Octave 3.8, jcobi/1

Percentage Accurate: 75.0% → 99.8%

Time: 10.4s

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: 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: 99.8% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9995)
     (/
      (+
       (+ beta 1.0)
       (* 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)) <= -0.9995) {
		tmp = ((beta + 1.0) + (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)) <= (-0.9995d0)) then
        tmp = ((beta + 1.0d0) + (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)) <= -0.9995) {
		tmp = ((beta + 1.0) + (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)) <= -0.9995:
		tmp = ((beta + 1.0) + (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)) <= -0.9995)
		tmp = Float64(Float64(Float64(beta + 1.0) + 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)) <= -0.9995)
		tmp = ((beta + 1.0) + (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], -0.9995], N[(N[(N[(beta + 1.0), $MachinePrecision] + N[(0.5 * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $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))) < -0.99950000000000006

   1. Initial program 6.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 99.7%

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

   if -0.99950000000000006 < (/.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 99.8%

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

Alternative 2: 99.8% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9995)
     (/
      (+
       (+ beta 1.0)
       (* 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 <= -0.9995) {
		tmp = ((beta + 1.0) + (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 <= (-0.9995d0)) then
        tmp = ((beta + 1.0d0) + (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 <= -0.9995) {
		tmp = ((beta + 1.0) + (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 <= -0.9995:
		tmp = ((beta + 1.0) + (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 <= -0.9995)
		tmp = Float64(Float64(Float64(beta + 1.0) + 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 <= -0.9995)
		tmp = ((beta + 1.0) + (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, -0.9995], N[(N[(N[(beta + 1.0), $MachinePrecision] + N[(0.5 * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] * N[(N[(beta + 2.0), $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 6.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 99.7%

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

   if -0.99950000000000006 < (/.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%

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

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.999999:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.999999)
     (+ (/ beta alpha) (/ 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 <= -0.999999) {
		tmp = (beta / alpha) + (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 <= (-0.999999d0)) then
        tmp = (beta / alpha) + (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 <= -0.999999) {
		tmp = (beta / alpha) + (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 <= -0.999999:
		tmp = (beta / alpha) + (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 <= -0.999999)
		tmp = Float64(Float64(beta / alpha) + Float64(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 <= -0.999999)
		tmp = (beta / alpha) + (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, -0.999999], N[(N[(beta / alpha), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $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))) < -0.999998999999999971

   1. Initial program 6.0%

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

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. /-lowering-/.f64 99.6%

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

   8. Simplified 99.6%

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

   if -0.999998999999999971 < (/.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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 8 \cdot 10^{+27}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 8e+27)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (+ (/ beta alpha) (/ 1.0 alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 8e+27) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta / alpha) + (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 <= 8d+27) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (beta / alpha) + (1.0d0 / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 8e+27) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta / alpha) + (1.0 / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 8e+27:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (beta / alpha) + (1.0 / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 8e+27)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(beta / alpha) + Float64(1.0 / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 8e+27)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (beta / alpha) + (1.0 / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 8e+27], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta / alpha), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 8.0000000000000001e27

   1. Initial program 98.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 97.8%

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

   5. Simplified 97.8%

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

   if 8.0000000000000001e27 < alpha

   1. Initial program 19.7%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 86.0%

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

   5. Simplified 86.0%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. /-lowering-/.f64 86.0%

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

   8. Simplified 86.0%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 8 \cdot 10^{+27}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.9:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.9) (+ 0.5 (* alpha -0.25)) (+ (/ beta alpha) (/ 1.0 alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.9) {
		tmp = 0.5 + (alpha * -0.25);
	} else {
		tmp = (beta / alpha) + (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 <= 1.9d0) then
        tmp = 0.5d0 + (alpha * (-0.25d0))
    else
        tmp = (beta / alpha) + (1.0d0 / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.9) {
		tmp = 0.5 + (alpha * -0.25);
	} else {
		tmp = (beta / alpha) + (1.0 / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.9:
		tmp = 0.5 + (alpha * -0.25)
	else:
		tmp = (beta / alpha) + (1.0 / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.9)
		tmp = Float64(0.5 + Float64(alpha * -0.25));
	else
		tmp = Float64(Float64(beta / alpha) + Float64(1.0 / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.9)
		tmp = 0.5 + (alpha * -0.25);
	else
		tmp = (beta / alpha) + (1.0 / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.9], N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision], N[(N[(beta / alpha), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.8999999999999999

   1. Initial program 100.0%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. metadata-eval N/A

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

      4. *-rgt-identity N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 74.3%

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

   5. Taylor expanded in beta around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

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

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

      8. +-commutative N/A

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

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

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

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

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

      11. +-commutative N/A

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

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

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

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

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

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

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 74.7%

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

   7. Simplified 74.7%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{4} \cdot \alpha} \]
   9. 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. *-commutative N/A

