Octave 3.8, jcobi/1

Percentage Accurate: 75.4% → 99.9%

Time: 12.8s

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: 75.4% 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: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{\beta + \left(\beta + 2\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} + \left(1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99999)
     (/
      (fma
       (/ (- (- -2.0 beta) beta) alpha)
       (/ (+ beta 2.0) alpha)
       (/ (+ beta (+ beta 2.0)) alpha))
      2.0)
     (/ (+ (/ beta t_0) (- 1.0 (/ alpha t_0))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999) {
		tmp = fma((((-2.0 - beta) - beta) / alpha), ((beta + 2.0) / alpha), ((beta + (beta + 2.0)) / alpha)) / 2.0;
	} else {
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99999)
		tmp = Float64(fma(Float64(Float64(Float64(-2.0 - beta) - beta) / alpha), Float64(Float64(beta + 2.0) / alpha), Float64(Float64(beta + Float64(beta + 2.0)) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / t_0) + Float64(1.0 - Float64(alpha / t_0))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99999], N[(N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] + N[(1.0 - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999990000000000046

   1. Initial program 9.3%

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

      1. +-commutative 9.3%

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

   3. Simplified 9.3%

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

   4. Taylor expanded in alpha around -inf 91.1%

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

   6. Simplified 100.0%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

   3. Simplified 99.8%

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

      1. div-sub 99.9%

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

      2. associate-+l- 99.8%

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

      3. associate-+l+ 99.8%

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

      4. associate-+l+ 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{\beta + \left(\beta + 2\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \frac{\beta}{t_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{t_1 + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 + \left(1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))) (t_1 (/ beta t_0)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99999)
     (/ (+ t_1 (/ (- beta -2.0) alpha)) 2.0)
     (/ (+ t_1 (- 1.0 (/ alpha t_0))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double t_1 = beta / t_0;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999) {
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (t_1 + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = beta + (alpha + 2.0d0)
    t_1 = beta / t_0
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.99999d0)) then
        tmp = (t_1 + ((beta - (-2.0d0)) / alpha)) / 2.0d0
    else
        tmp = (t_1 + (1.0d0 - (alpha / t_0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double t_1 = beta / t_0;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999) {
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (t_1 + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	t_1 = beta / t_0
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999:
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0
	else:
		tmp = (t_1 + (1.0 - (alpha / t_0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	t_1 = Float64(beta / t_0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99999)
		tmp = Float64(Float64(t_1 + Float64(Float64(beta - -2.0) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(t_1 + Float64(1.0 - Float64(alpha / t_0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	t_1 = beta / t_0;
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999)
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0;
	else
		tmp = (t_1 + (1.0 - (alpha / t_0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta / t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99999], N[(N[(t$95$1 + N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$1 + N[(1.0 - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999990000000000046

   1. Initial program 9.3%

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

      1. +-commutative 9.3%

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

   3. Simplified 9.3%

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

      1. div-sub 9.3%

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

      2. associate-+l- 12.4%

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

      3. associate-+l+ 12.4%

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

      4. associate-+l+ 12.4%

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

   5. Applied egg-rr 12.4%

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

   6. Taylor expanded in alpha around inf 97.8%

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

      1. mul-1-neg 97.8%

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

      2. distribute-neg-frac 97.8%

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

      3. +-commutative 97.8%

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

      4. distribute-neg-in 97.8%

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

      5. metadata-eval 97.8%

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

      6. sub-neg 97.8%

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

   8. Simplified 97.8%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

   3. Simplified 99.8%

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

      1. div-sub 99.9%

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

      2. associate-+l- 99.8%

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

      3. associate-+l+ 99.8%

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

      4. associate-+l+ 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.99999:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.99999)
     (/ (+ (/ beta (+ beta (+ alpha 2.0))) (/ (- 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 <= -0.99999) {
		tmp = ((beta / (beta + (alpha + 2.0))) + ((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 <= (-0.99999d0)) then
        tmp = ((beta / (beta + (alpha + 2.0d0))) + ((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 <= -0.99999) {
		tmp = ((beta / (beta + (alpha + 2.0))) + ((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 <= -0.99999:
		tmp = ((beta / (beta + (alpha + 2.0))) + ((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 <= -0.99999)
		tmp = Float64(Float64(Float64(beta / Float64(beta + Float64(alpha + 2.0))) + Float64(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 <= -0.99999)
		tmp = ((beta / (beta + (alpha + 2.0))) + ((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, -0.99999], N[(N[(N[(beta / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(beta - -2.0), $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)) < -0.999990000000000046

