Octave 3.8, jcobi/1

Percentage Accurate: 75.3% → 99.7%

Time: 11.0s

Alternatives: 18

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: 75.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.3%

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

   3. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 99.6%

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

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

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 100.0

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{-2 - \left(\beta + \alpha\right)}, \beta - \alpha, 1\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

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

Alternative 2: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, 0.0625, -0.125\right), 0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 - \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ 2.0 (+ beta alpha)))))
   (if (<= t_0 -0.5)
     (/ (+ beta 1.0) alpha)
     (if (<= t_0 0.5)
       (fma beta (fma beta (fma beta 0.0625 -0.125) 0.25) 0.5)
       (+ 1.0 (/ (- -1.0 alpha) beta))))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (2.0 + (beta + alpha));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = (beta + 1.0) / alpha;
	} else if (t_0 <= 0.5) {
		tmp = fma(beta, fma(beta, fma(beta, 0.0625, -0.125), 0.25), 0.5);
	} else {
		tmp = 1.0 + ((-1.0 - alpha) / beta);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	elseif (t_0 <= 0.5)
		tmp = fma(beta, fma(beta, fma(beta, 0.0625, -0.125), 0.25), 0.5);
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(beta * N[(beta * N[(beta * 0.0625 + -0.125), $MachinePrecision] + 0.25), $MachinePrecision] + 0.5), $MachinePrecision], N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / 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 8.3%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   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-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \beta \cdot \left(\frac{1}{4} + \beta \cdot \left(\frac{1}{16} \cdot \beta - \frac{1}{8}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\beta \cdot \left(\frac{1}{4} + \beta \cdot \left(\frac{1}{16} \cdot \beta - \frac{1}{8}\right)\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\beta, \frac{1}{4} + \beta \cdot \left(\frac{1}{16} \cdot \beta - \frac{1}{8}\right), \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\beta \cdot \left(\frac{1}{16} \cdot \beta - \frac{1}{8}\right) + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\mathsf{fma}\left(\beta, \frac{1}{16} \cdot \beta - \frac{1}{8}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \color{blue}{\frac{1}{16} \cdot \beta + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \color{blue}{\beta \cdot \frac{1}{16}} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), \frac{1}{4}\right), \frac{1}{2}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \beta \cdot \frac{1}{16} + \color{blue}{\frac{-1}{8}}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      8. lower-fma.f64 97.1

         \[\leadsto \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \color{blue}{\mathsf{fma}\left(\beta, 0.0625, -0.125\right)}, 0.25\right), 0.5\right) \]

   8. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, 0.0625, -0.125\right), 0.25\right), 0.5\right)} \]

   if 0.5 < (/.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. 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. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. neg-mul-1 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in beta around inf

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

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 98.7

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

   8. Applied rewrites 98.7%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, 0.0625, -0.125\right), 0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 - \alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, 0.0625, -0.125\right), 0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta - \alpha\right) + -1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ 2.0 (+ beta alpha)))))
   (if (<= t_0 -0.5)
     (/ (+ beta 1.0) alpha)
     (if (<= t_0 0.5)
       (fma beta (fma beta (fma beta 0.0625 -0.125) 0.25) 0.5)
       (/ (+ (- beta alpha) -1.0) beta)))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (2.0 + (beta + alpha));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = (beta + 1.0) / alpha;
	} else if (t_0 <= 0.5) {
		tmp = fma(beta, fma(beta, fma(beta, 0.0625, -0.125), 0.25), 0.5);
	} else {
		tmp = ((beta - alpha) + -1.0) / beta;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	elseif (t_0 <= 0.5)
		tmp = fma(beta, fma(beta, fma(beta, 0.0625, -0.125), 0.25), 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[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(beta * N[(beta * N[(beta * 0.0625 + -0.125), $MachinePrecision] + 0.25), $MachinePrecision] + 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 8.3%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   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-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \beta \cdot \left(\frac{1}{4} + \beta \cdot \left(\frac{1}{16} \cdot \beta - \frac{1}{8}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\beta \cdot \left(\frac{1}{4} + \beta \cdot \left(\frac{1}{16} \cdot \beta - \frac{1}{8}\right)\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\beta, \frac{1}{4} + \beta \cdot \left(\frac{1}{16} \cdot \beta - \frac{1}{8}\right), \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\beta \cdot \left(\frac{1}{16} \cdot \beta - \frac{1}{8}\right) + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\mathsf{fma}\left(\beta, \frac{1}{16} \cdot \beta - \frac{1}{8}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \color{blue}{\frac{1}{16} \cdot \beta + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \color{blue}{\beta \cdot \frac{1}{16}} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), \frac{1}{4}\right), \frac{1}{2}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \beta \cdot \frac{1}{16} + \color{blue}{\frac{-1}{8}}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      8. lower-fma.f64 97.1

