Octave 3.8, jcobi/1

Percentage Accurate: 75.0% → 99.9%

Time: 8.8s

Alternatives: 9

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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{1 + \left(\beta + 0.5 \cdot \left(\left(\left(-2 - \beta\right) - \beta\right) \cdot \frac{\beta + 2}{\alpha}\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t\_0} - \left(\frac{\alpha}{t\_0} + -1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99999)
     (/
      (+
       1.0
       (+ beta (* 0.5 (* (- (- -2.0 beta) beta) (/ (+ beta 2.0) alpha)))))
      alpha)
     (/ (- (/ beta t_0) (+ (/ alpha t_0) -1.0)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999) {
		tmp = (1.0 + (beta + (0.5 * (((-2.0 - beta) - beta) * ((beta + 2.0) / alpha))))) / alpha;
	} else {
		tmp = ((beta / t_0) - ((alpha / 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 + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.99999d0)) then
        tmp = (1.0d0 + (beta + (0.5d0 * ((((-2.0d0) - beta) - beta) * ((beta + 2.0d0) / alpha))))) / alpha
    else
        tmp = ((beta / t_0) - ((alpha / 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 + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999) {
		tmp = (1.0 + (beta + (0.5 * (((-2.0 - beta) - beta) * ((beta + 2.0) / alpha))))) / alpha;
	} else {
		tmp = ((beta / t_0) - ((alpha / t_0) + -1.0)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999:
		tmp = (1.0 + (beta + (0.5 * (((-2.0 - beta) - beta) * ((beta + 2.0) / alpha))))) / alpha
	else:
		tmp = ((beta / t_0) - ((alpha / t_0) + -1.0)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99999)
		tmp = Float64(Float64(1.0 + Float64(beta + Float64(0.5 * Float64(Float64(Float64(-2.0 - beta) - beta) * Float64(Float64(beta + 2.0) / alpha))))) / alpha);
	else
		tmp = Float64(Float64(Float64(beta / t_0) - Float64(Float64(alpha / t_0) + -1.0)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999)
		tmp = (1.0 + (beta + (0.5 * (((-2.0 - beta) - beta) * ((beta + 2.0) / alpha))))) / alpha;
	else
		tmp = ((beta / t_0) - ((alpha / t_0) + -1.0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99999], N[(N[(1.0 + N[(beta + N[(0.5 * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] - N[(N[(alpha / t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $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))) < -0.999990000000000046

   1. Initial program 7.1%

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

   3. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right) + \frac{1}{2} \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha}\right), \color{blue}{\alpha}\right) \]

   5. Simplified 99.8%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) + 1\right), 2\right) \]

      2. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right), 2\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right), \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right), 2\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\beta, \left(\left(\alpha + \beta\right) + 2\right)\right), \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right), 2\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\beta, \left(\left(\beta + \alpha\right) + 2\right)\right), \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right), 2\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\beta, \left(\beta + \left(\alpha + 2\right)\right)\right), \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right), 2\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right), 2\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right), 2\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + \left(\mathsf{neg}\left(1\right)\right)\right)\right), 2\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right), 2\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\alpha, \left(\left(\alpha + \beta\right) + 2\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right), 2\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\alpha, \left(\left(\beta + \alpha\right) + 2\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right), 2\right) \]

      13. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\alpha, \left(\beta + \left(\alpha + 2\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right), 2\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right), 2\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right), 2\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), -1\right)\right), 2\right) \]

   4. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{1 + \left(\beta + 0.5 \cdot \left(\left(\left(-2 - \beta\right) - \beta\right) \cdot \frac{\beta + 2}{\alpha}\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} + -1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.99999:\\ \;\;\;\;\frac{1 + \left(\beta + 0.5 \cdot \left(\left(\left(-2 - \beta\right) - \beta\right) \cdot \frac{\beta + 2}{\alpha}\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.99999)
     (/
      (+
       1.0
       (+ beta (* 0.5 (* (- (- -2.0 beta) beta) (/ (+ beta 2.0) 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 <= -0.99999) {
		tmp = (1.0 + (beta + (0.5 * (((-2.0 - beta) - beta) * ((beta + 2.0) / 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 <= (-0.99999d0)) then
        tmp = (1.0d0 + (beta + (0.5d0 * ((((-2.0d0) - beta) - beta) * ((beta + 2.0d0) / 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 <= -0.99999) {
		tmp = (1.0 + (beta + (0.5 * (((-2.0 - beta) - beta) * ((beta + 2.0) / 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 <= -0.99999:
		tmp = (1.0 + (beta + (0.5 * (((-2.0 - beta) - beta) * ((beta + 2.0) / 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 <= -0.99999)
		tmp = Float64(Float64(1.0 + Float64(beta + Float64(0.5 * Float64(Float64(Float64(-2.0 - beta) - beta) * Float64(Float64(beta + 2.0) / 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 <= -0.99999)
		tmp = (1.0 + (beta + (0.5 * (((-2.0 - beta) - beta) * ((beta + 2.0) / 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, -0.99999], N[(N[(1.0 + N[(beta + N[(0.5 * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $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))) < -0.999990000000000046

