Octave 3.8, jcobi/1

Percentage Accurate: 74.8% → 99.8%

Time: 7.5s

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: 74.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.98:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\beta, 0.5, 1\right), \frac{\left(-2 - \beta\right) - \beta}{\alpha}, 1 + \beta\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\alpha - \beta, \frac{0.5}{-2 - \left(\alpha + \beta\right)}, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ alpha beta) 2.0)) -0.98)
   (/
    (fma (fma beta 0.5 1.0) (/ (- (- -2.0 beta) beta) alpha) (+ 1.0 beta))
    alpha)
   (fma (- alpha beta) (/ 0.5 (- -2.0 (+ alpha beta))) 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.98) {
		tmp = fma(fma(beta, 0.5, 1.0), (((-2.0 - beta) - beta) / alpha), (1.0 + beta)) / alpha;
	} else {
		tmp = fma((alpha - beta), (0.5 / (-2.0 - (alpha + beta))), 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) <= -0.98)
		tmp = Float64(fma(fma(beta, 0.5, 1.0), Float64(Float64(Float64(-2.0 - beta) - beta) / alpha), Float64(1.0 + beta)) / alpha);
	else
		tmp = fma(Float64(alpha - beta), Float64(0.5 / Float64(-2.0 - Float64(alpha + beta))), 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], -0.98], N[(N[(N[(beta * 0.5 + 1.0), $MachinePrecision] * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision] + N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(alpha - beta), $MachinePrecision] * N[(0.5 / N[(-2.0 - N[(alpha + beta), $MachinePrecision]), $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))) < -0.97999999999999998

   1. Initial program 6.9%

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

   3. Taylor expanded in alpha around 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. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 5.5

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 5.5%

      \[\leadsto 0.5 \]

   9. Taylor expanded in alpha around inf

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

   11. Applied rewrites 100.0%

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

   if -0.97999999999999998 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.98:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\beta, 0.5, 1\right), \frac{\left(-2 - \beta\right) - \beta}{\alpha}, 1 + \beta\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\alpha - \beta, \frac{0.5}{-2 - \left(\alpha + \beta\right)}, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.25, \alpha, -0.5\right) \cdot \alpha, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta - \alpha\right) - 1}{\beta}\\ \end{array} \end{array} \]

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 1e-11)
       (fma (* (fma 0.25 alpha -0.5) alpha) 0.5 0.5)
       (/ (- (- beta alpha) 1.0) beta)))))

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 <= 1e-11) {
		tmp = fma((fma(0.25, alpha, -0.5) * alpha), 0.5, 0.5);
	} else {
		tmp = ((beta - alpha) - 1.0) / beta;
	}
	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 <= 1e-11)
		tmp = fma(Float64(fma(0.25, alpha, -0.5) * alpha), 0.5, 0.5);
	else
		tmp = Float64(Float64(Float64(beta - alpha) - 1.0) / beta);
	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, 1e-11], N[(N[(N[(0.25 * alpha + -0.5), $MachinePrecision] * alpha), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] - 1.0), $MachinePrecision] / beta), $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 8.1%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 9.99999999999999939e-12

   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. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 99.2%

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

   if 9.99999999999999939e-12 < (/.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--r+ N/A

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

      7. sub-neg N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-out-- N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 99.1%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.25, \alpha, -0.5\right) \cdot \alpha, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta - \alpha\right) - 1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \alpha, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta - \alpha\right) - 1}{\beta}\\ \end{array} \end{array} \]

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 1e-11)
       (fma (* -0.5 alpha) 0.5 0.5)
       (/ (- (- beta alpha) 1.0) beta)))))

