Octave 3.8, jcobi/1

Percentage Accurate: 73.4% → 99.8%

Time: 9.1s

Alternatives: 11

Speedup: 0.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.4% 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: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{0.5 \cdot \left(\left(2 + \beta \cdot 2\right) + \left(\beta + 2\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9995)
   (/
    (*
     0.5
     (+
      (+ 2.0 (* beta 2.0))
      (* (+ beta 2.0) (/ (- (- -2.0 beta) beta) alpha))))
    alpha)
   (/ (fma (- beta alpha) (/ 1.0 (+ beta (+ alpha 2.0))) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995) {
		tmp = (0.5 * ((2.0 + (beta * 2.0)) + ((beta + 2.0) * (((-2.0 - beta) - beta) / alpha)))) / alpha;
	} else {
		tmp = fma((beta - alpha), (1.0 / (beta + (alpha + 2.0))), 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)) <= -0.9995)
		tmp = Float64(Float64(0.5 * Float64(Float64(2.0 + Float64(beta * 2.0)) + Float64(Float64(beta + 2.0) * Float64(Float64(Float64(-2.0 - beta) - beta) / alpha)))) / alpha);
	else
		tmp = Float64(fma(Float64(beta - alpha), Float64(1.0 / Float64(beta + Float64(alpha + 2.0))), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9995], N[(N[(0.5 * N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99950000000000006

   1. Initial program 7.8%

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

      1. add-sqr-sqrt 7.8%

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

      2. pow2 7.8%

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

      3. +-commutative 7.8%

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

      4. associate-+l+ 7.8%

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

   4. Applied egg-rr 7.8%

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

   5. Taylor expanded in alpha around inf 91.4%

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

   7. Simplified 100.0%

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

   if -0.99950000000000006 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.8%

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

      1. div-inv 99.8%

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

      2. fma-define 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l+ 99.8%

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{0.5 \cdot \left(\left(2 + \beta \cdot 2\right) + \left(\beta + 2\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.9995:\\ \;\;\;\;\frac{0.5 \cdot \left(\left(2 + \beta \cdot 2\right) + \left(\beta + 2\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9995)
     (/
      (*
       0.5
       (+
        (+ 2.0 (* beta 2.0))
        (* (+ beta 2.0) (/ (- (- -2.0 beta) beta) alpha))))
      alpha)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.9995) {
		tmp = (0.5 * ((2.0 + (beta * 2.0)) + ((beta + 2.0) * (((-2.0 - beta) - beta) / alpha)))) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.9995d0)) then
        tmp = (0.5d0 * ((2.0d0 + (beta * 2.0d0)) + ((beta + 2.0d0) * ((((-2.0d0) - beta) - beta) / alpha)))) / alpha
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.9995) {
		tmp = (0.5 * ((2.0 + (beta * 2.0)) + ((beta + 2.0) * (((-2.0 - beta) - beta) / alpha)))) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.9995:
		tmp = (0.5 * ((2.0 + (beta * 2.0)) + ((beta + 2.0) * (((-2.0 - beta) - beta) / alpha)))) / alpha
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.9995)
		tmp = Float64(Float64(0.5 * Float64(Float64(2.0 + Float64(beta * 2.0)) + Float64(Float64(beta + 2.0) * Float64(Float64(Float64(-2.0 - beta) - beta) / alpha)))) / alpha);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.9995)
		tmp = (0.5 * ((2.0 + (beta * 2.0)) + ((beta + 2.0) * (((-2.0 - beta) - beta) / alpha)))) / alpha;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.9995], N[(N[(0.5 * N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $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))) < -0.99950000000000006

   1. Initial program 7.8%

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

      1. add-sqr-sqrt 7.8%

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

      2. pow2 7.8%

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

      3. +-commutative 7.8%

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

      4. associate-+l+ 7.8%

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

   4. Applied egg-rr 7.8%

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

   5. Taylor expanded in alpha around inf 91.4%

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

   7. Simplified 100.0%

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

   if -0.99950000000000006 < (/.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 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{0.5 \cdot \left(\left(2 + \beta \cdot 2\right) + \left(\beta + 2\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.9995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9995)
     (/ (/ (+ 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 <= -0.9995) {
		tmp = ((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 <= (-0.9995d0)) then
        tmp = ((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 <= -0.9995) {
		tmp = ((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 <= -0.9995:
		tmp = ((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 <= -0.9995)
		tmp = Float64(Float64(Float64(2.0 + 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 <= -0.9995)
		tmp = ((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, -0.9995], N[(N[(N[(2.0 + 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))) < -0.99950000000000006