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

      3. *-lowering-*.f64 74.7%

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

   10. Simplified 74.7%

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

   if 1.8999999999999999 < alpha

   1. Initial program 25.4%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 80.8%

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

   5. Simplified 80.8%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. /-lowering-/.f64 80.8%

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

   8. Simplified 80.8%

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

Alternative 6: 74.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.95) {
		tmp = 0.5 + (alpha * -0.25);
	} 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 <= 1.95d0) then
        tmp = 0.5d0 + (alpha * (-0.25d0))
    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 <= 1.95) {
		tmp = 0.5 + (alpha * -0.25);
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if alpha < 1.94999999999999996

   1. Initial program 100.0%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. metadata-eval N/A

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

      4. *-rgt-identity N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 74.3%

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

   5. Taylor expanded in beta around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

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

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

      8. +-commutative N/A

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

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

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

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

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

      11. +-commutative N/A

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

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

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

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

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

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

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 74.7%

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

   7. Simplified 74.7%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{4} \cdot \alpha} \]
   9. 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. *-commutative N/A

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

      3. *-lowering-*.f64 74.7%

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

   10. Simplified 74.7%

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

   if 1.94999999999999996 < alpha

   1. Initial program 25.4%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 80.8%

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

   5. Simplified 80.8%

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

4. Final simplification 76.9%

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

Alternative 7: 72.2% 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 69.5%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 68.1%

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

   5. Simplified 68.1%

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

   6. Taylor expanded in beta around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 66.3%

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

   8. Simplified 66.3%

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

   if 2 < beta

   1. Initial program 79.5%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 78.8%

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

   5. Simplified 78.8%

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

   6. Taylor expanded in beta around inf

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

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

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{\beta}\right)}\right) \]

      2. /-lowering-/.f64 78.6%

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

   8. Simplified 78.6%

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

4. Final simplification 70.2%

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

Alternative 8: 68.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 0.8:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 0.8) (+ 0.5 (* alpha -0.25)) (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.8) {
		tmp = 0.5 + (alpha * -0.25);
	} 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 <= 0.8d0) then
        tmp = 0.5d0 + (alpha * (-0.25d0))
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.8) {
		tmp = 0.5 + (alpha * -0.25);
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 0.8:
		tmp = 0.5 + (alpha * -0.25)
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 0.8)
		tmp = Float64(0.5 + Float64(alpha * -0.25));
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 0.8)
		tmp = 0.5 + (alpha * -0.25);
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 0.8], N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision], N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 0.80000000000000004

   1. Initial program 100.0%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. metadata-eval N/A

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

      4. *-rgt-identity N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 74.3%

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

   5. Taylor expanded in beta around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

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

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

      8. +-commutative N/A

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

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

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

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

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

      11. +-commutative N/A

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

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

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

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

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

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

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 74.7%

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

   7. Simplified 74.7%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{4} \cdot \alpha} \]
   9. 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. *-commutative N/A

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

      3. *-lowering-*.f64 74.7%

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

   10. Simplified 74.7%

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

   if 0.80000000000000004 < alpha

   1. Initial program 25.4%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 80.8%

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

   5. Simplified 80.8%

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

   6. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 61.8%

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

   8. Simplified 61.8%

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

Alternative 9: 68.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.95:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.95) 0.5 (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.95) {
		tmp = 0.5;
	} 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 <= 1.95d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.95) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.95:
		tmp = 0.5
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.95)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.95)
		tmp = 0.5;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.95], 0.5, N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if 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 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 74.5%

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

   if 1.94999999999999996 < alpha

   1. Initial program 25.4%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 80.8%

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

   5. Simplified 80.8%

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

   6. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 61.8%

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

   8. Simplified 61.8%

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

Alternative 10: 71.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 10000.0) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 10000.0) {
		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 <= 10000.0d0) 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 <= 10000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 10000.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 10000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 10000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 10000.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 1e4

   1. Initial program 68.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 67.5%

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

   5. Simplified 67.5%

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

   6. Taylor expanded in beta around 0

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

   8. Simplified 65.0%

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

   if 1e4 < beta

   1. Initial program 81.4%

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

   3. Taylor expanded in beta around inf

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

   5. Simplified 78.9%

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

Alternative 11: 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 72.6%

   \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
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 71.5%

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

5. Simplified 71.5%

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

6. Taylor expanded in beta around 0

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

8. Simplified 50.1%

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

Reproduce

herbie shell --seed 2024158 
(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))