Octave 3.8, jcobi/1

Percentage Accurate: 74.9% → 99.9%

Time: 7.8s

Alternatives: 11

Speedup: 0.7×

Specification

\[\alpha > -1 \land \beta > -1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 7.6

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 7.6

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

   4. Applied rewrites 7.6%

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

   5. Taylor expanded in alpha around inf

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

   7. Applied rewrites 99.9%

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

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

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. distribute-frac-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 100.0

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

   6. Applied rewrites 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -1

   1. Initial program 4.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -1 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.0100000000000000002

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in alpha around inf

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 99.5%

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

   if 0.0100000000000000002 < (/.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

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

      1. +-commutative N/A

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

      2. div-add N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. div-add-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 99.3%

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

Alternative 3: 98.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / (2.0d0 + (alpha + beta))
    if (t_0 <= (-1.0d0)) then
        tmp = (beta - (-1.0d0)) / alpha
    else if (t_0 <= 0.01d0) then
        tmp = 1.0d0 / (2.0d0 + alpha)
    else
        tmp = 1.0d0 - (1.0d0 / beta)
    end if
    code = tmp
end function

Java

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

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / (2.0 + (alpha + beta))
	tmp = 0
	if t_0 <= -1.0:
		tmp = (beta - -1.0) / alpha
	elif t_0 <= 0.01:
		tmp = 1.0 / (2.0 + alpha)
	else:
		tmp = 1.0 - (1.0 / beta)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / (2.0 + (alpha + beta));
	tmp = 0.0;
	if (t_0 <= -1.0)
		tmp = (beta - -1.0) / alpha;
	elseif (t_0 <= 0.01)
		tmp = 1.0 / (2.0 + alpha);
	else
		tmp = 1.0 - (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], N[(N[(beta - -1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[(1.0 / N[(2.0 + alpha), $MachinePrecision]), $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))) < -1

   1. Initial program 4.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -1 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.0100000000000000002

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in alpha around inf

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 99.5%

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

   if 0.0100000000000000002 < (/.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

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

      1. +-commutative N/A

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

      2. div-add N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. div-add-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 97.8%

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

4. Final simplification 99.1%

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

Alternative 4: 97.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (2.0 + (alpha + beta));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = (beta - -1.0) / alpha;
	} else if (t_0 <= 0.01) {
		tmp = fma(fma(0.125, alpha, -0.25), alpha, 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(alpha + beta)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(beta - -1.0) / alpha);
	elseif (t_0 <= 0.01)
		tmp = fma(fma(0.125, alpha, -0.25), alpha, 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[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(beta - -1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[(N[(0.125 * alpha + -0.25), $MachinePrecision] * alpha + 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 7.6%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 97.8

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

   5. Applied rewrites 97.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 98.9%

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

   if 0.0100000000000000002 < (/.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

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

      1. +-commutative N/A

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

      2. div-add N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. div-add-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 97.8%

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

4. Final simplification 98.3%

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

Alternative 5: 92.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (2.0 + (alpha + beta));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.01) {
		tmp = fma(fma(0.125, alpha, -0.25), alpha, 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(alpha + beta)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.01)
		tmp = fma(fma(0.125, alpha, -0.25), alpha, 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[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[(N[(0.125 * alpha + -0.25), $MachinePrecision] * alpha + 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 7.6%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 7.6

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

   5. Applied rewrites 7.6%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 78.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 98.9%

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

   if 0.0100000000000000002 < (/.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

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

      1. +-commutative N/A

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

      2. div-add N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. div-add-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 97.8%

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

4. Final simplification 93.5%

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

Alternative 6: 92.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(2.0 + Float64(alpha + beta)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.01)
		tmp = fma(fma(0.125, alpha, -0.25), alpha, 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[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[(N[(0.125 * alpha + -0.25), $MachinePrecision] * alpha + 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 7.6%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 7.6

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

   5. Applied rewrites 7.6%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 78.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 98.9%

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

   if 0.0100000000000000002 < (/.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 96.7%

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

4. Final simplification 93.2%

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

Alternative 7: 92.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(2.0 + Float64(alpha + beta)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.01)
		tmp = fma(-0.25, alpha, 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[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[(-0.25 * alpha + 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 7.6%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 7.6

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

   5. Applied rewrites 7.6%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 78.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 98.3%

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

   if 0.0100000000000000002 < (/.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 96.7%

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

4. Final simplification 92.9%

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

Alternative 8: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -1

   1. Initial program 4.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -1 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. distribute-frac-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 99.9

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

   6. Applied rewrites 99.9%

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

4. Final simplification 99.9%

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

Alternative 9: 98.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.6%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 97.8

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

   5. Applied rewrites 97.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

4. Final simplification 98.0%

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

Alternative 10: 71.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)} \leq 0.01:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ 2.0 (+ alpha beta))) 0.01) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / (2.0 + (alpha + beta))) <= 0.01) {
		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 + (alpha + beta))) <= 0.01d0) 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 + (alpha + beta))) <= 0.01) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / (2.0 + (alpha + beta))) <= 0.01:
		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(alpha + beta))) <= 0.01)
		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 + (alpha + beta))) <= 0.01)
		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[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.01], 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))) < 0.0100000000000000002

   1. Initial program 67.0%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 66.7

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 64.6%

      \[\leadsto 0.5 \]

   if 0.0100000000000000002 < (/.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 96.7%

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

4. Final simplification 74.3%

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

Alternative 11: 37.7% accurate, 35.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 1.0

Julia

function code(alpha, beta)
	return 1.0
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 1.0

Derivation

1. Initial program 76.9%

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

3. Taylor expanded in beta around inf

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

5. Applied rewrites 38.9%

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

Reproduce

herbie shell --seed 2024298 
(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))