Octave 3.8, jcobi/1

Percentage Accurate: 74.5% → 99.9%

Time: 26.9s

Alternatives: 13

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: 74.5% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t\_0} \leq -0.99999:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, \beta, 6\right)}{\alpha} - 2, \beta, \frac{4}{\alpha} - 2\right) \cdot -0.5}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\beta - \alpha, \frac{0.5}{t\_0}, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= (/ (- beta alpha) t_0) -0.99999)
     (/
      (*
       (fma (- (/ (fma 2.0 beta 6.0) alpha) 2.0) beta (- (/ 4.0 alpha) 2.0))
       -0.5)
      alpha)
     (fma (- beta alpha) (/ 0.5 t_0) 0.5))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (((beta - alpha) / t_0) <= -0.99999) {
		tmp = (fma(((fma(2.0, beta, 6.0) / alpha) - 2.0), beta, ((4.0 / alpha) - 2.0)) * -0.5) / alpha;
	} else {
		tmp = fma((beta - alpha), (0.5 / t_0), 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / t_0) <= -0.99999)
		tmp = Float64(Float64(fma(Float64(Float64(fma(2.0, beta, 6.0) / alpha) - 2.0), beta, Float64(Float64(4.0 / alpha) - 2.0)) * -0.5) / alpha);
	else
		tmp = fma(Float64(beta - alpha), Float64(0.5 / t_0), 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision], -0.99999], N[(N[(N[(N[(N[(N[(2.0 * beta + 6.0), $MachinePrecision] / alpha), $MachinePrecision] - 2.0), $MachinePrecision] * beta + N[(N[(4.0 / alpha), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(beta - alpha), $MachinePrecision] * N[(0.5 / t$95$0), $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 7.4%

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

   3. Taylor expanded in alpha around -inf

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \frac{\frac{-1}{2} \cdot \left(\left(4 \cdot \frac{1}{\alpha} + \beta \cdot \left(\left(2 \cdot \frac{\beta}{\alpha} + 6 \cdot \frac{1}{\alpha}\right) - 2\right)\right) - 2\right)}{\alpha} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \frac{-0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\beta}{\alpha}, 2, \frac{6}{\alpha} - 2\right), \beta, \frac{4}{\alpha}\right) - 2\right)}{\alpha} \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, \beta, 6\right)}{\alpha} - 2, \beta, \frac{4}{\alpha} - 2\right) \cdot -0.5}{\color{blue}{\alpha}} \]

   if -0.999990000000000046 < (/.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. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

Alternative 2: 91.7% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))))
   (if (<= t_0 -0.5)
     (pow alpha -1.0)
     (if (<= t_0 0.5) (fma (fma -0.125 beta 0.25) beta 0.5) 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 = pow(alpha, -1.0);
	} else if (t_0 <= 0.5) {
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	} else {
		tmp = 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 = alpha ^ -1.0;
	elseif (t_0 <= 0.5)
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	else
		tmp = 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[Power[alpha, -1.0], $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(N[(-0.125 * beta + 0.25), $MachinePrecision] * beta + 0.5), $MachinePrecision], 1.0]]]

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 9.5%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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. *-lft-identity N/A

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

      8. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 81.9%

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

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

   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 98.1

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

   5. Applied rewrites 98.1%

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

   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 97.7%

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

   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 99.1%

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

4. Final simplification 93.5%

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

Alternative 3: 91.5% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))))
   (if (<= t_0 -0.5)
     (pow alpha -1.0)
     (if (<= t_0 0.5) (fma 0.25 beta 0.5) 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 = pow(alpha, -1.0);
	} else if (t_0 <= 0.5) {
		tmp = fma(0.25, beta, 0.5);
	} else {
		tmp = 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 = alpha ^ -1.0;
	elseif (t_0 <= 0.5)
		tmp = fma(0.25, beta, 0.5);
	else
		tmp = 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[Power[alpha, -1.0], $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(0.25 * beta + 0.5), $MachinePrecision], 1.0]]]

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 9.5%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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. *-lft-identity N/A

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

      8. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 81.9%

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

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

   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 98.1

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

   5. Applied rewrites 98.1%

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

   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 97.3%

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

   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 99.1%

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

4. Final simplification 93.4%

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

Alternative 4: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision], -0.99999], N[(N[(N[(N[(N[(2.0 + beta), $MachinePrecision] / alpha), $MachinePrecision] * N[(2.0 * beta + 2.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 * beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(beta - alpha), $MachinePrecision] * N[(0.5 / t$95$0), $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 7.4%