   1. Initial program 7.1%

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

   3. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right) + \frac{1}{2} \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha}\right), \color{blue}{\alpha}\right) \]

   5. Simplified 99.8%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{1 + \left(\beta + 0.5 \cdot \left(\left(\left(-2 - \beta\right) - \beta\right) \cdot \frac{\beta + 2}{\alpha}\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.9999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9999998) (/ (+ beta 1.0) 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 <= -0.9999998) {
		tmp = (beta + 1.0) / 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 <= (-0.9999998d0)) then
        tmp = (beta + 1.0d0) / 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 <= -0.9999998) {
		tmp = (beta + 1.0) / 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 <= -0.9999998:
		tmp = (beta + 1.0) / 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 <= -0.9999998)
		tmp = Float64(Float64(beta + 1.0) / 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 <= -0.9999998)
		tmp = (beta + 1.0) / 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, -0.9999998], N[(N[(beta + 1.0), $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))) < -0.999999799999999994

   1. Initial program 6.2%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right)\right), \color{blue}{\alpha}\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \alpha\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \alpha\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(\frac{1}{2} \cdot 2\right) \cdot \beta\right), \alpha\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + 1 \cdot \beta\right), \alpha\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \alpha\right) \]

      8. +-lowering-+.f64 99.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \alpha\right) \]

   5. Simplified 99.5%

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

   if -0.999999799999999994 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.25 \cdot 10^{-53}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq 980000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.25e-53)
   (+ 0.5 (* alpha -0.25))
   (if (<= alpha 980000.0) 1.0 (/ (+ beta 1.0) alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.25e-53) {
		tmp = 0.5 + (alpha * -0.25);
	} else if (alpha <= 980000.0) {
		tmp = 1.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.25d-53) then
        tmp = 0.5d0 + (alpha * (-0.25d0))
    else if (alpha <= 980000.0d0) then
        tmp = 1.0d0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.25e-53) {
		tmp = 0.5 + (alpha * -0.25);
	} else if (alpha <= 980000.0) {
		tmp = 1.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.25e-53:
		tmp = 0.5 + (alpha * -0.25)
	elif alpha <= 980000.0:
		tmp = 1.0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.25e-53)
		tmp = Float64(0.5 + Float64(alpha * -0.25));
	elseif (alpha <= 980000.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.25e-53)
		tmp = 0.5 + (alpha * -0.25);
	elseif (alpha <= 980000.0)
		tmp = 1.0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.25e-53], N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[alpha, 980000.0], 1.0, N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 1.25e-53

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 - \frac{\alpha}{2 + \alpha}\right)}, 2\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\alpha}{2 + \alpha}\right)\right), 2\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \left(2 + \alpha\right)\right)\right), 2\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \left(\alpha + 2\right)\right)\right), 2\right) \]

      4. +-lowering-+.f64 76.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 2\right) \]

   5. Simplified 76.7%

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

   6. Taylor expanded in alpha around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{4} \cdot \alpha\right)}\right) \]

      2. *-lowering-*.f64 76.7%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \color{blue}{\alpha}\right)\right) \]

   8. Simplified 76.7%

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

   if 1.25e-53 < alpha < 9.8e5

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 69.9%

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

   if 9.8e5 < alpha

   1. Initial program 20.1%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right)\right), \color{blue}{\alpha}\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \alpha\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \alpha\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(\frac{1}{2} \cdot 2\right) \cdot \beta\right), \alpha\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + 1 \cdot \beta\right), \alpha\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \alpha\right) \]

      8. +-lowering-+.f64 86.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \alpha\right) \]

   5. Simplified 86.2%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.25 \cdot 10^{-53}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq 980000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2600000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2600000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2600000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 2600000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2600000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2600000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2600000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2600000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2600000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.6e6

   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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 + \beta\right)\right), 1\right), 2\right) \]

      2. +-lowering-+.f64 99.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right), 1\right), 2\right) \]

   5. Simplified 99.2%

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

   if 2.6e6 < alpha

   1. Initial program 20.1%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right)\right), \color{blue}{\alpha}\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \alpha\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \alpha\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(\frac{1}{2} \cdot 2\right) \cdot \beta\right), \alpha\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + 1 \cdot \beta\right), \alpha\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \alpha\right) \]

      8. +-lowering-+.f64 86.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \alpha\right) \]

   5. Simplified 86.2%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2600000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.25 \cdot 10^{-53}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq 75000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.25e-53)
   (+ 0.5 (* alpha -0.25))
   (if (<= alpha 75000.0) 1.0 (/ 1.0 alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.25e-53) {
		tmp = 0.5 + (alpha * -0.25);
	} else if (alpha <= 75000.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.25d-53) then
        tmp = 0.5d0 + (alpha * (-0.25d0))
    else if (alpha <= 75000.0d0) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.25e-53) {
		tmp = 0.5 + (alpha * -0.25);
	} else if (alpha <= 75000.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.25e-53:
		tmp = 0.5 + (alpha * -0.25)
	elif alpha <= 75000.0:
		tmp = 1.0
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.25e-53)
		tmp = Float64(0.5 + Float64(alpha * -0.25));
	elseif (alpha <= 75000.0)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.25e-53)
		tmp = 0.5 + (alpha * -0.25);
	elseif (alpha <= 75000.0)
		tmp = 1.0;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.25e-53], N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[alpha, 75000.0], 1.0, N[(1.0 / alpha), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 1.25e-53

