Octave 3.8, jcobi/1

Percentage Accurate: 73.7% → 99.7%

Time: 7.3s

Alternatives: 11

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: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.9999:\\ \;\;\;\;\frac{\frac{\left(\frac{\frac{8}{\alpha} - 4}{\alpha} + \beta\right) \cdot -1 - \left(2 + \beta\right)}{\alpha} \cdot -1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta - \alpha}{\left(2 + \left(\alpha + \beta\right)\right) \cdot 2} + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ alpha beta) 2.0)) -0.9999)
   (/
    (*
     (/
      (- (* (+ (/ (- (/ 8.0 alpha) 4.0) alpha) beta) -1.0) (+ 2.0 beta))
      alpha)
     -1.0)
    2.0)
   (+ (/ (- beta alpha) (* (+ 2.0 (+ alpha beta)) 2.0)) 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.9999) {
		tmp = ((((((((8.0 / alpha) - 4.0) / alpha) + beta) * -1.0) - (2.0 + beta)) / alpha) * -1.0) / 2.0;
	} else {
		tmp = ((beta - alpha) / ((2.0 + (alpha + beta)) * 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) / ((alpha + beta) + 2.0d0)) <= (-0.9999d0)) then
        tmp = ((((((((8.0d0 / alpha) - 4.0d0) / alpha) + beta) * (-1.0d0)) - (2.0d0 + beta)) / alpha) * (-1.0d0)) / 2.0d0
    else
        tmp = ((beta - alpha) / ((2.0d0 + (alpha + beta)) * 2.0d0)) + 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.9999) {
		tmp = ((((((((8.0 / alpha) - 4.0) / alpha) + beta) * -1.0) - (2.0 + beta)) / alpha) * -1.0) / 2.0;
	} else {
		tmp = ((beta - alpha) / ((2.0 + (alpha + beta)) * 2.0)) + 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.9999:
		tmp = ((((((((8.0 / alpha) - 4.0) / alpha) + beta) * -1.0) - (2.0 + beta)) / alpha) * -1.0) / 2.0
	else:
		tmp = ((beta - alpha) / ((2.0 + (alpha + beta)) * 2.0)) + 0.5
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) <= -0.9999)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(8.0 / alpha) - 4.0) / alpha) + beta) * -1.0) - Float64(2.0 + beta)) / alpha) * -1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta - alpha) / Float64(Float64(2.0 + Float64(alpha + beta)) * 2.0)) + 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.9999)
		tmp = ((((((((8.0 / alpha) - 4.0) / alpha) + beta) * -1.0) - (2.0 + beta)) / alpha) * -1.0) / 2.0;
	else
		tmp = ((beta - alpha) / ((2.0 + (alpha + beta)) * 2.0)) + 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999], N[(N[(N[(N[(N[(N[(N[(N[(N[(8.0 / alpha), $MachinePrecision] - 4.0), $MachinePrecision] / alpha), $MachinePrecision] + beta), $MachinePrecision] * -1.0), $MachinePrecision] - N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] * -1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * 2.0), $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))) < -0.99990000000000001

   1. Initial program 7.0%

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

   3. Taylor expanded in alpha around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(-1 \cdot \beta + -1 \cdot \frac{\left(\frac{\beta \cdot {\left(2 + \beta\right)}^{2}}{\alpha} + \frac{{\left(2 + \beta\right)}^{3}}{\alpha}\right) - \left(\beta \cdot \left(2 + \beta\right) + {\left(2 + \beta\right)}^{2}\right)}{\alpha}\right) - \left(2 + \beta\right)}{\alpha}}}{2} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 99.9%

      \[\leadsto \frac{\frac{\left(\frac{\frac{8}{\alpha} - 4}{\alpha} + \beta\right) \cdot -1 - \left(2 + \beta\right)}{\alpha} \cdot -1}{2} \]

   if -0.99990000000000001 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lower-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

Alternative 2: 97.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.99999:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, \beta, 1\right)}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha}{2 + \alpha}, -0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - {\beta}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))))
   (if (<= t_0 -0.99999)
     (/ (fma 1.0 beta 1.0) alpha)
     (if (<= t_0 0.01)
       (fma (/ alpha (+ 2.0 alpha)) -0.5 0.5)
       (- 1.0 (pow beta -1.0))))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((alpha + beta) + 2.0);
	double tmp;
	if (t_0 <= -0.99999) {
		tmp = fma(1.0, beta, 1.0) / alpha;
	} else if (t_0 <= 0.01) {
		tmp = fma((alpha / (2.0 + alpha)), -0.5, 0.5);
	} else {
		tmp = 1.0 - pow(beta, -1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.99999)
		tmp = Float64(fma(1.0, beta, 1.0) / alpha);
	elseif (t_0 <= 0.01)
		tmp = fma(Float64(alpha / Float64(2.0 + alpha)), -0.5, 0.5);
	else
		tmp = Float64(1.0 - (beta ^ -1.0));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.99999], N[(N[(1.0 * beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[(N[(alpha / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision], N[(1.0 - N[Power[beta, -1.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   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

