Octave 3.8, jcobi/1

Percentage Accurate: 75.0% → 99.7%

Time: 10.2s

Alternatives: 10

Speedup: 0.9×

Specification

\[\alpha > -1 \land \beta > -1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999:
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0
	else:
		tmp = (((beta - alpha) / (-1.0 + ((beta + alpha) + 3.0))) + 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)) <= -0.9999999)
		tmp = Float64(Float64(Float64(beta + Float64(beta - -2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(-1.0 + Float64(Float64(beta + alpha) + 3.0))) + 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)) <= -0.9999999)
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0;
	else
		tmp = (((beta - alpha) / (-1.0 + ((beta + alpha) + 3.0))) + 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], -0.9999999], N[(N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(-1.0 + N[(N[(beta + alpha), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.5%

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

      1. +-commutative 6.5%

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

   3. Simplified 6.5%

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

   5. Taylor expanded in alpha around -inf 99.4%

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

      1. mul-1-neg 99.4%

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

      2. distribute-neg-frac2 99.4%

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

      3. sub-neg 99.4%

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

      4. +-commutative 99.4%

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

      5. mul-1-neg 99.4%

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

      6. sub-neg 99.4%

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

      7. mul-1-neg 99.4%

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

      8. distribute-lft-in 99.4%

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

      9. metadata-eval 99.4%

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

      10. mul-1-neg 99.4%

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

      11. unsub-neg 99.4%

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

   7. Simplified 99.4%

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

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

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

   3. Simplified 99.8%

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

      1. expm1-log1p-u 96.6%

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

      2. expm1-undefine 96.6%

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

      3. associate-+l+ 96.6%

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

   6. Applied egg-rr 96.6%

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

      1. sub-neg 96.6%

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

      2. metadata-eval 96.6%

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

      3. +-commutative 96.6%

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

      4. log1p-undefine 96.6%

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

      5. rem-exp-log 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+r+ 99.8%

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

      9. metadata-eval 99.8%

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

      10. +-commutative 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 99.6%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.9999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9999999)
     (/ (/ (+ 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 <= -0.9999999) {
		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 <= (-0.9999999d0)) 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 <= -0.9999999) {
		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 <= -0.9999999:
		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 <= -0.9999999)
		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 <= -0.9999999)
		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, -0.9999999], 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) #s(literal 2 binary64))) < -0.999999900000000053

   1. Initial program 6.5%

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

      1. +-commutative 6.5%

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

   3. Simplified 6.5%

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

   5. Taylor expanded in alpha around -inf 99.4%

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

      1. mul-1-neg 99.4%

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

      2. distribute-neg-frac2 99.4%

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

      3. sub-neg 99.4%

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

      4. +-commutative 99.4%

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

      5. mul-1-neg 99.4%

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

      6. sub-neg 99.4%

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

      7. mul-1-neg 99.4%

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

      8. distribute-lft-in 99.4%

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

      9. metadata-eval 99.4%

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

      10. mul-1-neg 99.4%

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

      11. unsub-neg 99.4%

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

   7. Simplified 99.4%

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

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

   1. Initial program 99.8%

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

4. Final simplification 99.6%

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

Alternative 3: 64.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\alpha \leq -9 \cdot 10^{-202}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\alpha \leq -1.25 \cdot 10^{-260}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 0.4:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (* beta 0.5)) 2.0)))
   (if (<= alpha -9e-202)
     t_0
     (if (<= alpha -1.25e-260)
       1.0
       (if (<= alpha 0.4) t_0 (/ (/ 2.0 alpha) 2.0))))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double tmp;
	if (alpha <= -9e-202) {
		tmp = t_0;
	} else if (alpha <= -1.25e-260) {
		tmp = 1.0;
	} else if (alpha <= 0.4) {
		tmp = t_0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (beta * 0.5d0)) / 2.0d0
    if (alpha <= (-9d-202)) then
        tmp = t_0
    else if (alpha <= (-1.25d-260)) then
        tmp = 1.0d0
    else if (alpha <= 0.4d0) then
        tmp = t_0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double tmp;
	if (alpha <= -9e-202) {
		tmp = t_0;
	} else if (alpha <= -1.25e-260) {
		tmp = 1.0;
	} else if (alpha <= 0.4) {
		tmp = t_0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 + (beta * 0.5)) / 2.0
	tmp = 0
	if alpha <= -9e-202:
		tmp = t_0
	elif alpha <= -1.25e-260:
		tmp = 1.0
	elif alpha <= 0.4:
		tmp = t_0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + Float64(beta * 0.5)) / 2.0)
	tmp = 0.0
	if (alpha <= -9e-202)
		tmp = t_0;
	elseif (alpha <= -1.25e-260)
		tmp = 1.0;
	elseif (alpha <= 0.4)
		tmp = t_0;
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 + (beta * 0.5)) / 2.0;
	tmp = 0.0;
	if (alpha <= -9e-202)
		tmp = t_0;
	elseif (alpha <= -1.25e-260)
		tmp = 1.0;
	elseif (alpha <= 0.4)
		tmp = t_0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, -9e-202], t$95$0, If[LessEqual[alpha, -1.25e-260], 1.0, If[LessEqual[alpha, 0.4], t$95$0, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -9.00000000000000078e-202 or -1.2500000000000001e-260 < alpha < 0.40000000000000002