         \[\leadsto \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \color{blue}{\mathsf{fma}\left(\beta, 0.0625, -0.125\right)}, 0.25\right), 0.5\right) \]

   8. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, 0.0625, -0.125\right), 0.25\right), 0.5\right)} \]

   if 0.5 < (/.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. 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. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. neg-mul-1 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. lower-+.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 98.7

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

   8. Applied rewrites 98.7%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, \mathsf{fma}\left(\beta, 0.0625, -0.125\right), 0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta - \alpha\right) + -1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	elseif (t_0 <= 0.5)
		tmp = fma(beta, fma(beta, -0.125, 0.25), 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[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(beta * N[(beta * -0.125 + 0.25), $MachinePrecision] + 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 8.3%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   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-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\beta, \frac{1}{4} + \frac{-1}{8} \cdot \beta, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\frac{-1}{8} \cdot \beta + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\beta \cdot \frac{-1}{8}} + \frac{1}{4}, \frac{1}{2}\right) \]

      5. lower-fma.f64 96.9

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

   8. Applied rewrites 96.9%

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

   if 0.5 < (/.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. 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. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. neg-mul-1 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. lower-+.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 98.7

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

   8. Applied rewrites 98.7%

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

4. Final simplification 97.8%

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

Alternative 5: 97.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(beta * N[(beta * -0.125 + 0.25), $MachinePrecision] + 0.5), $MachinePrecision], N[(1.0 - N[(alpha / 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 8.3%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   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-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\beta, \frac{1}{4} + \frac{-1}{8} \cdot \beta, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\frac{-1}{8} \cdot \beta + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\beta \cdot \frac{-1}{8}} + \frac{1}{4}, \frac{1}{2}\right) \]

      5. lower-fma.f64 96.9

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

   8. Applied rewrites 96.9%

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

   if 0.5 < (/.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. 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. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. neg-mul-1 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in beta around inf

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

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 98.7

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in alpha around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 98.5

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

   11. Applied rewrites 98.5%

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

4. Final simplification 97.7%

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

Alternative 6: 97.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	elseif (t_0 <= 0.5)
		tmp = fma(beta, fma(beta, -0.125, 0.25), 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[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(beta * N[(beta * -0.125 + 0.25), $MachinePrecision] + 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 8.3%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   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-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\beta, \frac{1}{4} + \frac{-1}{8} \cdot \beta, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\frac{-1}{8} \cdot \beta + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\beta \cdot \frac{-1}{8}} + \frac{1}{4}, \frac{1}{2}\right) \]

      5. lower-fma.f64 96.9

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

   8. Applied rewrites 96.9%

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

   if 0.5 < (/.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. 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-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in beta around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 98.0

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

   8. Applied rewrites 98.0%

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

4. Final simplification 97.6%

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

Alternative 7: 92.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.5)
		tmp = fma(beta, fma(beta, -0.125, 0.25), 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[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(beta * N[(beta * -0.125 + 0.25), $MachinePrecision] + 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 8.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 8.2

