Octave 3.8, jcobi/1

Percentage Accurate: 74.9% → 99.9%

Time: 10.6s

Alternatives: 15

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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999995:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \left(\beta + 2\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}, \beta + 1\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999995)
   (/
    (fma 0.5 (* (+ beta 2.0) (/ (- (- -2.0 beta) beta) alpha)) (+ 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) / ((beta + alpha) + 2.0)) <= -0.999995) {
		tmp = fma(0.5, ((beta + 2.0) * (((-2.0 - beta) - beta) / alpha)), (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(Float64(beta + alpha) + 2.0)) <= -0.999995)
		tmp = Float64(fma(0.5, Float64(Float64(beta + 2.0) * Float64(Float64(Float64(-2.0 - beta) - beta) / alpha)), 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[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999995], N[(N[(0.5 * N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(beta + 1.0), $MachinePrecision]), $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))) < -0.99999499999999997

   1. Initial program 6.5%

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

   3. Taylor expanded in alpha around inf

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

      1. /-lowering-/.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. Simplified 100.0%

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

   if -0.99999499999999997 < (/.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. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. +-lowering-+.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999995:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \left(\beta + 2\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}, \beta + 1\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.999995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha}{-2 - \alpha}, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\beta + \alpha}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.999995)
     (/ (+ beta 1.0) alpha)
     (if (<= t_0 0.8)
       (fma (/ alpha (- -2.0 alpha)) 0.5 0.5)
       (fma (/ (- beta alpha) (+ beta alpha)) 0.5 0.5)))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.999995) {
		tmp = (beta + 1.0) / alpha;
	} else if (t_0 <= 0.8) {
		tmp = fma((alpha / (-2.0 - alpha)), 0.5, 0.5);
	} else {
		tmp = fma(((beta - alpha) / (beta + alpha)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.999995)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	elseif (t_0 <= 0.8)
		tmp = fma(Float64(alpha / Float64(-2.0 - alpha)), 0.5, 0.5);
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(beta + alpha)), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.999995], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[(N[(alpha / N[(-2.0 - alpha), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99999499999999997

   1. Initial program 6.5%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.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. +-lowering-+.f64 99.4

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

   5. Simplified 99.4%

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

   if -0.99999499999999997 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.80000000000000004

   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. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. +-lowering-+.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. neg-mul-1 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. unsub-neg N/A

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

      9. --lowering--.f64 99.0

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

   7. Simplified 99.0%

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

   if 0.80000000000000004 < (/.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. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. +-lowering-+.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in alpha around inf

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

   7. Simplified 99.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha}{-2 - \alpha}, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\beta + \alpha}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, -0.0625, 0.125\right), -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.5)
     (/ (+ beta 1.0) alpha)
     (if (<= t_0 0.8)
       (fma alpha (fma alpha (fma alpha -0.0625 0.125) -0.25) 0.5)
       (- 1.0 (/ alpha beta))))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = (beta + 1.0) / alpha;
	} else if (t_0 <= 0.8) {
		tmp = fma(alpha, fma(alpha, fma(alpha, -0.0625, 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(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	elseif (t_0 <= 0.8)
		tmp = fma(alpha, fma(alpha, fma(alpha, -0.0625, 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[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[(alpha * N[(alpha * N[(alpha * -0.0625 + 0.125), $MachinePrecision] + -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 8.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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. /-lowering-/.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. +-lowering-+.f64 97.6

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

   5. Simplified 97.6%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.80000000000000004

   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. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. +-lowering-+.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. neg-mul-1 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. unsub-neg N/A

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

      9. --lowering--.f64 99.2

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

   7. Simplified 99.2%

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

   8. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 98.6

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

   10. Simplified 98.6%

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

   if 0.80000000000000004 < (/.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. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. +-lowering-+.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in alpha around inf

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

   7. Simplified 99.9%

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

   8. Taylor expanded in alpha around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 98.2

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

   10. Simplified 98.2%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, -0.0625, 0.125\right), -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.5)
     (/ (+ beta 1.0) alpha)
     (if (<= t_0 0.8)
       (fma alpha (fma alpha 0.125 -0.25) 0.5)
       (- 1.0 (/ alpha beta))))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = (beta + 1.0) / alpha;
	} else if (t_0 <= 0.8) {
		tmp = fma(alpha, fma(alpha, 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(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	elseif (t_0 <= 0.8)
		tmp = fma(alpha, fma(alpha, 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[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[(alpha * N[(alpha * 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 8.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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. /-lowering-/.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. +-lowering-+.f64 97.6

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

   5. Simplified 97.6%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.80000000000000004

   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. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. +-lowering-+.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. neg-mul-1 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. unsub-neg N/A