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 <= 1e-11) {
		tmp = fma((-0.5 * alpha), 0.5, 0.5);
	} else {
		tmp = ((beta - alpha) - 1.0) / beta;
	}
	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 <= 1e-11)
		tmp = fma(Float64(-0.5 * alpha), 0.5, 0.5);
	else
		tmp = Float64(Float64(Float64(beta - alpha) - 1.0) / beta);
	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, 1e-11], N[(N[(-0.5 * alpha), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] - 1.0), $MachinePrecision] / beta), $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 8.1%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 9.99999999999999939e-12

   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. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 99.0%

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

   if 9.99999999999999939e-12 < (/.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--r+ N/A

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

      7. sub-neg N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-out-- N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 99.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \alpha, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta - \alpha\right) - 1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \alpha, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \end{array} \]

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 1e-11) (fma (* -0.5 alpha) 0.5 0.5) (- 1.0 (/ 1.0 beta))))))

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 <= 1e-11) {
		tmp = fma((-0.5 * alpha), 0.5, 0.5);
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	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 <= 1e-11)
		tmp = fma(Float64(-0.5 * alpha), 0.5, 0.5);
	else
		tmp = Float64(1.0 - Float64(1.0 / beta));
	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, 1e-11], N[(N[(-0.5 * alpha), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision], N[(1.0 - N[(1.0 / beta), $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 8.1%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 9.99999999999999939e-12

   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. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 99.0%

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

   if 9.99999999999999939e-12 < (/.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--r+ N/A

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

      7. sub-neg N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-out-- N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 98.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \alpha, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.98:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\alpha - \beta, \frac{0.5}{-2 - \left(\alpha + \beta\right)}, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ alpha beta) 2.0)) -0.98)
   (/ (+ 1.0 beta) alpha)
   (fma (- alpha beta) (/ 0.5 (- -2.0 (+ alpha beta))) 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.98) {
		tmp = (1.0 + beta) / alpha;
	} else {
		tmp = fma((alpha - beta), (0.5 / (-2.0 - (alpha + beta))), 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) <= -0.98)
		tmp = Float64(Float64(1.0 + beta) / alpha);
	else
		tmp = fma(Float64(alpha - beta), Float64(0.5 / Float64(-2.0 - Float64(alpha + beta))), 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], -0.98], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(alpha - beta), $MachinePrecision] * N[(0.5 / N[(-2.0 - N[(alpha + beta), $MachinePrecision]), $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))) < -0.97999999999999998

   1. Initial program 6.9%

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

   3. Taylor expanded in alpha around inf

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

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-+.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if -0.97999999999999998 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.98:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\alpha - \beta, \frac{0.5}{-2 - \left(\alpha + \beta\right)}, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ alpha beta) 2.0)) -0.5)
   (/ (+ 1.0 beta) alpha)
   (fma (/ beta (- beta -2.0)) 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 / (beta - -2.0)), 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(beta - -2.0)), 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[(beta - -2.0), $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 8.1%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   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. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 98.4%

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

Alternative 7: 71.7% 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 62.5%

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 60.7

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 60.3%

      \[\leadsto 0.5 \]

   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 97.9%

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

Alternative 8: 72.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 65.1%

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 63.2

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

   5. Applied rewrites 63.2%

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

   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 63.1%

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

   if 2 < beta

   1. Initial program 91.3%

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

   3. Taylor expanded in beta around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--r+ N/A

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

      7. sub-neg N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-out-- N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval 89.9

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 89.8%

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

Alternative 9: 72.2% accurate, 1.8× speedup

Math

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

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 65.1%

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 63.2

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

   5. Applied rewrites 63.2%

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

   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 63.1%

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

   if 2 < beta

   1. Initial program 91.3%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 89.3%

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

Alternative 10: 72.1% 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 65.1%

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 63.2

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

   5. Applied rewrites 63.2%

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

   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 63.0%

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

   if 2 < beta

   1. Initial program 91.3%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 89.3%

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

Alternative 11: 37.2% 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 73.8%

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

3. Taylor expanded in beta around inf

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

5. Applied rewrites 38.8%

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

Reproduce

herbie shell --seed 2024266 
(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))