   1. Initial program 9.3%

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

      1. +-commutative 9.3%

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

   3. Simplified 9.3%

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

      1. div-sub 9.3%

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

      2. associate-+l- 12.4%

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

      3. associate-+l+ 12.4%

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

      4. associate-+l+ 12.4%

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

   5. Applied egg-rr 12.4%

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

   6. Taylor expanded in alpha around inf 97.8%

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

      1. mul-1-neg 97.8%

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

      2. distribute-neg-frac 97.8%

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

      3. +-commutative 97.8%

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

      4. distribute-neg-in 97.8%

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

      5. metadata-eval 97.8%

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

      6. sub-neg 97.8%

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

   8. Simplified 97.8%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]

Alternative 4: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.99999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.99999)
     (/ (/ (+ 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 <= -0.99999) {
		tmp = ((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 <= (-0.99999d0)) then
        tmp = ((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 <= -0.99999) {
		tmp = ((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 <= -0.99999:
		tmp = ((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 <= -0.99999)
		tmp = Float64(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 <= -0.99999)
		tmp = ((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, -0.99999], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $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)) < -0.999990000000000046

   1. Initial program 9.3%

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

      1. +-commutative 9.3%

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

   3. Simplified 9.3%

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

   4. Taylor expanded in alpha around -inf 97.8%

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

      1. associate-*r/ 97.8%

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

      2. sub-neg 97.8%

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

      3. mul-1-neg 97.8%

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

      4. distribute-lft-in 97.8%

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

      5. neg-mul-1 97.8%

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

      6. mul-1-neg 97.8%

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

      7. remove-double-neg 97.8%

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

      8. neg-mul-1 97.8%

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

      9. mul-1-neg 97.8%

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

      10. remove-double-neg 97.8%

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

      11. +-commutative 97.8%

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

   6. Simplified 97.8%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]

Alternative 5: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq 3.5 \cdot 10^{-219}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 3.25 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 0.42:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 (* alpha 0.5)) 2.0)))
   (if (<= beta 3.5e-219)
     t_0
     (if (<= beta 3.25e-139) (/ 1.0 alpha) (if (<= beta 0.42) t_0 1.0)))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	double tmp;
	if (beta <= 3.5e-219) {
		tmp = t_0;
	} else if (beta <= 3.25e-139) {
		tmp = 1.0 / alpha;
	} else if (beta <= 0.42) {
		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 - (alpha * 0.5d0)) / 2.0d0
    if (beta <= 3.5d-219) then
        tmp = t_0
    else if (beta <= 3.25d-139) then
        tmp = 1.0d0 / alpha
    else if (beta <= 0.42d0) 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 - (alpha * 0.5)) / 2.0;
	double tmp;
	if (beta <= 3.5e-219) {
		tmp = t_0;
	} else if (beta <= 3.25e-139) {
		tmp = 1.0 / alpha;
	} else if (beta <= 0.42) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 - (alpha * 0.5)) / 2.0
	tmp = 0
	if beta <= 3.5e-219:
		tmp = t_0
	elif beta <= 3.25e-139:
		tmp = 1.0 / alpha
	elif beta <= 0.42:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 - Float64(alpha * 0.5)) / 2.0)
	tmp = 0.0
	if (beta <= 3.5e-219)
		tmp = t_0;
	elseif (beta <= 3.25e-139)
		tmp = Float64(1.0 / alpha);
	elseif (beta <= 0.42)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	tmp = 0.0;
	if (beta <= 3.5e-219)
		tmp = t_0;
	elseif (beta <= 3.25e-139)
		tmp = 1.0 / alpha;
	elseif (beta <= 0.42)
		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[(alpha * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[beta, 3.5e-219], t$95$0, If[LessEqual[beta, 3.25e-139], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[beta, 0.42], t$95$0, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 3.50000000000000011e-219 or 3.25e-139 < beta < 0.419999999999999984