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

      9. --lowering--.f64 99.2

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

   7. Simplified 99.2%

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

   8. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 98.4

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

   10. Simplified 98.4%

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

   if 0.80000000000000004 < (/.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. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. +-lowering-+.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in alpha around inf

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

   7. Simplified 99.9%

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

   8. Taylor expanded in alpha around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 98.2

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

   10. Simplified 98.2%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.5)
     (/ 1.0 alpha)
     (if (<= t_0 0.8)
       (fma alpha (fma alpha 0.125 -0.25) 0.5)
       (- 1.0 (/ alpha beta))))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.8) {
		tmp = fma(alpha, fma(alpha, 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(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.8)
		tmp = fma(alpha, fma(alpha, 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[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[(alpha * N[(alpha * 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 8.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. +-lowering-+.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 8.8

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

   4. Applied egg-rr 8.8%

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

   5. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. neg-mul-1 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. unsub-neg N/A

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

      9. --lowering--.f64 8.3

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

   7. Simplified 8.3%

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

   8. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 81.2

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

   10. Simplified 81.2%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.80000000000000004

   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. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. +-lowering-+.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. neg-mul-1 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. unsub-neg N/A

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

      9. --lowering--.f64 99.2

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

   7. Simplified 99.2%

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

   8. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 98.4

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

   10. Simplified 98.4%

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

   if 0.80000000000000004 < (/.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. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. +-lowering-+.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in alpha around inf

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

   7. Simplified 99.9%

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

   8. Taylor expanded in alpha around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 98.2

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

   10. Simplified 98.2%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.5)
     (/ 1.0 alpha)
     (if (<= t_0 0.8)
       (fma alpha (fma alpha 0.125 -0.25) 0.5)
       (+ 1.0 (/ -1.0 beta))))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.8) {
		tmp = fma(alpha, fma(alpha, 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(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.8)
		tmp = fma(alpha, fma(alpha, 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[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[(alpha * N[(alpha * 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 8.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. +-lowering-+.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 8.8

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

   4. Applied egg-rr 8.8%

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

   5. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. neg-mul-1 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. unsub-neg N/A

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

      9. --lowering--.f64 8.3

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

   7. Simplified 8.3%

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

   8. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 81.2

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

   10. Simplified 81.2%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.80000000000000004

   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. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. +-lowering-+.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. neg-mul-1 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. unsub-neg N/A

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

      9. --lowering--.f64 99.2

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

   7. Simplified 99.2%

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

   8. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 98.4

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

   10. Simplified 98.4%

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

   if 0.80000000000000004 < (/.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. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-lowering-+.f64 97.2

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

   5. Simplified 97.2%

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

   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. +-lowering-+.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. /-lowering-/.f64 97.3

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

   8. Simplified 97.3%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.5)
     (/ 1.0 alpha)
     (if (<= t_0 0.8) (fma alpha (fma alpha 0.125 -0.25) 0.5) 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.8) {
		tmp = fma(alpha, fma(alpha, 0.125, -0.25), 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.8)
		tmp = fma(alpha, fma(alpha, 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[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[(alpha * N[(alpha * 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 8.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. +-lowering-+.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 8.8

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

   4. Applied egg-rr 8.8%

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

   5. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. neg-mul-1 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. unsub-neg N/A

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

      9. --lowering--.f64 8.3

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

   7. Simplified 8.3%

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

   8. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 81.2

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

   10. Simplified 81.2%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.80000000000000004

   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. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. +-lowering-+.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. neg-mul-1 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. unsub-neg N/A

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

      9. --lowering--.f64 99.2

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

   7. Simplified 99.2%

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

   8. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 98.4

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

   10. Simplified 98.4%

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

   if 0.80000000000000004 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

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

   5. Simplified 97.2%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999995:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \frac{-2 - \beta}{\alpha}, \beta + 1\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999995)
   (/ (fma beta (/ (- -2.0 beta) alpha) (+ 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) / ((beta + alpha) + 2.0)) <= -0.999995) {
		tmp = fma(beta, ((-2.0 - beta) / alpha), (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(Float64(beta + alpha) + 2.0)) <= -0.999995)
		tmp = Float64(fma(beta, Float64(Float64(-2.0 - beta) / alpha), 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[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999995], N[(N[(beta * N[(N[(-2.0 - beta), $MachinePrecision] / alpha), $MachinePrecision] + N[(beta + 1.0), $MachinePrecision]), $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))) < -0.99999499999999997

   1. Initial program 6.5%

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

   3. Taylor expanded in alpha around -inf

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

   5. Simplified 99.9%

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

   6. Taylor expanded in beta around inf

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

   8. Simplified 99.6%

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

   9. Taylor expanded in alpha around inf

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

       1. mul-1-neg N/A

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

       2. associate-/l* N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. distribute-frac-neg N/A

