Octave 3.8, jcobi/1

Percentage Accurate: 74.9% → 99.8%

Time: 8.3s

Alternatives: 10

Speedup: 1.2×

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: 74.9% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999:\\ \;\;\;\;\frac{\left(\frac{1}{\alpha} \cdot \frac{\beta + 2}{\alpha}\right) \cdot \left(\left(-2 - \beta\right) - \beta\right) + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta - \alpha\right) \cdot \frac{-1}{-2 - \left(\beta + \alpha\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999)
   (/
    (+
     (* (* (/ 1.0 alpha) (/ (+ beta 2.0) alpha)) (- (- -2.0 beta) beta))
     (/ (+ beta (- beta -2.0)) alpha))
    2.0)
   (/ (+ (* (- beta alpha) (/ -1.0 (- -2.0 (+ beta alpha)))) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999) {
		tmp = ((((1.0 / alpha) * ((beta + 2.0) / alpha)) * ((-2.0 - beta) - beta)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = (((beta - alpha) * (-1.0 / (-2.0 - (beta + alpha)))) + 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) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.999d0)) then
        tmp = ((((1.0d0 / alpha) * ((beta + 2.0d0) / alpha)) * (((-2.0d0) - beta) - beta)) + ((beta + (beta - (-2.0d0))) / alpha)) / 2.0d0
    else
        tmp = (((beta - alpha) * ((-1.0d0) / ((-2.0d0) - (beta + alpha)))) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999) {
		tmp = ((((1.0 / alpha) * ((beta + 2.0) / alpha)) * ((-2.0 - beta) - beta)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = (((beta - alpha) * (-1.0 / (-2.0 - (beta + alpha)))) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999:
		tmp = ((((1.0 / alpha) * ((beta + 2.0) / alpha)) * ((-2.0 - beta) - beta)) + ((beta + (beta - -2.0)) / alpha)) / 2.0
	else:
		tmp = (((beta - alpha) * (-1.0 / (-2.0 - (beta + alpha)))) + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999)
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 / alpha) * Float64(Float64(beta + 2.0) / alpha)) * Float64(Float64(-2.0 - beta) - beta)) + Float64(Float64(beta + Float64(beta - -2.0)) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) * Float64(-1.0 / Float64(-2.0 - Float64(beta + alpha)))) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999)
		tmp = ((((1.0 / alpha) * ((beta + 2.0) / alpha)) * ((-2.0 - beta) - beta)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	else
		tmp = (((beta - alpha) * (-1.0 / (-2.0 - (beta + alpha)))) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999], N[(N[(N[(N[(N[(1.0 / alpha), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $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[(N[(N[(beta - alpha), $MachinePrecision] * N[(-1.0 / N[(-2.0 - N[(beta + alpha), $MachinePrecision]), $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) 2)) < -0.998999999999999999