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 8.2

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

   4. Applied rewrites 8.2%

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

   5. Taylor expanded in alpha around inf

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 98.3

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in beta around 0

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

      1. lower-/.f64 75.4

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

   10. Applied rewrites 75.4%

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

   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-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\beta, \frac{1}{4} + \frac{-1}{8} \cdot \beta, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\frac{-1}{8} \cdot \beta + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\beta \cdot \frac{-1}{8}} + \frac{1}{4}, \frac{1}{2}\right) \]

      5. lower-fma.f64 96.9

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

   8. Applied rewrites 96.9%

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

   if 0.5 < (/.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. 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-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in beta around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 98.0

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

   8. Applied rewrites 98.0%

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

4. Final simplification 91.3%

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

Alternative 8: 92.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(beta * N[(beta * -0.125 + 0.25), $MachinePrecision] + 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 8.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 8.2

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 8.2

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

   4. Applied rewrites 8.2%

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

   5. Taylor expanded in alpha around inf

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 98.3

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in beta around 0

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

      1. lower-/.f64 75.4

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

   10. Applied rewrites 75.4%

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

   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-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\beta, \frac{1}{4} + \frac{-1}{8} \cdot \beta, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\frac{-1}{8} \cdot \beta + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\beta, \color{blue}{\beta \cdot \frac{-1}{8}} + \frac{1}{4}, \frac{1}{2}\right) \]

      5. lower-fma.f64 96.9

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

   8. Applied rewrites 96.9%

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

   if 0.5 < (/.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. Taylor expanded in beta around inf

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

   5. Applied rewrites 97.8%

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

4. Final simplification 91.3%

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

Alternative 9: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\beta - \alpha}}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ 2.0 (+ beta alpha))) -0.999998)
   (/ (+ beta 1.0) alpha)
   (fma (/ 1.0 (/ (+ alpha (+ beta 2.0)) (- beta alpha))) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / (2.0 + (beta + alpha))) <= -0.999998) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = fma((1.0 / ((alpha + (beta + 2.0)) / (beta - alpha))), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha))) <= -0.999998)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = fma(Float64(1.0 / Float64(Float64(alpha + Float64(beta + 2.0)) / Float64(beta - alpha))), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.999998], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(1.0 / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $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))) < -0.999998000000000054

   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 99.1

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

   5. Applied rewrites 99.1%

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

   if -0.999998000000000054 < (/.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

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. 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 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 99.8

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. associate-+l+ N/A

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

      11. lift-+.f64 N/A

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

      12. lower-+.f64 99.8

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.8

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

   6. Applied rewrites 99.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\beta - \alpha}}}, 0.5, 0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\beta - \alpha}}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-1}{-2 - \left(\beta + \alpha\right)}, \beta - \alpha, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ 2.0 (+ beta alpha))) -0.999998)
   (/ (+ beta 1.0) alpha)
   (/ (fma (/ -1.0 (- -2.0 (+ beta alpha))) (- beta alpha) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / (2.0 + (beta + alpha))) <= -0.999998) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = fma((-1.0 / (-2.0 - (beta + alpha))), (beta - alpha), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha))) <= -0.999998)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = Float64(fma(Float64(-1.0 / Float64(-2.0 - Float64(beta + alpha))), Float64(beta - alpha), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.999998], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(-1.0 / N[(-2.0 - N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta - alpha), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999998000000000054

   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 99.1

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

   5. Applied rewrites 99.1%

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

   if -0.999998000000000054 < (/.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

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{-2 - \left(\beta + \alpha\right)}, \beta - \alpha, 1\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-1}{-2 - \left(\beta + \alpha\right)}, \beta - \alpha, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(beta * 0.25 + 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 8.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 8.2