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 - \frac{\alpha}{2 + \alpha}\right)}, 2\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\alpha}{2 + \alpha}\right)\right), 2\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \left(2 + \alpha\right)\right)\right), 2\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \left(\alpha + 2\right)\right)\right), 2\right) \]

      4. +-lowering-+.f64 76.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\alpha, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 2\right) \]

   5. Simplified 76.7%

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

   6. Taylor expanded in alpha around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{4} \cdot \alpha\right)}\right) \]

      2. *-lowering-*.f64 76.7%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \color{blue}{\alpha}\right)\right) \]

   8. Simplified 76.7%

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

   if 1.25e-53 < alpha < 75000

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 69.9%

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

   if 75000 < alpha

   1. Initial program 20.1%

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

   3. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right) + \frac{1}{2} \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha}\right), \color{blue}{\alpha}\right) \]

   5. Simplified 86.6%

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

   6. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 - 2 \cdot \frac{1}{\alpha}\right), \color{blue}{\alpha}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(2 \cdot \frac{1}{\alpha}\right)\right)\right), \alpha\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(2 \cdot \frac{1}{\alpha}\right)\right)\right), \alpha\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{2 \cdot 1}{\alpha}\right)\right)\right), \alpha\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{2}{\alpha}\right)\right)\right), \alpha\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(2\right)}{\alpha}\right)\right), \alpha\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-2}{\alpha}\right)\right), \alpha\right) \]

      8. /-lowering-/.f64 75.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-2, \alpha\right)\right), \alpha\right) \]

   8. Simplified 75.0%

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

   9. Taylor expanded in alpha around inf

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

       1. /-lowering-/.f64 74.5%

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\alpha}\right) \]

   11. Simplified 74.5%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.25 \cdot 10^{-53}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq 75000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.5 \cdot 10^{-54}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 10500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.5e-54) 0.5 (if (<= alpha 10500.0) 1.0 (/ 1.0 alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.5e-54) {
		tmp = 0.5;
	} else if (alpha <= 10500.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 2.5d-54) then
        tmp = 0.5d0
    else if (alpha <= 10500.0d0) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.5e-54) {
		tmp = 0.5;
	} else if (alpha <= 10500.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2.5e-54:
		tmp = 0.5
	elif alpha <= 10500.0:
		tmp = 1.0
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.5e-54)
		tmp = 0.5;
	elseif (alpha <= 10500.0)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2.5e-54)
		tmp = 0.5;
	elseif (alpha <= 10500.0)
		tmp = 1.0;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.5e-54], 0.5, If[LessEqual[alpha, 10500.0], 1.0, N[(1.0 / alpha), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 2.50000000000000008e-54

   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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 + \beta\right)\right), 1\right), 2\right) \]

      2. +-lowering-+.f64 99.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right), 1\right), 2\right) \]

   5. Simplified 99.8%

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

   6. Taylor expanded in beta around 0

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

   8. Simplified 76.5%

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

   if 2.50000000000000008e-54 < alpha < 10500

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 69.9%

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

   if 10500 < alpha

   1. Initial program 20.1%

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

   3. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right) + \frac{1}{2} \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha}\right), \color{blue}{\alpha}\right) \]

   5. Simplified 86.6%

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

   6. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 - 2 \cdot \frac{1}{\alpha}\right), \color{blue}{\alpha}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(2 \cdot \frac{1}{\alpha}\right)\right)\right), \alpha\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(2 \cdot \frac{1}{\alpha}\right)\right)\right), \alpha\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{2 \cdot 1}{\alpha}\right)\right)\right), \alpha\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{2}{\alpha}\right)\right)\right), \alpha\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(2\right)}{\alpha}\right)\right), \alpha\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-2}{\alpha}\right)\right), \alpha\right) \]

      8. /-lowering-/.f64 75.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-2, \alpha\right)\right), \alpha\right) \]

   8. Simplified 75.0%

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

   9. Taylor expanded in alpha around inf

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

       1. /-lowering-/.f64 74.5%

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\alpha}\right) \]

   11. Simplified 74.5%

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

Alternative 8: 71.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 70.1%

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

   3. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 + \beta\right)\right), 1\right), 2\right) \]

      2. +-lowering-+.f64 68.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right), 1\right), 2\right) \]

   5. Simplified 68.8%

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

   6. Taylor expanded in beta around 0

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

   8. Simplified 67.6%

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

   if 2 < beta

   1. Initial program 89.2%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 82.7%

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

Alternative 9: 49.3% 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 76.0%

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

3. Taylor expanded in alpha around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 + \beta\right)\right), 1\right), 2\right) \]

   2. +-lowering-+.f64 74.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right), 1\right), 2\right) \]

5. Simplified 74.4%

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

6. Taylor expanded in beta around 0

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

8. Simplified 52.3%

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

Reproduce

herbie shell --seed 2024140 
(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))