   1. Initial program 7.8%

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

   3. Taylor expanded in alpha around inf 98.8%

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

   if -0.99950000000000006 < (/.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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq -2.5 \cdot 10^{-26}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \mathbf{elif}\;\alpha \leq 300000000:\\ \;\;\;\;0.5 + \frac{0.5}{\frac{-2}{\alpha} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha -2.5e-26)
   (+ 1.0 (/ -1.0 beta))
   (if (<= alpha 300000000.0)
     (+ 0.5 (/ 0.5 (+ (/ -2.0 alpha) -1.0)))
     (+ (/ 1.0 alpha) (/ beta alpha)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= -2.5e-26) {
		tmp = 1.0 + (-1.0 / beta);
	} else if (alpha <= 300000000.0) {
		tmp = 0.5 + (0.5 / ((-2.0 / alpha) + -1.0));
	} else {
		tmp = (1.0 / alpha) + (beta / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= (-2.5d-26)) then
        tmp = 1.0d0 + ((-1.0d0) / beta)
    else if (alpha <= 300000000.0d0) then
        tmp = 0.5d0 + (0.5d0 / (((-2.0d0) / alpha) + (-1.0d0)))
    else
        tmp = (1.0d0 / alpha) + (beta / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= -2.5e-26) {
		tmp = 1.0 + (-1.0 / beta);
	} else if (alpha <= 300000000.0) {
		tmp = 0.5 + (0.5 / ((-2.0 / alpha) + -1.0));
	} else {
		tmp = (1.0 / alpha) + (beta / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= -2.5e-26:
		tmp = 1.0 + (-1.0 / beta)
	elif alpha <= 300000000.0:
		tmp = 0.5 + (0.5 / ((-2.0 / alpha) + -1.0))
	else:
		tmp = (1.0 / alpha) + (beta / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= -2.5e-26)
		tmp = Float64(1.0 + Float64(-1.0 / beta));
	elseif (alpha <= 300000000.0)
		tmp = Float64(0.5 + Float64(0.5 / Float64(Float64(-2.0 / alpha) + -1.0)));
	else
		tmp = Float64(Float64(1.0 / alpha) + Float64(beta / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= -2.5e-26)
		tmp = 1.0 + (-1.0 / beta);
	elseif (alpha <= 300000000.0)
		tmp = 0.5 + (0.5 / ((-2.0 / alpha) + -1.0));
	else
		tmp = (1.0 / alpha) + (beta / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, -2.5e-26], N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision], If[LessEqual[alpha, 300000000.0], N[(0.5 + N[(0.5 / N[(N[(-2.0 / alpha), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / alpha), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -2.5000000000000001e-26