   1. Initial program 9.1%

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

   2. Taylor expanded in alpha around -inf 94.4%

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

      1. mul-1-neg 94.4%

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

      2. unsub-neg 94.4%

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

   4. Simplified 99.1%

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

      1. *-un-lft-identity 99.1%

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

      2. unpow2 99.1%

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

      3. times-frac 99.8%

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

   6. Applied egg-rr 99.8%

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

   if -0.998999999999999999 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

      1. frac-2neg 99.9%

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

      2. div-inv 99.9%

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

      3. *-commutative 99.9%

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

      4. neg-sub0 99.9%

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

      5. +-commutative 99.9%

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

      6. associate--r+ 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

      9. add-log-exp 65.5%

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

      10. neg-log 65.5%

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

      11. exp-diff 65.4%

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

      12. associate-/l* 65.4%

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

      13. *-un-lft-identity 65.4%

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

      14. diff-log 65.4%

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

      15. add-log-exp 66.0%

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

      16. add-log-exp 99.9%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999:\\ \;\;\;\;\frac{\left(\frac{1}{\alpha} \cdot \frac{\beta + 2}{\alpha}\right) \cdot \left(\left(-2 - \beta\right) - \beta\right) + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta - \alpha\right) \cdot \frac{-1}{-2 - \left(\beta + \alpha\right)} + 1}{2}\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta - \alpha\right) \cdot \frac{-1}{-2 - \left(\beta + \alpha\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999)
   (/ (+ (* 2.0 (/ beta alpha)) (* 2.0 (/ 1.0 alpha))) 2.0)
   (/ (+ (* (- beta alpha) (/ -1.0 (- -2.0 (+ beta alpha)))) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999) {
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	} else {
		tmp = (((beta - alpha) * (-1.0 / (-2.0 - (beta + alpha)))) + 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) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.999d0)) then
        tmp = ((2.0d0 * (beta / alpha)) + (2.0d0 * (1.0d0 / alpha))) / 2.0d0
    else
        tmp = (((beta - alpha) * ((-1.0d0) / ((-2.0d0) - (beta + alpha)))) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999) {
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	} else {
		tmp = (((beta - alpha) * (-1.0 / (-2.0 - (beta + alpha)))) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999:
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0
	else:
		tmp = (((beta - alpha) * (-1.0 / (-2.0 - (beta + alpha)))) + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999)
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(2.0 * Float64(1.0 / alpha))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) * Float64(-1.0 / Float64(-2.0 - Float64(beta + alpha)))) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999)
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	else
		tmp = (((beta - alpha) * (-1.0 / (-2.0 - (beta + alpha)))) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999], N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(-1.0 / N[(-2.0 - N[(beta + alpha), $MachinePrecision]), $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) 2)) < -0.998999999999999999

   1. Initial program 9.1%

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

   2. Taylor expanded in alpha around inf 98.1%

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

   3. Taylor expanded in beta around 0 98.1%

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

   if -0.998999999999999999 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

      1. frac-2neg 99.9%

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

      2. div-inv 99.9%

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

      3. *-commutative 99.9%

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

      4. neg-sub0 99.9%

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

      5. +-commutative 99.9%

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

      6. associate--r+ 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

      9. add-log-exp 65.5%

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

      10. neg-log 65.5%

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

      11. exp-diff 65.4%

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

      12. associate-/l* 65.4%

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

      13. *-un-lft-identity 65.4%

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

      14. diff-log 65.4%

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

      15. add-log-exp 66.0%

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

      16. add-log-exp 99.9%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta - \alpha\right) \cdot \frac{-1}{-2 - \left(\beta + \alpha\right)} + 1}{2}\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.999:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{t_0}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= (/ (- beta alpha) t_0) -0.999)
     (/ (+ (* 2.0 (/ beta alpha)) (* 2.0 (/ 1.0 alpha))) 2.0)
     (/ (- 1.0 (/ (- alpha beta) t_0)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (((beta - alpha) / t_0) <= -0.999) {
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	} else {
		tmp = (1.0 - ((alpha - beta) / t_0)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta + alpha) + 2.0d0
    if (((beta - alpha) / t_0) <= (-0.999d0)) then
        tmp = ((2.0d0 * (beta / alpha)) + (2.0d0 * (1.0d0 / alpha))) / 2.0d0
    else
        tmp = (1.0d0 - ((alpha - beta) / t_0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (((beta - alpha) / t_0) <= -0.999) {
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	} else {
		tmp = (1.0 - ((alpha - beta) / t_0)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	tmp = 0
	if ((beta - alpha) / t_0) <= -0.999:
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0
	else:
		tmp = (1.0 - ((alpha - beta) / t_0)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / t_0) <= -0.999)
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(2.0 * Float64(1.0 / alpha))) / 2.0);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(alpha - beta) / t_0)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	tmp = 0.0;
	if (((beta - alpha) / t_0) <= -0.999)
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	else
		tmp = (1.0 - ((alpha - beta) / t_0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision], -0.999], N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 - N[(N[(alpha - beta), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.998999999999999999

   1. Initial program 9.1%

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

   2. Taylor expanded in alpha around inf 98.1%

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

   3. Taylor expanded in beta around 0 98.1%

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

   if -0.998999999999999999 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}{2}\\ \end{array} \]