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 98.2%

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

   6. Taylor expanded in beta around 0 68.6%

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

   if -9.00000000000000078e-202 < alpha < -1.2500000000000001e-260

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 74.6%

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

   if 0.40000000000000002 < alpha

   1. Initial program 19.6%

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

      1. +-commutative 19.6%

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

   3. Simplified 19.6%

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

   5. Taylor expanded in alpha around -inf 86.6%

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

      1. mul-1-neg 86.6%

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

      2. distribute-neg-frac2 86.6%

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

      3. sub-neg 86.6%

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

      4. +-commutative 86.6%

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

      5. mul-1-neg 86.6%

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

      6. sub-neg 86.6%

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

      7. mul-1-neg 86.6%

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

      8. distribute-lft-in 86.6%

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

      9. metadata-eval 86.6%

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

      10. mul-1-neg 86.6%

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

      11. unsub-neg 86.6%

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

   7. Simplified 86.6%

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

   8. Taylor expanded in beta around 0 65.0%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -9 \cdot 10^{-202}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\alpha \leq -1.25 \cdot 10^{-260}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 0.4:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq -1.42 \cdot 10^{-258}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2.5 \cdot 10^{-206}:\\ \;\;\;\;\frac{1 + \frac{\alpha}{\beta}}{2}\\ \mathbf{elif}\;\alpha \leq 10800000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha -1.42e-258)
   1.0
   (if (<= alpha 2.5e-206)
     (/ (+ 1.0 (/ alpha beta)) 2.0)
     (if (<= alpha 10800000000.0) 1.0 (/ (/ 2.0 alpha) 2.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= -1.42e-258) {
		tmp = 1.0;
	} else if (alpha <= 2.5e-206) {
		tmp = (1.0 + (alpha / beta)) / 2.0;
	} else if (alpha <= 10800000000.0) {
		tmp = 1.0;
	} 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 <= (-1.42d-258)) then
        tmp = 1.0d0
    else if (alpha <= 2.5d-206) then
        tmp = (1.0d0 + (alpha / beta)) / 2.0d0
    else if (alpha <= 10800000000.0d0) then
        tmp = 1.0d0
    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 <= -1.42e-258) {
		tmp = 1.0;
	} else if (alpha <= 2.5e-206) {
		tmp = (1.0 + (alpha / beta)) / 2.0;
	} else if (alpha <= 10800000000.0) {
		tmp = 1.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= -1.42e-258:
		tmp = 1.0
	elif alpha <= 2.5e-206:
		tmp = (1.0 + (alpha / beta)) / 2.0
	elif alpha <= 10800000000.0:
		tmp = 1.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= -1.42e-258)
		tmp = 1.0;
	elseif (alpha <= 2.5e-206)
		tmp = Float64(Float64(1.0 + Float64(alpha / beta)) / 2.0);
	elseif (alpha <= 10800000000.0)
		tmp = 1.0;
	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 <= -1.42e-258)
		tmp = 1.0;
	elseif (alpha <= 2.5e-206)
		tmp = (1.0 + (alpha / beta)) / 2.0;
	elseif (alpha <= 10800000000.0)
		tmp = 1.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, -1.42e-258], 1.0, If[LessEqual[alpha, 2.5e-206], N[(N[(1.0 + N[(alpha / beta), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 10800000000.0], 1.0, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -1.42000000000000001e-258 or 2.5e-206 < alpha < 1.08e10