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

       5. mul-1-neg N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. /-lowering-/.f64 N/A

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

       8. distribute-lft-in N/A

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

       9. metadata-eval N/A

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

       10. mul-1-neg N/A

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

       11. unsub-neg N/A

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

       12. --lowering--.f64 N/A

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

       13. distribute-lft-in N/A

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

       14. metadata-eval N/A

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

       15. mul-1-neg N/A

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

       16. associate--l+ N/A

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

       17. distribute-rgt-in N/A

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

       18. metadata-eval N/A

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

       19. sub-neg N/A

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

       20. count-2 N/A

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

       21. distribute-rgt-neg-in N/A

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

   11. Simplified 99.6%

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

   if -0.99999499999999997 < (/.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. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. +-lowering-+.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999995:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \frac{-2 - \beta}{\alpha}, \beta + 1\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \alpha, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.5)
     (/ 1.0 alpha)
     (if (<= t_0 0.8) (fma -0.25 alpha 0.5) 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.8) {
		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(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.8)
		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[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.8], 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 8.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. +-lowering-+.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 8.8

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

   4. Applied egg-rr 8.8%

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

   5. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. neg-mul-1 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. unsub-neg N/A

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

      9. --lowering--.f64 8.3

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

   7. Simplified 8.3%

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

   8. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 81.2

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

   10. Simplified 81.2%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.80000000000000004

   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. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. +-lowering-+.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. neg-mul-1 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. unsub-neg N/A

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

      9. --lowering--.f64 99.2

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

   7. Simplified 99.2%

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

   8. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 97.9

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

   10. Simplified 97.9%

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

   if 0.80000000000000004 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

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

   5. Simplified 97.2%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \alpha, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999995)
   (/ (+ 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) / ((beta + alpha) + 2.0)) <= -0.999995) {
		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(Float64(beta + alpha) + 2.0)) <= -0.999995)
		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[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999995], 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))) < -0.99999499999999997

   1. Initial program 6.5%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.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. +-lowering-+.f64 99.4

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

   5. Simplified 99.4%

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

   if -0.99999499999999997 < (/.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. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. +-lowering-+.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.2% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -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) / ((beta + alpha) + 2.0)) <= -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(Float64(beta + alpha) + 2.0)) <= -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[(N[(beta + alpha), $MachinePrecision] + 2.0), $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 8.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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. /-lowering-/.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. +-lowering-+.f64 97.6

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

   5. Simplified 97.6%

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

   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. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-lowering-+.f64 97.4

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

   5. Simplified 97.4%

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

4. Final simplification 97.5%

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

Alternative 12: 71.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.0002:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) 0.0002) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= 0.0002) {
		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) / ((beta + alpha) + 2.0d0)) <= 0.0002d0) 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) / ((beta + alpha) + 2.0)) <= 0.0002) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= 0.0002:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= 0.0002)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= 0.0002)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], 0.0002], 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))) < 2.0000000000000001e-4

   1. Initial program 65.1%

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

   3. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. distribute-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. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-lowering-+.f64 62.7

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

   5. Simplified 62.7%

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

   6. Taylor expanded in beta around 0

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

   8. Simplified 62.2%

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

   if 2.0000000000000001e-4 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

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

   5. Simplified 96.1%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.0002:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.0% 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 68.5%

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

   3. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. distribute-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. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-lowering-+.f64 65.6

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

   5. Simplified 65.6%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{4} \cdot \beta} \]
   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. accelerator-lowering-fma.f64 65.6

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

   8. Simplified 65.6%

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

   if 2 < beta

   1. Initial program 88.0%

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

   3. Taylor expanded in beta around inf

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

   5. Simplified 85.4%

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

Alternative 14: 49.0% 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 74.3%

   \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
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. accelerator-lowering-fma.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. +-lowering-+.f64 71.6

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

5. Simplified 71.6%

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

6. Taylor expanded in beta around 0

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

8. Simplified 50.7%

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

Alternative 15: 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 74.3%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. /-lowering-/.f64 N/A

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

   9. --lowering--.f64 N/A

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

   10. +-commutative N/A

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

   11. associate-+l+ N/A

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

   12. +-lowering-+.f64 N/A

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

   13. +-lowering-+.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval 74.3

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

4. Applied egg-rr 74.3%

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

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. Simplified 3.8%

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

   1. metadata-eval N/A

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

   2. metadata-eval 3.8

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

9. Applied egg-rr 3.8%

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

Reproduce

herbie shell --seed 2024198 
(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))