Alternative 4: 93.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 1.42 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{1}{\alpha \cdot 0.5 + 1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 5e-20)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (if (<= alpha 1.42e+54)
     (/ (/ 1.0 (+ (* alpha 0.5) 1.0)) 2.0)
     (/ (+ (* 2.0 (/ beta alpha)) (* 2.0 (/ 1.0 alpha))) 2.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 5e-20) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else if (alpha <= 1.42e+54) {
		tmp = (1.0 / ((alpha * 0.5) + 1.0)) / 2.0;
	} else {
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 5d-20) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else if (alpha <= 1.42d+54) then
        tmp = (1.0d0 / ((alpha * 0.5d0) + 1.0d0)) / 2.0d0
    else
        tmp = ((2.0d0 * (beta / alpha)) + (2.0d0 * (1.0d0 / alpha))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 5e-20) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else if (alpha <= 1.42e+54) {
		tmp = (1.0 / ((alpha * 0.5) + 1.0)) / 2.0;
	} else {
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 5e-20:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	elif alpha <= 1.42e+54:
		tmp = (1.0 / ((alpha * 0.5) + 1.0)) / 2.0
	else:
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 5e-20)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	elseif (alpha <= 1.42e+54)
		tmp = Float64(Float64(1.0 / Float64(Float64(alpha * 0.5) + 1.0)) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(2.0 * Float64(1.0 / alpha))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 5e-20)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	elseif (alpha <= 1.42e+54)
		tmp = (1.0 / ((alpha * 0.5) + 1.0)) / 2.0;
	else
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 5e-20], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 1.42e+54], N[(N[(1.0 / N[(N[(alpha * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 4.9999999999999999e-20

   1. Initial program 100.0%

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

   2. Taylor expanded in alpha around 0 98.5%

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

   if 4.9999999999999999e-20 < alpha < 1.41999999999999995e54

   1. Initial program 64.1%

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

   2. Taylor expanded in beta around 0 49.2%

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

      1. +-commutative 49.2%

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

   4. Simplified 49.2%

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

      1. flip-- 49.0%

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

      2. clear-num 48.9%

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

      3. metadata-eval 48.9%

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

      4. pow2 48.9%

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

   6. Applied egg-rr 48.9%

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

   7. Taylor expanded in alpha around 0 84.4%

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

      1. *-commutative 84.4%

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

   9. Simplified 84.4%

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

   if 1.41999999999999995e54 < alpha

   1. Initial program 16.7%

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

   2. Taylor expanded in alpha around inf 90.0%

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

   3. Taylor expanded in beta around 0 90.0%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 1.42 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{1}{\alpha \cdot 0.5 + 1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \]

Alternative 5: 93.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 1.42 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{1}{\alpha \cdot 0.5 + 1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 5e-20)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (if (<= alpha 1.42e+54)
     (/ (/ 1.0 (+ (* alpha 0.5) 1.0)) 2.0)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 5e-20) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else if (alpha <= 1.42e+54) {
		tmp = (1.0 / ((alpha * 0.5) + 1.0)) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 5d-20) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else if (alpha <= 1.42d+54) then
        tmp = (1.0d0 / ((alpha * 0.5d0) + 1.0d0)) / 2.0d0
    else
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 5e-20) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else if (alpha <= 1.42e+54) {
		tmp = (1.0 / ((alpha * 0.5) + 1.0)) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 5e-20:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	elif alpha <= 1.42e+54:
		tmp = (1.0 / ((alpha * 0.5) + 1.0)) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 5e-20)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	elseif (alpha <= 1.42e+54)
		tmp = Float64(Float64(1.0 / Float64(Float64(alpha * 0.5) + 1.0)) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 5e-20)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	elseif (alpha <= 1.42e+54)
		tmp = (1.0 / ((alpha * 0.5) + 1.0)) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 5e-20], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 1.42e+54], N[(N[(1.0 / N[(N[(alpha * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 4.9999999999999999e-20