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 50.8%

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

   if -1.42000000000000001e-258 < alpha < 2.5e-206

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 35.9%

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

   6. Taylor expanded in beta around 0 70.8%

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

      1. neg-mul-1 70.8%

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

      2. distribute-neg-frac2 70.8%

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

   8. Simplified 70.8%

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

      1. +-commutative 70.8%

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

      2. distribute-frac-neg2 70.8%

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

      3. unsub-neg 70.8%

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

      4. add-sqr-sqrt 30.2%

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

      5. sqrt-unprod 38.3%

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

      6. sqr-neg 38.3%

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

      7. sqrt-unprod 40.4%

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

      8. add-sqr-sqrt 70.3%

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

      9. *-rgt-identity 70.3%

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

      10. *-un-lft-identity 70.3%

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

      11. distribute-frac-neg2 70.3%

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

      12. cancel-sign-sub 70.3%

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

      13. add-sqr-sqrt 29.9%

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

      14. sqrt-unprod 38.5%

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

      15. sqr-neg 38.5%

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

      16. sqrt-unprod 40.6%

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

      17. add-sqr-sqrt 70.8%

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

      18. *-rgt-identity 70.8%

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

      19. add-sqr-sqrt 40.6%

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

      20. sqrt-unprod 38.5%

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

      21. sqr-neg 38.5%

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

      22. sqrt-unprod 29.9%

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

      23. add-sqr-sqrt 70.3%

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

   10. Applied egg-rr 70.3%

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

       1. *-lft-identity 70.3%

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

   12. Simplified 70.3%

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

   if 1.08e10 < alpha

   1. Initial program 18.9%

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

      1. +-commutative 18.9%

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

   3. Simplified 18.9%

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

   5. Taylor expanded in alpha around -inf 87.4%

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

      1. mul-1-neg 87.4%

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

      2. distribute-neg-frac2 87.4%

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

      3. sub-neg 87.4%

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

      4. +-commutative 87.4%

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

      5. mul-1-neg 87.4%

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

      6. sub-neg 87.4%

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

      7. mul-1-neg 87.4%

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

      8. distribute-lft-in 87.4%

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

      9. metadata-eval 87.4%

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

      10. mul-1-neg 87.4%

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

      11. unsub-neg 87.4%

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

   7. Simplified 87.4%

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

   8. Taylor expanded in beta around 0 65.5%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -1.42 \cdot 10^{-258}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2.5 \cdot 10^{-206}:\\ \;\;\;\;\frac{1 + \frac{\alpha}{\beta}}{2}\\ \mathbf{elif}\;\alpha \leq 10800000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1200000000:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1200000000.0)
   (/ (- 1.0 (/ (- alpha beta) (+ beta 2.0))) 2.0)
   (/ (/ (+ beta (- beta -2.0)) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1200000000.0) {
		tmp = (1.0 - ((alpha - 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 <= 1200000000.0d0) then
        tmp = (1.0d0 - ((alpha - 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 <= 1200000000.0) {
		tmp = (1.0 - ((alpha - 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 <= 1200000000.0:
		tmp = (1.0 - ((alpha - 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 <= 1200000000.0)
		tmp = Float64(Float64(1.0 - Float64(Float64(alpha - 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 <= 1200000000.0)
		tmp = (1.0 - ((alpha - 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, 1200000000.0], N[(N[(1.0 - N[(N[(alpha - beta), $MachinePrecision] / 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 < 1.2e9

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 99.0%

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

   if 1.2e9 < alpha

   1. Initial program 18.9%

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

      1. +-commutative 18.9%

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

   3. Simplified 18.9%

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

   5. Taylor expanded in alpha around -inf 87.4%

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

      1. mul-1-neg 87.4%

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

      2. distribute-neg-frac2 87.4%

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

      3. sub-neg 87.4%

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

      4. +-commutative 87.4%

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

      5. mul-1-neg 87.4%

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

      6. sub-neg 87.4%

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

      7. mul-1-neg 87.4%

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

      8. distribute-lft-in 87.4%

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

      9. metadata-eval 87.4%

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

      10. mul-1-neg 87.4%

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

      11. unsub-neg 87.4%

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

   7. Simplified 87.4%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1200000000:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5100000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 5100000000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 5100000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 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 <= 5100000000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 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 <= 5100000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 5100000000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 5100000000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 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 <= 5100000000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 5100000000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 5.1e9

