Octave 3.8, jcobi/1

Percentage Accurate: 74.9% → 99.9%

Time: 7.1s

Alternatives: 10

Speedup: 0.7×

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: 74.9% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision], -0.99999], N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] * N[(N[(2.0 * beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] + N[(2.0 * beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 - N[(N[(alpha - beta), $MachinePrecision] / t$95$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 7.2%

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

   3. Taylor expanded in alpha around -inf

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

   5. Applied rewrites 100.0%

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

   if -0.999990000000000046 < (/.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 99.8%

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

Alternative 2: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{t\_0} \leq -0.999999995:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(2, \beta, 2\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{t\_0}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= (/ (- beta alpha) t_0) -0.999999995)
     (* 0.5 (/ (fma 2.0 beta 2.0) alpha))
     (/ (- 1.0 (/ (- alpha beta) t_0)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (((beta - alpha) / t_0) <= -0.999999995) {
		tmp = 0.5 * (fma(2.0, beta, 2.0) / alpha);
	} else {
		tmp = (1.0 - ((alpha - beta) / t_0)) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / t_0) <= -0.999999995)
		tmp = Float64(0.5 * Float64(fma(2.0, beta, 2.0) / alpha));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(alpha - beta) / t_0)) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision], -0.999999995], N[(0.5 * N[(N[(2.0 * beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(alpha - beta), $MachinePrecision] / t$95$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))) < -0.99999999500000003

   1. Initial program 6.4%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. div-add N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. div-add N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 99.5

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(2, \beta, 2\right)}}{\alpha} \cdot 0.5 \]

   8. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(2, \beta, 2\right)}{\alpha} \cdot 0.5} \]

   if -0.99999999500000003 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)} \leq -0.999999995:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(2, \beta, 2\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{2 + \left(\alpha + \beta\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.5

   1. Initial program 10.4%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 5.8%

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

   6. Taylor expanded in alpha around inf

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

      1. *-commutative N/A

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

      2. div-add N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. div-add N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 96.4

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(2, \beta, 2\right)}}{\alpha} \cdot 0.5 \]

   8. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(2, \beta, 2\right)}{\alpha} \cdot 0.5} \]

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   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

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

   5. Applied rewrites 50.7%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 98.6

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

   8. Applied rewrites 98.6%

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

   10. Applied rewrites 98.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\beta}{\beta + 2}, 0.5, 0.5\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 98.6%

       \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\frac{0.5}{2 + \beta}}, 0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

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

Alternative 4: 71.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.5

   1. Initial program 63.7%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 13.5%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 60.6

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

   8. Applied rewrites 60.6%

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

   9. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 60.2%

       \[\leadsto 0.5 \]

   if 0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   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

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

   5. Applied rewrites 95.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval 95.7

         \[\leadsto 2 \cdot \color{blue}{0.5} \]

   7. Applied rewrites 95.7%

      \[\leadsto \color{blue}{2 \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

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

Alternative 5: 73.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.2 \cdot 10^{-74}:\\ \;\;\;\;\left(1 - \frac{\alpha}{2 + \alpha}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\beta, \frac{0.5}{2 + \beta}, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.2e-74)
   (* (- 1.0 (/ alpha (+ 2.0 alpha))) 0.5)
   (fma beta (/ 0.5 (+ 2.0 beta)) 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.2e-74) {
		tmp = (1.0 - (alpha / (2.0 + alpha))) * 0.5;
	} else {
		tmp = fma(beta, (0.5 / (2.0 + beta)), 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.2e-74)
		tmp = Float64(Float64(1.0 - Float64(alpha / Float64(2.0 + alpha))) * 0.5);
	else
		tmp = fma(beta, Float64(0.5 / Float64(2.0 + beta)), 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 7.2e-74], N[(N[(1.0 - N[(alpha / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(beta * N[(0.5 / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.2000000000000005e-74

   1. Initial program 70.2%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 14.3%

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

   6. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 69.7

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

   8. Applied rewrites 69.7%

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

   if 7.2000000000000005e-74 < beta

   1. Initial program 80.4%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 79.4

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

   8. Applied rewrites 79.4%

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

   10. Applied rewrites 79.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\beta}{\beta + 2}, 0.5, 0.5\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 79.4%

       \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\frac{0.5}{2 + \beta}}, 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 72.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0) (fma (fma -0.125 beta 0.25) beta 0.5) (- 1.0 (/ 1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	else
		tmp = Float64(1.0 - Float64(1.0 / beta));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(N[(-0.125 * beta + 0.25), $MachinePrecision] * beta + 0.5), $MachinePrecision], N[(1.0 - N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 68.8%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 14.1%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 65.9

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

   8. Applied rewrites 65.9%

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

   9. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\beta \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot \beta\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 65.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \color{blue}{\beta}, 0.5\right) \]

   if 2 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 80.2

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

   8. Applied rewrites 80.2%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 80.7%

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

Alternative 7: 72.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\beta, \frac{0.5}{2 + \beta}, 0.5\right) \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (fma beta (/ 0.5 (+ 2.0 beta)) 0.5))

C

double code(double alpha, double beta) {
	return fma(beta, (0.5 / (2.0 + beta)), 0.5);
}

Julia

function code(alpha, beta)
	return fma(beta, Float64(0.5 / Float64(2.0 + beta)), 0.5)
end

Wolfram

code[alpha_, beta_] := N[(beta * N[(0.5 / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 74.4%

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

3. Taylor expanded in beta around inf

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

5. Applied rewrites 37.9%

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

6. Taylor expanded in alpha around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 72.1

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

8. Applied rewrites 72.1%

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

10. Applied rewrites 72.1%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\beta}{\beta + 2}, 0.5, 0.5\right)} \]
11. Step-by-step derivation

12. Applied rewrites 72.1%

    \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\frac{0.5}{2 + \beta}}, 0.5\right) \]
13. Add Preprocessing

Alternative 8: 72.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0) (fma (fma -0.125 beta 0.25) beta 0.5) (* 2.0 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	} else {
		tmp = 2.0 * 0.5;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	else
		tmp = Float64(2.0 * 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(N[(-0.125 * beta + 0.25), $MachinePrecision] * beta + 0.5), $MachinePrecision], N[(2.0 * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 68.8%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 14.1%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 65.9

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

   8. Applied rewrites 65.9%

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

   9. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\beta \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot \beta\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 65.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \color{blue}{\beta}, 0.5\right) \]

   if 2 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 80.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval 80.2

         \[\leadsto 2 \cdot \color{blue}{0.5} \]

   7. Applied rewrites 80.2%

      \[\leadsto \color{blue}{2 \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 72.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\mathsf{fma}\left(0.25, \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 68.8%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 14.1%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 65.9

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

   8. Applied rewrites 65.9%

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

   9. Taylor expanded in beta around 0

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

   11. Applied rewrites 65.9%

       \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{\beta}, 0.5\right) \]

   if 2 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 80.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval 80.2

         \[\leadsto 2 \cdot \color{blue}{0.5} \]

   7. Applied rewrites 80.2%

      \[\leadsto \color{blue}{2 \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 49.8% accurate, 35.0× 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 74.4%

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

3. Taylor expanded in beta around inf

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

5. Applied rewrites 37.9%

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

6. Taylor expanded in alpha around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 72.1

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

8. Applied rewrites 72.1%

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

9. Taylor expanded in beta around 0

   \[\leadsto \frac{1}{2} \]
10. Step-by-step derivation

11. Applied rewrites 47.9%

    \[\leadsto 0.5 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024312 
(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))