   1. Initial program 72.9%

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

      1. +-commutative 72.9%

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

   3. Simplified 72.9%

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

   4. Taylor expanded in beta around 0 70.7%

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

      1. +-commutative 70.7%

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

   6. Simplified 70.7%

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

   7. Taylor expanded in alpha around 0 66.4%

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

      1. *-commutative 66.4%

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

   9. Simplified 66.4%

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

   if 3.50000000000000011e-219 < beta < 3.25e-139

   1. Initial program 39.8%

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

      1. +-commutative 39.8%

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

   3. Simplified 39.8%

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

      1. div-sub 39.8%

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

      2. associate-+l- 39.9%

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

      3. associate-+l+ 39.9%

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

      4. associate-+l+ 39.9%

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

   5. Applied egg-rr 39.9%

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

   6. Taylor expanded in alpha around inf 65.9%

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

      1. mul-1-neg 65.9%

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

      2. distribute-neg-frac 65.9%

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

      3. +-commutative 65.9%

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

      4. distribute-neg-in 65.9%

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

      5. metadata-eval 65.9%

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

      6. sub-neg 65.9%

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

   8. Simplified 65.9%

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

   9. Taylor expanded in beta around 0 65.9%

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

   if 0.419999999999999984 < beta

   1. Initial program 84.6%

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

      1. +-commutative 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in beta around inf 82.2%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{-219}:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 3.25 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 0.42:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq 1.7 \cdot 10^{-222}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 8.5 \cdot 10^{-141}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 1.75:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 (* alpha 0.5)) 2.0)))
   (if (<= beta 1.7e-222)
     t_0
     (if (<= beta 8.5e-141)
       (/ 1.0 alpha)
       (if (<= beta 1.75) t_0 (/ (- 2.0 (/ 2.0 beta)) 2.0))))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	double tmp;
	if (beta <= 1.7e-222) {
		tmp = t_0;
	} else if (beta <= 8.5e-141) {
		tmp = 1.0 / alpha;
	} else if (beta <= 1.75) {
		tmp = t_0;
	} else {
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - (alpha * 0.5d0)) / 2.0d0
    if (beta <= 1.7d-222) then
        tmp = t_0
    else if (beta <= 8.5d-141) then
        tmp = 1.0d0 / alpha
    else if (beta <= 1.75d0) then
        tmp = t_0
    else
        tmp = (2.0d0 - (2.0d0 / beta)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	double tmp;
	if (beta <= 1.7e-222) {
		tmp = t_0;
	} else if (beta <= 8.5e-141) {
		tmp = 1.0 / alpha;
	} else if (beta <= 1.75) {
		tmp = t_0;
	} else {
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 - (alpha * 0.5)) / 2.0
	tmp = 0
	if beta <= 1.7e-222:
		tmp = t_0
	elif beta <= 8.5e-141:
		tmp = 1.0 / alpha
	elif beta <= 1.75:
		tmp = t_0
	else:
		tmp = (2.0 - (2.0 / beta)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 - Float64(alpha * 0.5)) / 2.0)
	tmp = 0.0
	if (beta <= 1.7e-222)
		tmp = t_0;
	elseif (beta <= 8.5e-141)
		tmp = Float64(1.0 / alpha);
	elseif (beta <= 1.75)
		tmp = t_0;
	else
		tmp = Float64(Float64(2.0 - Float64(2.0 / beta)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	tmp = 0.0;
	if (beta <= 1.7e-222)
		tmp = t_0;
	elseif (beta <= 8.5e-141)
		tmp = 1.0 / alpha;
	elseif (beta <= 1.75)
		tmp = t_0;
	else
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 - N[(alpha * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[beta, 1.7e-222], t$95$0, If[LessEqual[beta, 8.5e-141], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[beta, 1.75], t$95$0, N[(N[(2.0 - N[(2.0 / beta), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 1.7000000000000001e-222 or 8.50000000000000021e-141 < beta < 1.75

   1. Initial program 72.9%

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

      1. +-commutative 72.9%

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

   3. Simplified 72.9%

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

   4. Taylor expanded in beta around 0 70.7%

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

      1. +-commutative 70.7%

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

   6. Simplified 70.7%

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

   7. Taylor expanded in alpha around 0 66.4%

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

      1. *-commutative 66.4%

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

   9. Simplified 66.4%

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

   if 1.7000000000000001e-222 < beta < 8.50000000000000021e-141

   1. Initial program 39.8%

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

      1. +-commutative 39.8%

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

   3. Simplified 39.8%

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

      1. div-sub 39.8%

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

      2. associate-+l- 39.9%

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

      3. associate-+l+ 39.9%

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

      4. associate-+l+ 39.9%

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

   5. Applied egg-rr 39.9%

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

   6. Taylor expanded in alpha around inf 65.9%

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

      1. mul-1-neg 65.9%

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

      2. distribute-neg-frac 65.9%

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

      3. +-commutative 65.9%

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

      4. distribute-neg-in 65.9%

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

      5. metadata-eval 65.9%

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

      6. sub-neg 65.9%

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

   8. Simplified 65.9%

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

   9. Taylor expanded in beta around 0 65.9%

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

   if 1.75 < beta

   1. Initial program 84.6%

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

      1. +-commutative 84.6%

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

   3. Simplified 84.6%

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

      1. div-sub 84.7%

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

      2. associate-+l- 86.9%

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

      3. associate-+l+ 86.9%

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

      4. associate-+l+ 86.9%

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

   5. Applied egg-rr 86.9%

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

   6. Taylor expanded in beta around -inf 81.5%

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

      1. +-commutative 81.5%

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

      2. mul-1-neg 81.5%

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

      3. unsub-neg 81.5%

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

      4. associate--l+ 81.5%

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

      5. sub-neg 81.5%

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

      6. mul-1-neg 81.5%

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

      7. remove-double-neg 81.5%

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

   8. Simplified 81.5%

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

   9. Taylor expanded in alpha around 0 82.4%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.7 \cdot 10^{-222}:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 8.5 \cdot 10^{-141}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 1.75:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \]