   1. Initial program 100.0%

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

   2. Taylor expanded in alpha around 0 98.5%

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

   if 4.9999999999999999e-20 < alpha < 1.41999999999999995e54

   1. Initial program 64.1%

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

   2. Taylor expanded in beta around 0 49.2%

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

      1. +-commutative 49.2%

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

   4. Simplified 49.2%

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

      1. flip-- 49.0%

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

      2. clear-num 48.9%

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

      3. metadata-eval 48.9%

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

      4. pow2 48.9%

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

   6. Applied egg-rr 48.9%

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

   7. Taylor expanded in alpha around 0 84.4%

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

      1. *-commutative 84.4%

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

   9. Simplified 84.4%

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

   if 1.41999999999999995e54 < alpha

   1. Initial program 16.7%

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

   2. Taylor expanded in alpha around inf 90.0%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 1.42 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{1}{\alpha \cdot 0.5 + 1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 6: 87.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1200:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1200.0)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1200.0) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1200.0d0) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1200.0) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1200.0:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1200.0)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1200.0)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1200.0], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1200

   1. Initial program 100.0%

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

   2. Taylor expanded in alpha around 0 96.4%

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

   if 1200 < alpha

   1. Initial program 20.9%

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

   2. Taylor expanded in alpha around inf 86.6%

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

   3. Taylor expanded in beta around 0 64.7%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1200:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 93.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{\alpha \cdot 0.5 + 1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5e-11)
   (/ (/ 1.0 (+ (* alpha 0.5) 1.0)) 2.0)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e-11) {
		tmp = (1.0 / ((alpha * 0.5) + 1.0)) / 2.0;
	} else {
		tmp = ((beta / (beta + 2.0)) + 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) :: tmp
    if (beta <= 5d-11) then
        tmp = (1.0d0 / ((alpha * 0.5d0) + 1.0d0)) / 2.0d0
    else
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5e-11) {
		tmp = (1.0 / ((alpha * 0.5) + 1.0)) / 2.0;
	} else {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 5e-11:
		tmp = (1.0 / ((alpha * 0.5) + 1.0)) / 2.0
	else:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5e-11)
		tmp = Float64(Float64(1.0 / Float64(Float64(alpha * 0.5) + 1.0)) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5e-11)
		tmp = (1.0 / ((alpha * 0.5) + 1.0)) / 2.0;
	else
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 5e-11], N[(N[(1.0 / N[(N[(alpha * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.00000000000000018e-11

   1. Initial program 70.8%

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

   2. Taylor expanded in beta around 0 69.4%

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

      1. +-commutative 69.4%

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

   4. Simplified 69.4%

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

      1. flip-- 69.3%

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

      2. clear-num 69.3%

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

      3. metadata-eval 69.3%

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

      4. pow2 69.3%

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

   6. Applied egg-rr 69.3%

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

   7. Taylor expanded in alpha around 0 97.6%

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

      1. *-commutative 97.6%

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

   9. Simplified 97.6%

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

   if 5.00000000000000018e-11 < beta

   1. Initial program 80.2%

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

   2. Taylor expanded in alpha around 0 77.6%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{\alpha \cdot 0.5 + 1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \end{array} \]

Alternative 8: 71.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{\beta \cdot 0.5 + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

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

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = ((beta * 0.5) + 1.0) / 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 <= 2.0d0) then
        tmp = ((beta * 0.5d0) + 1.0d0) / 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 <= 2.0) {
		tmp = ((beta * 0.5) + 1.0) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 71.1%

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

   2. Taylor expanded in alpha around 0 66.0%

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

   3. Taylor expanded in beta around 0 65.2%

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

      1. *-commutative 65.2%

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

   5. Simplified 65.2%

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

   if 2 < beta

   1. Initial program 80.2%

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

   2. Taylor expanded in beta around inf 77.4%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{\beta \cdot 0.5 + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 71.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 70000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 70000.0) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 70000.0) {
		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 <= 70000.0d0) 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 <= 70000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 70000.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 70000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 70000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 70000.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 7e4

   1. Initial program 70.0%

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

   2. Taylor expanded in beta around 0 67.9%

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

      1. +-commutative 67.9%

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

   4. Simplified 67.9%

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

   5. Taylor expanded in alpha around 0 62.8%

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

   if 7e4 < beta

   1. Initial program 83.0%

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

   2. Taylor expanded in beta around inf 80.2%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 70000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 49.2% 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 74.0%

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

2. Taylor expanded in beta around 0 51.3%

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

   1. +-commutative 51.3%

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

4. Simplified 51.3%

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

5. Taylor expanded in alpha around 0 48.5%

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

6. Final simplification 48.5%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023301 
(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))