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 8.2

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

   4. Applied rewrites 8.2%

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

   5. Taylor expanded in alpha around inf

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 98.3

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in beta around 0

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

      1. lower-/.f64 75.4

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

   10. Applied rewrites 75.4%

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

   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-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{4} \cdot \beta + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\beta \cdot \frac{1}{4}} + \frac{1}{2} \]

      3. lower-fma.f64 96.4

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

   8. Applied rewrites 96.4%

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

   if 0.5 < (/.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. Taylor expanded in beta around inf

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

   5. Applied rewrites 97.8%

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

4. Final simplification 91.1%

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

Alternative 12: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ 2.0 (+ beta alpha))) -0.999998)
   (/ (+ beta 1.0) alpha)
   (fma (/ (- beta alpha) (+ beta (+ alpha 2.0))) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / (2.0 + (beta + alpha))) <= -0.999998) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = fma(((beta - alpha) / (beta + (alpha + 2.0))), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha))) <= -0.999998)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0))), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.999998], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $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))) < -0.999998000000000054

   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 99.1

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

   5. Applied rewrites 99.1%

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

   if -0.999998000000000054 < (/.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

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. 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 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 98.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(0.5 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * beta + 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 8.3%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   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-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.4

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

   5. Applied rewrites 98.4%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. div-inv N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 98.4

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 98.4

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

   7. Applied rewrites 98.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. clear-num N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. un-div-inv N/A

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

      12. lower-/.f64 98.4

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

   9. Applied rewrites 98.4%

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

4. Final simplification 98.4%

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

Alternative 14: 98.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.3%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   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-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 98.4%

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

Alternative 15: 71.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / (2.0 + (beta + alpha))) <= 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(2.0 + Float64(beta + alpha))) <= 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) / (2.0 + (beta + alpha))) <= 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[(2.0 + N[(beta + alpha), $MachinePrecision]), $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 64.7%

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

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 62.6

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

   5. Applied rewrites 62.6%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 60.5%

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

   if 0.5 < (/.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. Taylor expanded in beta around inf

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

   5. Applied rewrites 97.8%

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

4. Final simplification 71.3%

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

Alternative 16: 72.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 71.0%

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

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in beta around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{4} \cdot \beta + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\beta \cdot \frac{1}{4}} + \frac{1}{2} \]

      3. lower-fma.f64 67.2

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

   8. Applied rewrites 67.2%

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

   if 2 < beta

   1. Initial program 82.0%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 80.7%

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

Alternative 17: 49.7% accurate, 35.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 0.5)

C

double code(double alpha, double beta) {
	return 0.5;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.5d0
end function

Java

public static double code(double alpha, double beta) {
	return 0.5;
}

Python

def code(alpha, beta):
	return 0.5

Julia

function code(alpha, beta)
	return 0.5
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.5;
end

Wolfram

code[alpha_, beta_] := 0.5

Derivation

1. Initial program 74.9%

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

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 73.1

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

5. Applied rewrites 73.1%

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

6. Taylor expanded in beta around 0

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

8. Applied rewrites 48.4%

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

Alternative 18: 3.7% accurate, 35.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 0.0)

C

double code(double alpha, double beta) {
	return 0.0;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.0d0
end function

Java

public static double code(double alpha, double beta) {
	return 0.0;
}

Python

def code(alpha, beta):
	return 0.0

Julia

function code(alpha, beta)
	return 0.0
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.0;
end

Wolfram

code[alpha_, beta_] := 0.0

Derivation

1. Initial program 74.9%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. lift-+.f64 N/A

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

   9. distribute-rgt-in N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. 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 74.9%

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

5. Taylor expanded in alpha around inf

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

7. Applied rewrites 3.7%

   \[\leadsto \mathsf{fma}\left(\color{blue}{-1}, 0.5, 0.5\right) \]
8. Step-by-step derivation

   1. metadata-eval N/A

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

   2. metadata-eval 3.7

      \[\leadsto \color{blue}{0} \]

9. Applied rewrites 3.7%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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))