Alternative 7: 88.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 114000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 114000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ beta 2.0) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 114000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((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 <= 114000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((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 <= 114000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 114000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((beta + 2.0) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 114000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta + 2.0) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 114000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((beta + 2.0) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 114000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 114000

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in alpha around 0 97.4%

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

   if 114000 < alpha

   1. Initial program 26.2%

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

      1. +-commutative 26.2%

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

   3. Simplified 26.2%

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

      1. div-sub 26.3%

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

      2. associate-+l- 28.9%

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

      3. associate-+l+ 28.9%

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

      4. associate-+l+ 28.9%

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

   5. Applied egg-rr 28.9%

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

   6. Taylor expanded in alpha around inf 81.2%

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

      1. mul-1-neg 81.2%

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

      2. distribute-neg-frac 81.2%

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

      3. +-commutative 81.2%

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

      4. distribute-neg-in 81.2%

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

      5. metadata-eval 81.2%

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

      6. sub-neg 81.2%

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

   8. Simplified 81.2%

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

   9. Taylor expanded in alpha around 0 66.2%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 114000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 93.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 104000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 104000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 104000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (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 <= 104000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((beta + (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 <= 104000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 104000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 104000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 104000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 104000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 104000

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in alpha around 0 97.4%

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

   if 104000 < alpha

   1. Initial program 26.2%

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

      1. +-commutative 26.2%

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

   3. Simplified 26.2%

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

   4. Taylor expanded in alpha around -inf 81.1%

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

      1. associate-*r/ 81.1%

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

      2. sub-neg 81.1%

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

      3. mul-1-neg 81.1%

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

      4. distribute-lft-in 81.1%

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

      5. neg-mul-1 81.1%

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

      6. mul-1-neg 81.1%

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

      7. remove-double-neg 81.1%

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

      8. neg-mul-1 81.1%

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

      9. mul-1-neg 81.1%

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

      10. remove-double-neg 81.1%

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

      11. +-commutative 81.1%

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

   6. Simplified 81.1%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 104000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 9: 51.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 110000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 110000.0) 1.0 (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 110000.0) {
		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 <= 110000.0d0) 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 <= 110000.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 110000.0:
		tmp = 1.0
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 110000.0)
		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 <= 110000.0)
		tmp = 1.0;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 110000.0], 1.0, N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.1e5

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in beta around inf 48.0%

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

   if 1.1e5 < alpha

   1. Initial program 26.2%

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

      1. +-commutative 26.2%

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

   3. Simplified 26.2%

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

      1. div-sub 26.3%

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

      2. associate-+l- 28.9%

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

      3. associate-+l+ 28.9%

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

      4. associate-+l+ 28.9%

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

   5. Applied egg-rr 28.9%

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

   6. Taylor expanded in alpha around inf 81.2%

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

      1. mul-1-neg 81.2%

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

      2. distribute-neg-frac 81.2%

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

      3. +-commutative 81.2%

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

      4. distribute-neg-in 81.2%

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

      5. metadata-eval 81.2%

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

      6. sub-neg 81.2%

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

   8. Simplified 81.2%

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

   9. Taylor expanded in beta around 0 63.4%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 110000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 10: 23.8% 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 75.8%

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

   1. +-commutative 75.8%

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

3. Simplified 75.8%

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

   1. div-sub 75.8%

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

   2. associate-+l- 76.6%

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

   3. associate-+l+ 76.6%

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

   4. associate-+l+ 76.6%

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

5. Applied egg-rr 76.6%

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

6. Taylor expanded in alpha around inf 28.6%

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

   1. mul-1-neg 28.6%

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

   2. distribute-neg-frac 28.6%

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

   3. +-commutative 28.6%

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

   4. distribute-neg-in 28.6%

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

   5. metadata-eval 28.6%

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

   6. sub-neg 28.6%

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

8. Simplified 28.6%

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

9. Taylor expanded in beta around 0 23.0%

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

10. Final simplification 23.0%

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

Reproduce

herbie shell --seed 2023274 
(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))