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

   3. Taylor expanded in alpha around -inf

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

   5. Applied rewrites 95.8%

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

   7. Applied rewrites 99.9%

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

   if -0.999990000000000046 < (/.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. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

Alternative 5: 98.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))))
   (if (<= t_0 -0.99999)
     (/ (+ 1.0 beta) alpha)
     (if (<= t_0 0.5)
       (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 = (1.0 + beta) / alpha;
	} else if (t_0 <= 0.5) {
		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(Float64(1.0 + beta) / alpha);
	elseif (t_0 <= 0.5)
		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), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.5], 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 7.4%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      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. *-lft-identity N/A

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

      8. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.5

   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. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   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 96.9

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

   7. Applied rewrites 96.9%

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

   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

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot \alpha}{\beta}} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 6: 97.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))))
   (if (<= t_0 -0.99999)
     (/ (+ 1.0 beta) alpha)
     (if (<= t_0 0.5) (fma (/ alpha (+ 2.0 alpha)) -0.5 0.5) 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 = (1.0 + beta) / alpha;
	} else if (t_0 <= 0.5) {
		tmp = fma((alpha / (2.0 + alpha)), -0.5, 0.5);
	} else {
		tmp = 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(Float64(1.0 + beta) / alpha);
	elseif (t_0 <= 0.5)
		tmp = fma(Float64(alpha / Float64(2.0 + alpha)), -0.5, 0.5);
	else
		tmp = 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), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(N[(alpha / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision], 1.0]]]

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 7.4%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      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. *-lft-identity N/A

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

      8. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.5

   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. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   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 96.9

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

   7. Applied rewrites 96.9%

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

   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 99.1%

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

Alternative 7: 97.3% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))))
   (if (<= t_0 -0.5)
     (/ (+ 1.0 beta) alpha)
     (if (<= t_0 0.5) (fma (fma -0.125 beta 0.25) beta 0.5) 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 = (1.0 + beta) / alpha;
	} else if (t_0 <= 0.5) {
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	} else {
		tmp = 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(Float64(1.0 + beta) / alpha);
	elseif (t_0 <= 0.5)
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	else
		tmp = 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), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(N[(-0.125 * beta + 0.25), $MachinePrecision] * beta + 0.5), $MachinePrecision], 1.0]]]

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 9.5%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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. *-lft-identity N/A

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

      8. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

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

   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 98.1

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

   5. Applied rewrites 98.1%

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

   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 97.7%

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

   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 99.1%

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

Alternative 8: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision], -0.99999], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(beta - alpha), $MachinePrecision] * N[(0.5 / t$95$0), $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 7.4%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      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. *-lft-identity N/A

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

      8. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -0.999990000000000046 < (/.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. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

Alternative 9: 98.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ alpha beta) 2.0)) -0.5)
   (/ (+ 1.0 beta) alpha)
   (fma (/ (- beta alpha) (+ 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 = (1.0 + beta) / alpha;
	} else {
		tmp = fma(((beta - alpha) / (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(Float64(1.0 + beta) / alpha);
	else
		tmp = fma(Float64(Float64(beta - alpha) / 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), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / 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 9.5%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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. *-lft-identity N/A

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

      8. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   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. 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. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-+.f64 99.2

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

   7. Applied rewrites 99.2%

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

Alternative 10: 98.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ alpha beta) 2.0)) -0.5)
   (/ (+ 1.0 beta) 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 = (1.0 + beta) / 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(Float64(1.0 + beta) / 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), $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 9.5%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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. *-lft-identity N/A

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

      8. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   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 11: 71.1% 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 65.6%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 62.8

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

   5. Applied rewrites 62.8%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 61.3%

      \[\leadsto 0.5 \]

   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 99.1%

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

Alternative 12: 71.6% 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 69.2%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 66.6

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

   5. Applied rewrites 66.6%

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

   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 66.1%

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

   if 2 < beta

   1. Initial program 85.9%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 84.5%

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

Alternative 13: 35.9% 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 74.2%

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

3. Taylor expanded in beta around inf

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

5. Applied rewrites 35.0%

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

Reproduce

herbie shell --seed 2024331 
(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))