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 98.4%

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

   if 5.1e9 < alpha

   1. Initial program 18.9%

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

      1. +-commutative 18.9%

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

   3. Simplified 18.9%

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

   5. Taylor expanded in alpha around -inf 87.4%

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

      1. mul-1-neg 87.4%

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

      2. distribute-neg-frac2 87.4%

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

      3. sub-neg 87.4%

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

      4. +-commutative 87.4%

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

      5. mul-1-neg 87.4%

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

      6. sub-neg 87.4%

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

      7. mul-1-neg 87.4%

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

      8. distribute-lft-in 87.4%

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

      9. metadata-eval 87.4%

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

      10. mul-1-neg 87.4%

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

      11. unsub-neg 87.4%

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

   7. Simplified 87.4%

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

   8. Taylor expanded in beta around 0 65.5%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5100000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 900000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 900000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 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 <= 900000000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 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 <= 900000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

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

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 900000000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 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 <= 900000000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 900000000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $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 2 regimes

2. if alpha < 9e8

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 98.4%

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

   if 9e8 < alpha

   1. Initial program 18.9%

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

      1. +-commutative 18.9%

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

   3. Simplified 18.9%

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

   5. Taylor expanded in alpha around inf 87.4%

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

4. Final simplification 94.0%

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

Alternative 8: 93.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 900000000.0)
   (/ (+ 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 <= 900000000.0) {
		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 <= 900000000.0d0) 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 <= 900000000.0) {
		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 <= 900000000.0:
		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 <= 900000000.0)
		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 <= 900000000.0)
		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, 900000000.0], 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 < 9e8

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 98.4%

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

   if 9e8 < alpha

   1. Initial program 18.9%

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

      1. +-commutative 18.9%

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

   3. Simplified 18.9%

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

   5. Taylor expanded in alpha around -inf 87.4%

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

      1. mul-1-neg 87.4%

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

      2. distribute-neg-frac2 87.4%

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

      3. sub-neg 87.4%

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

      4. +-commutative 87.4%

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

      5. mul-1-neg 87.4%

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

      6. sub-neg 87.4%

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

      7. mul-1-neg 87.4%

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

      8. distribute-lft-in 87.4%

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

      9. metadata-eval 87.4%

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

      10. mul-1-neg 87.4%

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

      11. unsub-neg 87.4%

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

   7. Simplified 87.4%

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

4. Final simplification 94.0%

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

Alternative 9: 52.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1600000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1600000000.0) 1.0 (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1600000000.0) {
		tmp = 1.0;
	} 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 <= 1600000000.0d0) then
        tmp = 1.0d0
    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 <= 1600000000.0) {
		tmp = 1.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1600000000.0:
		tmp = 1.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1600000000.0)
		tmp = 1.0;
	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 <= 1600000000.0)
		tmp = 1.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1600000000.0], 1.0, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.6e9

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 47.0%

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

   if 1.6e9 < alpha

   1. Initial program 18.9%

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

      1. +-commutative 18.9%

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

   3. Simplified 18.9%

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

   5. Taylor expanded in alpha around -inf 87.4%

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

      1. mul-1-neg 87.4%

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

      2. distribute-neg-frac2 87.4%

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

      3. sub-neg 87.4%

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

      4. +-commutative 87.4%

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

      5. mul-1-neg 87.4%

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

      6. sub-neg 87.4%

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

      7. mul-1-neg 87.4%

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

      8. distribute-lft-in 87.4%

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

      9. metadata-eval 87.4%

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

      10. mul-1-neg 87.4%

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

      11. unsub-neg 87.4%

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

   7. Simplified 87.4%

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

   8. Taylor expanded in beta around 0 65.5%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1600000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 37.7% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 1.0

Julia

function code(alpha, beta)
	return 1.0
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 1.0

Derivation

1. Initial program 67.3%

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

   1. +-commutative 67.3%

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

3. Simplified 67.3%

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

5. Taylor expanded in beta around inf 34.5%

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

6. Final simplification 34.5%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024079 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))