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.0100000000000000002

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lower-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. metadata-eval 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 97.3

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

   7. Applied rewrites 97.3%

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

   if 0.0100000000000000002 < (/.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

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

      1. +-commutative N/A

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

      2. div-add N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. div-add-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 99.2%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, \beta, 1\right)}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha}{2 + \alpha}, -0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - {\beta}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, \beta, 1\right)}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - {\beta}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))))
   (if (<= t_0 -0.5)
     (/ (fma 1.0 beta 1.0) alpha)
     (if (<= t_0 0.01)
       (fma (fma -0.125 beta 0.25) beta 0.5)
       (- 1.0 (pow beta -1.0))))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((alpha + beta) + 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = fma(1.0, beta, 1.0) / alpha;
	} else if (t_0 <= 0.01) {
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	} else {
		tmp = 1.0 - pow(beta, -1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(fma(1.0, beta, 1.0) / alpha);
	elseif (t_0 <= 0.01)
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	else
		tmp = Float64(1.0 - (beta ^ -1.0));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(1.0 * beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[(N[(-0.125 * beta + 0.25), $MachinePrecision] * beta + 0.5), $MachinePrecision], N[(1.0 - N[Power[beta, -1.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 9.2%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 96.9

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

   5. Applied rewrites 96.9%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.0100000000000000002

   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

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 96.8%

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

   if 0.0100000000000000002 < (/.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

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

      1. +-commutative N/A

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

      2. div-add N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. div-add-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 99.2%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, \beta, 1\right)}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - {\beta}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.1% accurate, 0.3× speedup

Math

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

FPCore

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

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 - pow(beta, -1.0);
	}
	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 - (beta ^ -1.0));
	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[Power[beta, -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 70.4%

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

   3. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 66.6%

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

   if 2 < beta

   1. Initial program 82.8%

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

   3. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. div-add N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. div-add-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 82.0

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 82.5%

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

4. Final simplification 71.8%

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

Alternative 5: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.99999:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, \beta, 1\right)}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha}{2 + \alpha}, -0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{1 + \alpha}{\beta}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))))
   (if (<= t_0 -0.99999)
     (/ (fma 1.0 beta 1.0) alpha)
     (if (<= t_0 0.01)
       (fma (/ alpha (+ 2.0 alpha)) -0.5 0.5)
       (fma -1.0 (/ (+ 1.0 alpha) beta) 1.0)))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((alpha + beta) + 2.0);
	double tmp;
	if (t_0 <= -0.99999) {
		tmp = fma(1.0, beta, 1.0) / alpha;
	} else if (t_0 <= 0.01) {
		tmp = fma((alpha / (2.0 + alpha)), -0.5, 0.5);
	} else {
		tmp = fma(-1.0, ((1.0 + alpha) / beta), 1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.99999)
		tmp = Float64(fma(1.0, beta, 1.0) / alpha);
	elseif (t_0 <= 0.01)
		tmp = fma(Float64(alpha / Float64(2.0 + alpha)), -0.5, 0.5);
	else
		tmp = fma(-1.0, Float64(Float64(1.0 + alpha) / beta), 1.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.99999], N[(N[(1.0 * beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[(N[(alpha / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision], N[(-1.0 * N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   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

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.0100000000000000002

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lower-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. metadata-eval 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 97.3

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

   7. Applied rewrites 97.3%

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

   if 0.0100000000000000002 < (/.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

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

      1. +-commutative N/A

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

      2. div-add N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. div-add-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

Alternative 6: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99999], N[(N[(1.0 * beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] * 2.0), $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))) < -0.999990000000000046

   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

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 99.5

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

   5. Applied rewrites 99.5%

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

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lower-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

Alternative 7: 98.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(1.0 * beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(beta / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * 0.5 + 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 9.2%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 96.9

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

   5. Applied rewrites 96.9%

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

   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 alpha around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.5

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

   5. Applied rewrites 98.5%

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

Alternative 8: 70.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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 65.0%

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

   3. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 62.4

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 61.0%

      \[\leadsto 0.5 \]

   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

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

   5. Applied rewrites 98.8%

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

Alternative 9: 70.9% accurate, 1.8× speedup

Math

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

FPCore

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

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;
	}
	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 = 1.0;
	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], 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 70.4%

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

   3. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 66.6%

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

   if 2 < beta

   1. Initial program 82.8%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 82.3%

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

Alternative 10: 70.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 70.4%

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

   3. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 66.3%

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

   if 2 < beta

   1. Initial program 82.8%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 82.3%

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

Alternative 11: 36.2% accurate, 35.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 1.0

Julia

function code(alpha, beta)
	return 1.0
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 1.0

Derivation

1. Initial program 74.4%

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

3. Taylor expanded in beta around inf

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

5. Applied rewrites 36.7%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024313 
(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))