alpha} \cdot e^{2 \cdot \frac{1}{\alpha}}}}}{2} \]

       4. rem-exp-log 37.1%

          \[\leadsto \frac{\frac{2}{\color{blue}{\alpha} \cdot e^{2 \cdot \frac{1}{\alpha}}}}{2} \]

       5. associate-*r/ 37.1%

          \[\leadsto \frac{\frac{2}{\alpha \cdot e^{\color{blue}{\frac{2 \cdot 1}{\alpha}}}}}{2} \]

       6. metadata-eval 37.1%

          \[\leadsto \frac{\frac{2}{\alpha \cdot e^{\frac{\color{blue}{2}}{\alpha}}}}{2} \]

   12. Simplified 37.1%

       \[\leadsto \frac{\color{blue}{\frac{2}{\alpha \cdot e^{\frac{2}{\alpha}}}}}{2} \]

   13. Taylor expanded in alpha around inf 99.6%

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

   if 1.40000000000000008e-10 < beta

   1. Initial program 93.3%

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

   3. Taylor expanded in alpha around 0 92.2%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{2}{\alpha + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 24.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 24.5) 1.0 (+ (/ 1.0 alpha) (/ beta alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 24.5) {
		tmp = 1.0;
	} else {
		tmp = (1.0 / alpha) + (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 <= 24.5d0) then
        tmp = 1.0d0
    else
        tmp = (1.0d0 / alpha) + (beta / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 24.5) {
		tmp = 1.0;
	} else {
		tmp = (1.0 / alpha) + (beta / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 24.5:
		tmp = 1.0
	else:
		tmp = (1.0 / alpha) + (beta / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 24.5)
		tmp = 1.0;
	else
		tmp = Float64(Float64(1.0 / alpha) + Float64(beta / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 24.5)
		tmp = 1.0;
	else
		tmp = (1.0 / alpha) + (beta / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 24.5], 1.0, N[(N[(1.0 / alpha), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 24.5

   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 inf 47.9%

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

   if 24.5 < alpha

   1. Initial program 27.4%

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

   3. Taylor expanded in alpha around inf 79.8%

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

   4. Taylor expanded in beta around 0 79.8%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 24.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.6:\\ \;\;\;\;\frac{\frac{2}{\alpha + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.6) (/ (/ 2.0 (+ alpha 2.0)) 2.0) 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.6) {
		tmp = (2.0 / (alpha + 2.0)) / 2.0;
	} 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) :: tmp
    if (beta <= 6.6d0) then
        tmp = (2.0d0 / (alpha + 2.0d0)) / 2.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.6) {
		tmp = (2.0 / (alpha + 2.0)) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 6.6:
		tmp = (2.0 / (alpha + 2.0)) / 2.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.6)
		tmp = Float64(Float64(2.0 / Float64(alpha + 2.0)) / 2.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.6)
		tmp = (2.0 / (alpha + 2.0)) / 2.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 6.6], N[(N[(2.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5999999999999996

   1. Initial program 68.1%

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

      1. flip3-+ 68.0%

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

      2. add-exp-log 68.0%

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

      3. flip3-+ 68.1%

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

      4. +-commutative 68.1%

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

      5. log1p-udef 68.1%

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

      6. +-commutative 68.1%

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

      7. associate-+l+ 68.1%

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

   4. Applied egg-rr 68.1%

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

      1. +-commutative 68.1%

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

      2. +-commutative 68.1%

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

      3. +-commutative 68.1%

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

   6. Simplified 68.1%

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

   7. Taylor expanded in alpha around inf 33.6%

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

      1. sub-neg 33.6%

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

      2. mul-1-neg 33.6%

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

      3. remove-double-neg 33.6%

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

      4. +-commutative 33.6%

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

      5. log-rec 33.6%

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

      6. unsub-neg 33.6%

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

      7. associate-*r/ 33.6%

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

      8. distribute-lft-in 33.6%

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

      9. metadata-eval 33.6%

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

      10. neg-mul-1 33.6%

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

      11. unsub-neg 33.6%

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

   9. Simplified 33.6%

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

   10. Taylor expanded in beta around 0 33.5%

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

       1. exp-diff 33.8%

          \[\leadsto \frac{\color{blue}{\frac{e^{\log 2}}{e^{\log \alpha + 2 \cdot \frac{1}{\alpha}}}}}{2} \]

       2. rem-exp-log 33.8%

          \[\leadsto \frac{\frac{\color{blue}{2}}{e^{\log \alpha + 2 \cdot \frac{1}{\alpha}}}}{2} \]

       3. exp-sum 33.8%

          \[\leadsto \frac{\frac{2}{\color{blue}{e^{\log \alpha} \cdot e^{2 \cdot \frac{1}{\alpha}}}}}{2} \]

       4. rem-exp-log 36.9%

          \[\leadsto \frac{\frac{2}{\color{blue}{\alpha} \cdot e^{2 \cdot \frac{1}{\alpha}}}}{2} \]

       5. associate-*r/ 36.9%

          \[\leadsto \frac{\frac{2}{\alpha \cdot e^{\color{blue}{\frac{2 \cdot 1}{\alpha}}}}}{2} \]

       6. metadata-eval 36.9%

          \[\leadsto \frac{\frac{2}{\alpha \cdot e^{\frac{\color{blue}{2}}{\alpha}}}}{2} \]

   12. Simplified 36.9%

       \[\leadsto \frac{\color{blue}{\frac{2}{\alpha \cdot e^{\frac{2}{\alpha}}}}}{2} \]

   13. Taylor expanded in alpha around inf 99.2%

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

   if 6.5999999999999996 < beta

   1. Initial program 93.2%

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

   3. Taylor expanded in beta around inf 90.7%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.6:\\ \;\;\;\;\frac{\frac{2}{\alpha + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 24.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 24.5) 1.0 (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 24.5) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 24.5d0) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 24.5) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 24.5:
		tmp = 1.0
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 24.5)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 24.5)
		tmp = 1.0;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 24.5], 1.0, N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 24.5

   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 inf 47.9%

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

   if 24.5 < alpha

   1. Initial program 27.4%

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

   3. Taylor expanded in alpha around inf 79.8%

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

   4. Taylor expanded in beta around 0 72.7%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 24.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 24.6% accurate, 4.3× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 (/ 1.0 alpha))

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 1.0 / alpha

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

3. Taylor expanded in alpha around inf 28.0%

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

4. Taylor expanded in beta around 0 26.0%

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

5. Final simplification 26.0%

   \[\leadsto \frac{1}{\alpha} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(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))