   1. Initial program 99.9%

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

   3. Taylor expanded in alpha around 0 92.2%

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

   4. Taylor expanded in beta around inf 73.0%

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

   if -2.5000000000000001e-26 < alpha < 3e8

   1. Initial program 99.8%

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

   3. Taylor expanded in alpha around -inf 90.9%

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

      1. mul-1-neg 90.9%

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

      2. *-commutative 90.9%

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

      3. distribute-rgt-neg-in 90.9%

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

      4. sub-neg 90.9%

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

      5. metadata-eval 90.9%

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

      6. +-commutative 90.9%

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

      7. associate-*r/ 90.9%

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

      8. distribute-lft-in 90.9%

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

      9. metadata-eval 90.9%

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

      10. mul-1-neg 90.9%

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

      11. unsub-neg 90.9%

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

   5. Simplified 90.9%

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

   6. Taylor expanded in beta around 0 73.9%

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

      1. associate-*r/ 73.9%

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

      2. metadata-eval 73.9%

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

   8. Simplified 73.9%

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

      1. div-sub 73.9%

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

      2. metadata-eval 73.9%

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

      3. sub-neg 73.9%

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

      4. div-inv 73.9%

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

      5. metadata-eval 73.9%

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

   10. Applied egg-rr 73.9%

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

       1. sub-neg 73.9%

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

       2. associate-*l/ 73.9%

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

       3. metadata-eval 73.9%

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

       4. remove-double-neg 73.9%

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

       5. unsub-neg 73.9%

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

       6. distribute-neg-frac 73.9%

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

       7. metadata-eval 73.9%

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

   12. Simplified 73.9%

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

   if 3e8 < alpha

   1. Initial program 18.5%

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

   3. Taylor expanded in alpha around inf 88.2%

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

   4. Taylor expanded in beta around 0 88.2%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -2.5 \cdot 10^{-26}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \mathbf{elif}\;\alpha \leq 300000000:\\ \;\;\;\;0.5 + \frac{0.5}{\frac{-2}{\alpha} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq -5.5 \cdot 10^{-26}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \mathbf{elif}\;\alpha \leq 1.95:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha -5.5e-26)
   (+ 1.0 (/ -1.0 beta))
   (if (<= alpha 1.95)
     (+ 0.5 (* alpha -0.25))
     (+ (/ 1.0 alpha) (/ beta alpha)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= -5.5e-26) {
		tmp = 1.0 + (-1.0 / beta);
	} else if (alpha <= 1.95) {
		tmp = 0.5 + (alpha * -0.25);
	} else {
		tmp = (1.0 / alpha) + (beta / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= (-5.5d-26)) then
        tmp = 1.0d0 + ((-1.0d0) / beta)
    else if (alpha <= 1.95d0) then
        tmp = 0.5d0 + (alpha * (-0.25d0))
    else
        tmp = (1.0d0 / alpha) + (beta / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= -5.5e-26) {
		tmp = 1.0 + (-1.0 / beta);
	} else if (alpha <= 1.95) {
		tmp = 0.5 + (alpha * -0.25);
	} else {
		tmp = (1.0 / alpha) + (beta / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= -5.5e-26:
		tmp = 1.0 + (-1.0 / beta)
	elif alpha <= 1.95:
		tmp = 0.5 + (alpha * -0.25)
	else:
		tmp = (1.0 / alpha) + (beta / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= -5.5e-26)
		tmp = Float64(1.0 + Float64(-1.0 / beta));
	elseif (alpha <= 1.95)
		tmp = Float64(0.5 + Float64(alpha * -0.25));
	else
		tmp = Float64(Float64(1.0 / alpha) + Float64(beta / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= -5.5e-26)
		tmp = 1.0 + (-1.0 / beta);
	elseif (alpha <= 1.95)
		tmp = 0.5 + (alpha * -0.25);
	else
		tmp = (1.0 / alpha) + (beta / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, -5.5e-26], N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision], If[LessEqual[alpha, 1.95], N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / alpha), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -5.5000000000000005e-26

   1. Initial program 99.9%

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

   3. Taylor expanded in alpha around 0 92.2%

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

   4. Taylor expanded in beta around inf 73.0%

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

   if -5.5000000000000005e-26 < alpha < 1.94999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in alpha around -inf 90.8%

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

      1. mul-1-neg 90.8%

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

      2. *-commutative 90.8%

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

      3. distribute-rgt-neg-in 90.8%

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

      4. sub-neg 90.8%

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

      5. metadata-eval 90.8%

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

      6. +-commutative 90.8%

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

      7. associate-*r/ 90.8%

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

      8. distribute-lft-in 90.8%

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

      9. metadata-eval 90.8%

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

      10. mul-1-neg 90.8%

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

      11. unsub-neg 90.8%

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

   5. Simplified 90.8%

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

   6. Taylor expanded in beta around 0 73.9%

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

      1. associate-*r/ 73.9%

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

      2. metadata-eval 73.9%

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

   8. Simplified 73.9%

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

   9. Taylor expanded in alpha around 0 73.4%

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

       1. *-commutative 73.4%

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

   11. Simplified 73.4%

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

   if 1.94999999999999996 < alpha

   1. Initial program 22.1%

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

   3. Taylor expanded in alpha around inf 85.0%

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

   4. Taylor expanded in beta around 0 85.0%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -5.5 \cdot 10^{-26}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \mathbf{elif}\;\alpha \leq 1.95:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1700:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1700.0)
   (/ (+ 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 <= 1700.0) {
		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 <= 1700.0d0) 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 <= 1700.0) {
		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 <= 1700.0:
		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 <= 1700.0)
		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 <= 1700.0)
		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, 1700.0], 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 < 1700

   1. Initial program 100.0%

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

   3. Taylor expanded in alpha around 0 98.0%

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

   if 1700 < alpha

   1. Initial program 20.2%

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

   3. Taylor expanded in alpha around inf 86.9%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1700:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1700:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1700.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (+ (/ 1.0 alpha) (/ beta alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1700.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (1.0 / alpha) + (beta / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1700.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (1.0d0 / alpha) + (beta / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1700.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (1.0 / alpha) + (beta / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1700.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (1.0 / alpha) + (beta / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1700.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(1.0 / alpha) + Float64(beta / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1700.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (1.0 / alpha) + (beta / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1700.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 / alpha), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1700

   1. Initial program 100.0%

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

   3. Taylor expanded in alpha around 0 98.0%

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

   if 1700 < alpha

   1. Initial program 20.2%

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

   3. Taylor expanded in alpha around inf 86.9%

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

   4. Taylor expanded in beta around 0 86.8%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1700:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.5 + \beta \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0) (+ 0.5 (* beta 0.25)) (+ 1.0 (/ -1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5 + (beta * 0.25);
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.0d0) then
        tmp = 0.5d0 + (beta * 0.25d0)
    else
        tmp = 1.0d0 + ((-1.0d0) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5 + (beta * 0.25);
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.5 + (beta * 0.25)
	else:
		tmp = 1.0 + (-1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(0.5 + Float64(beta * 0.25));
	else
		tmp = Float64(1.0 + Float64(-1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.5 + (beta * 0.25);
	else
		tmp = 1.0 + (-1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(0.5 + N[(beta * 0.25), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 72.6%

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

   3. Taylor expanded in alpha around 0 68.8%

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

   4. Taylor expanded in beta around 0 67.8%

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

      1. *-commutative 67.8%

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

   6. Simplified 67.8%

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

   if 2 < beta

   1. Initial program 79.7%

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

   3. Taylor expanded in alpha around 0 79.6%

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

   4. Taylor expanded in beta around inf 78.8%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.5 + \beta \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.5 + \beta \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0) (+ 0.5 (* beta 0.25)) 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5 + (beta * 0.25);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.0d0) then
        tmp = 0.5d0 + (beta * 0.25d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5 + (beta * 0.25);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.5 + (beta * 0.25)
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(0.5 + Float64(beta * 0.25));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.5 + (beta * 0.25);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(0.5 + N[(beta * 0.25), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 72.6%

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

   3. Taylor expanded in alpha around 0 68.8%

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

   4. Taylor expanded in beta around 0 67.8%

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

      1. *-commutative 67.8%

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

   6. Simplified 67.8%

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

   if 2 < beta

   1. Initial program 79.7%

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

      1. div-inv 79.9%

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

      2. fma-define 79.9%

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

      3. +-commutative 79.9%

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

      4. associate-+l+ 79.9%

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

   4. Applied egg-rr 79.9%

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

   5. Taylor expanded in beta around inf 76.5%

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

Alternative 10: 69.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2050:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 2050.0) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2050.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2050.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2050.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2050.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2050.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2050.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2050.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2050

   1. Initial program 71.4%

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

   3. Taylor expanded in alpha around 0 67.7%

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

   4. Taylor expanded in beta around 0 65.8%

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

   if 2050 < beta

   1. Initial program 82.4%

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

      1. div-inv 82.5%

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

      2. fma-define 82.5%

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

      3. +-commutative 82.5%

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

      4. associate-+l+ 82.5%

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

   4. Applied egg-rr 82.5%

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

   5. Taylor expanded in beta around inf 79.0%

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

Alternative 11: 48.3% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.5

Julia

function code(alpha, beta)
	return 0.5
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.5

Derivation

1. Initial program 75.0%

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

3. Taylor expanded in alpha around 0 72.5%

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

4. Taylor expanded in beta around 0 49.6%

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

Reproduce

herbie shell --seed 2024170 
(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))